├── .gitignore ├── PHYS 408 - Optics └── phys408fs.pdf ├── MECH 466 - Automatic Control ├── mech466fs.pdf ├── SecondOrderFreqResponse.jpg └── mech466fs.tex ├── ELEC 221 - Signals and Systems ├── elec221fs.pdf └── elec221fs.tex ├── PHYS 401 - Electromagnetic Theory ├── BoldR.pdf ├── ScriptR.pdf └── phys401fs.pdf ├── APSC 278 - Engineering Materials ├── apsc278fs.pdf └── apsc278fs.tex ├── ELEC 404 - RF Integrated Circuits ├── elec404fs.pdf ├── scattering_parameters.png └── elec404fs.tex ├── MECH 360 - Mechanics of Materials ├── mech360fs.pdf └── mech360fs.tex ├── MECH 431 - Engineering Economics ├── mech431fs.pdf └── mech431fs.tex ├── PHYS 301 - Electricity and Magnetism ├── BoldR.pdf ├── ScriptR.pdf ├── phys301fs.pdf └── phys301fs.tex ├── PHYS 403 - Statistical Mechanics ├── phys403fs.pdf └── phys403fs.tex ├── MATH 305 - Applied Complex Analysis ├── math305fs.pdf └── math305fs.tex ├── CPSC 340 - Machine Learning and Data Mining ├── cpsc340fs.pdf └── cpsc340fs.tex ├── MECH 280 - Introduction to Fluid Mechanics ├── mech280fs.pdf └── mech280fs.tex ├── PHYS 350 - Applications of Classical Mechanics ├── BoldR.pdf ├── ScriptR.pdf ├── phys350fs.pdf └── phys350fs.tex ├── MATH 217 - Multivariable and Vector Calculus ├── math217fs.pdf └── math217fs.tex ├── PHYS 250 - Introduction to Modern Physics ├── phys250finalfs.pdf ├── phys250mt1fs.pdf ├── phys250mt2fs.pdf ├── phys250mt1fs.tex ├── phys250mt2fs.tex └── phys250finalfs.tex ├── PHYS 304 - Introduction to Quantum Mechanics ├── phys304fs.pdf └── phys304fs.tex ├── ELEC 401 - Analog CMOS Integrated Circuit Design ├── elec401fs.pdf └── elec401fs.tex ├── MATH 318 - Probability with Physical Applications ├── math318fs.pdf └── math318fs.tex ├── MATH 400 - Applied Partial Differential Equations ├── math400fs.pdf └── math400fs.tex ├── ELEC 433 - Error Control Coding for Communications and Computers └── elec433fs.pdf └── README.md /.gitignore: -------------------------------------------------------------------------------- 1 | *.synctex.gz 2 | *.ini 3 | *.aux 4 | *.log 5 | *.out 6 | *.DS_Store 7 | -------------------------------------------------------------------------------- /PHYS 408 - Optics/phys408fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 408 - Optics/phys408fs.pdf -------------------------------------------------------------------------------- /MECH 466 - Automatic Control/mech466fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MECH 466 - Automatic Control/mech466fs.pdf -------------------------------------------------------------------------------- /ELEC 221 - Signals and Systems/elec221fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/ELEC 221 - Signals and Systems/elec221fs.pdf -------------------------------------------------------------------------------- /PHYS 401 - Electromagnetic Theory/BoldR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 401 - Electromagnetic Theory/BoldR.pdf -------------------------------------------------------------------------------- /PHYS 401 - Electromagnetic Theory/ScriptR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 401 - Electromagnetic Theory/ScriptR.pdf -------------------------------------------------------------------------------- /APSC 278 - Engineering Materials/apsc278fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/APSC 278 - Engineering Materials/apsc278fs.pdf -------------------------------------------------------------------------------- /ELEC 404 - RF Integrated Circuits/elec404fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/ELEC 404 - RF Integrated Circuits/elec404fs.pdf -------------------------------------------------------------------------------- /MECH 360 - Mechanics of Materials/mech360fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MECH 360 - Mechanics of Materials/mech360fs.pdf -------------------------------------------------------------------------------- /MECH 431 - Engineering Economics/mech431fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MECH 431 - Engineering Economics/mech431fs.pdf -------------------------------------------------------------------------------- /PHYS 301 - Electricity and Magnetism/BoldR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 301 - Electricity and Magnetism/BoldR.pdf -------------------------------------------------------------------------------- /PHYS 401 - Electromagnetic Theory/phys401fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 401 - Electromagnetic Theory/phys401fs.pdf -------------------------------------------------------------------------------- /PHYS 403 - Statistical Mechanics/phys403fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 403 - Statistical Mechanics/phys403fs.pdf -------------------------------------------------------------------------------- /MATH 305 - Applied Complex Analysis/math305fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MATH 305 - Applied Complex Analysis/math305fs.pdf -------------------------------------------------------------------------------- /PHYS 301 - Electricity and Magnetism/ScriptR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 301 - Electricity and Magnetism/ScriptR.pdf -------------------------------------------------------------------------------- /PHYS 301 - Electricity and Magnetism/phys301fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 301 - Electricity and Magnetism/phys301fs.pdf -------------------------------------------------------------------------------- /CPSC 340 - Machine Learning and Data Mining/cpsc340fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/CPSC 340 - Machine Learning and Data Mining/cpsc340fs.pdf -------------------------------------------------------------------------------- /MECH 280 - Introduction to Fluid Mechanics/mech280fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MECH 280 - Introduction to Fluid Mechanics/mech280fs.pdf -------------------------------------------------------------------------------- /MECH 466 - Automatic Control/SecondOrderFreqResponse.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MECH 466 - Automatic Control/SecondOrderFreqResponse.jpg -------------------------------------------------------------------------------- /PHYS 350 - Applications of Classical Mechanics/BoldR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 350 - Applications of Classical Mechanics/BoldR.pdf -------------------------------------------------------------------------------- /ELEC 404 - RF Integrated Circuits/scattering_parameters.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/ELEC 404 - RF Integrated Circuits/scattering_parameters.png -------------------------------------------------------------------------------- /MATH 217 - Multivariable and Vector Calculus/math217fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MATH 217 - Multivariable and Vector Calculus/math217fs.pdf -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250finalfs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 250 - Introduction to Modern Physics/phys250finalfs.pdf -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250mt1fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 250 - Introduction to Modern Physics/phys250mt1fs.pdf -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250mt2fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 250 - Introduction to Modern Physics/phys250mt2fs.pdf -------------------------------------------------------------------------------- /PHYS 304 - Introduction to Quantum Mechanics/phys304fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 304 - Introduction to Quantum Mechanics/phys304fs.pdf -------------------------------------------------------------------------------- /PHYS 350 - Applications of Classical Mechanics/ScriptR.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 350 - Applications of Classical Mechanics/ScriptR.pdf -------------------------------------------------------------------------------- /PHYS 350 - Applications of Classical Mechanics/phys350fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/PHYS 350 - Applications of Classical Mechanics/phys350fs.pdf -------------------------------------------------------------------------------- /ELEC 401 - Analog CMOS Integrated Circuit Design/elec401fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/ELEC 401 - Analog CMOS Integrated Circuit Design/elec401fs.pdf -------------------------------------------------------------------------------- /MATH 318 - Probability with Physical Applications/math318fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MATH 318 - Probability with Physical Applications/math318fs.pdf -------------------------------------------------------------------------------- /MATH 400 - Applied Partial Differential Equations/math400fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/MATH 400 - Applied Partial Differential Equations/math400fs.pdf -------------------------------------------------------------------------------- /ELEC 433 - Error Control Coding for Communications and Computers/elec433fs.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/DonneyF/formula-sheets/HEAD/ELEC 433 - Error Control Coding for Communications and Computers/elec433fs.pdf -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Formula Sheets 2 | A collection of formula sheets and quick review notes from some UBC courses. Few courses allowed such sheets in an examination environment. 3 | 4 | | Course | Title | Term | Formula Sheet | 5 | | --- | --- | --- | --- | 6 | | APSC 278 | Engineering Materials | 2020W1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/APSC%20278%20-%20Engineering%20Materials/apsc278fs.pdf) | 7 | | CPSC 340 | Machine Learning and Data Mining | 2020W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/CPSC%20340%20-%20Machine%20Learning%20and%20Data%20Mining/cpsc340fs.pdf) | 8 | | ELEC 221 | Signals and Systems | 2018W1| [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/ELEC%20221%20-%20Signals%20and%20Systems/elec221fs.pdf) | 9 | | ELEC 401 | Analog CMOS Integrated Circuit Design | 2020W1| [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/ELEC%20401%20-%20Analog%20CMOS%20Integrated%20Circuit%20Design/elec401fs.pdf) | 10 | | ELEC 404 | RF Integrated Circuits | 2020W2| [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/ELEC%20404%20-%20RF%20Integrated%20Circuits/elec404fs.pdf) | 11 | | ELEC 433 | Error Control Coding for Communications and Computers | 2021W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/ELEC%20433%20-%20Error%20Control%20Coding%20for%20Communications%20and%20Computers/elec433fs.pdf) | 12 | | MATH 217 | Multivariable and Vector Calculus | 2017W1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MATH%20217%20-%20Multivariable%20and%20Vector%20Calculus/math217fs.pdf) | 13 | | MATH 305 | Applied Complex Analysis | 2018W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MATH%20305%20-%20Applied%20Complex%20Analysis/math305fs.pdf) | 14 | | MATH 318 | Probability with Physical Applications | 2019W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MATH%20318%20-%20Probability%20with%20Physical%20Applications/math318fs.pdf) | 15 | | MATH 400 | Applied Partial Differential Equations | 2020W1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MATH%20400%20-%20Applied%20Partial%20Differential%20Equations/math400fs.pdf) | 16 | | MECH 280 | Introduction to Fluid Mechanics | 2018W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MECH%20280%20-%20Introduction%20to%20Fluid%20Mechanics/mech280fs.pdf) | 17 | | MECH 360 | Mechanics of Materials | 2018W1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MECH%20360%20-%20Mechanics%20of%20Materials/mech360fs.pdf) | 18 | | MECH 431 | Engineering Economics | 2019S1-2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MECH%20431%20-%20Engineering%20Economics/mech431fs.pdf) | 19 | | MECH 466 | Automatic Control | 2019W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/MECH%20466%20-%20Automatic%20Control/mech466fs.pdf) | 20 | | PHYS 250 | Introduction to Modern Physics | 2018S1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20250%20-%20Introduction%20to%20Modern%20Physics/phys250finalfs.pdf) | 21 | | PHYS 304 | Introduction to Quantum Mechanics | 2018W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20304%20-%20Introduction%20to%20Quantum%20Mechanics/phys304fs.pdf) | 22 | | PHYS 301 | Electricity and Magnetism | 2018W1 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20301%20-%20Electricity%20and%20Magnetism/phys301fs.pdf) | 23 | | PHYS 350 | Applications of Classical Mechanics | 2019W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20350%20-%20Applications%20of%20Classical%20Mechanics/phys350fs.pdf) | 24 | | PHYS 401 | Electromagnetic Theory | 2020W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20401%20-%20Electromagnetic%20Theory/phys401fs.pdf) | 25 | | PHYS 403 | Statistical Mechanics | 2020W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20403%20-%20Statistical%20Mechanics/phys403fs.pdf) | 26 | | PHYS 408 | Optics | 2021W2 | [View](https://docs.google.com/viewer?url=https://github.com/DonneyF/formula-sheets/raw/master/PHYS%20408%20-%20Optics/phys408fs.pdf) | 27 | 28 | 29 | Adapted from [this great template](https://wch.github.io/latexsheet/). 30 | 31 | ## Editing 32 | 33 | To edit this formula sheet, you will need to download the appropriate `.tex` file and open it in your favorite LaTeX edting software (I used [TexStudio](https://www.texstudio.org/) for this, but [OverLeaf](https://www.overleaf.com/) is a nice online solution). 34 | -------------------------------------------------------------------------------- /MECH 431 - Engineering Economics/mech431fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | 14 | % This sets page margins to .5 inch if using letter paper, and to 1cm 15 | % if using A4 paper. (This probably isn't strictly necessary.) 16 | % If using another size paper, use default 1cm margins. 17 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 18 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 19 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | } 23 | 24 | % Turn off header and footer 25 | \pagestyle{empty} 26 | 27 | 28 | % Redefine section commands to use less space 29 | \makeatletter 30 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 31 | {-1ex plus -.5ex minus -.2ex}% 32 | {0.5ex plus .2ex}%x 33 | {\normalfont\normalsize\bfseries}} 34 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 35 | {-1explus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}% 37 | {\normalfont\small\bfseries}} 38 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 39 | {-1ex plus -.5ex minus -.2ex}% 40 | {1ex plus .2ex}% 41 | {\normalfont\footnotessize\bfseries}} 42 | \makeatother 43 | 44 | % Define BibTeX command 45 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 46 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 47 | 48 | % Don't print section numbers 49 | \setcounter{secnumdepth}{0} 50 | 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | \newcommand{\ap}{a_+} 58 | \newcommand{\am}{a_-} 59 | 60 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 61 | 62 | % Redefine some commands for newtxmath boldness 63 | \renewcommand{\grad}{\nabla} 64 | \renewcommand{\curl}[1]{\nabla\times#1} 65 | \renewcommand{\div}[1]{\nabla\cdot#1} 66 | \renewcommand{\cross}{\times} 67 | 68 | % ----------------------------------------------------------------------- 69 | 70 | \begin{document} 71 | 72 | \raggedright 73 | \footnotesize 74 | \begin{multicols*}{3} 75 | 76 | % multicol parameters 77 | % These lengths are set only within the two main columns 78 | %\setlength{\columnseprule}{0.25pt} 79 | \setlength{\premulticols}{1pt} 80 | \setlength{\postmulticols}{1pt} 81 | \setlength{\multicolsep}{1pt} 82 | \setlength{\columnsep}{2pt} 83 | 84 | \begin{center} 85 | \Large{\underline{MECH 431 Formula Sheet}} 86 | \end{center} 87 | 88 | \section{Engineering Costs and Cost Estimating} 89 | Engineering Costs: 90 | \begin{itemize} 91 | \itemsep 0em 92 | \item Fixed - Constant regardless of output activity 93 | \item Variable - Depends on output activity 94 | \item Marginal - Variable cost of one more unit 95 | \item Average - Total cost divided by number of units 96 | \item Total - Total Fixed + Total Variable 97 | \item Sunk - Money already spend, result of a past decision 98 | \item Opportunity - Next best benefit forgone 99 | \item Recurring - Repeating expense that is known, anticipated 100 | \item Non-Recurring - One-of-a-kind, Irregular 101 | \item Incremental - Cost differences between alternatives 102 | \item Cash - Costs associated with cash transactions 103 | \item Book - Cost effects from past decisions 104 | \item Life-Cycle - Costs over various phases of a product's life 105 | \end{itemize} 106 | 107 | Estimating Models: 108 | \begin{itemize} 109 | \itemsep 0em 110 | \item Per-Unit - Per-unit factor 111 | \item Segmenting - Divide \& conquer 112 | \item Cost Indices - Historical changes based on ratio 113 | $$\frac{\text{Cost at time A}}{\text{Cost at time B}} = \frac{\text{Index value at time A}}{\text{Index value at time B}}$$ 114 | \item Power-Sizing - Accounts for Economies of Scale 115 | $$\frac{\text{Cost of equipment A}}{\text{Cost of equipment B}} = \left(\frac{\text{Capacity of equipment A}}{\text{Capacity of equipment B}}\right)^x$$ 116 | \item Learning Curve - Relationship between repetition and performance 117 | $$T_N = T_i \times N^b$$ 118 | $$b = \log_2(\text{learning curve expressed as a decimal})$$ 119 | for N completed units. 120 | \end{itemize} 121 | 122 | \section{Interest and Equivalence} 123 | Simple Interest:\\ 124 | \tab $F = P(1 + in)$ 125 | 126 | Single-Payment Compound Interest:\\ 127 | \tab $F = P(1 + i)^n$ 128 | 129 | Single-Payment Present Worth:\\ 130 | \tab $P = F(1 + i)^{-n}$ 131 | 132 | Effective Annual Interest Rate for a nominal interest rate ($r$) and $m$ compounding subperiods:\\ 133 | \tab $\ds i_a = \left(1+ \frac{r}{m}\right)^m -1$ \qquad $i_a = (1 + i)^m-1$ 134 | 135 | Uniform Series Compound Amount/Sinking Fund:\\ 136 | \tab $\ds F = A\left[\frac{(1+i)^n-1}{i}\right]$ \qquad $\ds A = F\left[\frac{i}{(1+i)^n-1}\right]$ 137 | 138 | Uniform Series Capital Recovery/Present Worth:\\ 139 | \tab $\ds A = P\left[\frac{i(1+i)^n}{(1+i)^n-1}\right]$ \qquad $\ds P = A\left[\frac{(1+i)^n-1}{i(1+i)^n}\right]$ 140 | 141 | Arithmetic Gradient Present Worth:\\ 142 | \tab $\ds P = G\left[\frac{(1+i)^n-in-1}{i^2(1+i)^n}\right]$ 143 | 144 | Arithmetic Gradient Uniform Series:\\ 145 | \tab $\ds A = G\left[\frac{(1+i)^n-in-1}{i(1+i)^n-i}\right]$ 146 | 147 | Geometric Gradient Present Worth:\\ 148 | \tab $\ds P = A_1\left[\frac{1-(1+g)^n(1+i)^{-n}}{i-g}\right]$ \quad for $i\neq g$\\ 149 | \tab $\ds P = A_1n(1+i)^{-1}$ \hspace{2.27cm} for $i = g$ 150 | 151 | 152 | % Footer content 153 | \rule{0.3\linewidth}{0.25pt} 154 | \scriptsize\\ 155 | Updated \today\\ 156 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 157 | \end{multicols*} 158 | \end{document} 159 | -------------------------------------------------------------------------------- /PHYS 403 - Statistical Mechanics/phys403fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | \usepackage{booktabs} 14 | 15 | % This sets page margins to .5 inch if using letter paper, and to 1cm 16 | % if using A4 paper. (This probably isn't strictly necessary.) 17 | % If using another size paper, use default 1cm margins. 18 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 19 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 20 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 23 | } 24 | 25 | % Turn off header and footer 26 | \pagestyle{empty} 27 | 28 | 29 | % Redefine section commands to use less space 30 | \makeatletter 31 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 32 | {-1ex plus -.5ex minus -.2ex}% 33 | {0.5ex plus .2ex}%x 34 | {\normalfont\normalsize\bfseries}} 35 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 36 | {-1explus -.5ex minus -.2ex}% 37 | {0.5ex plus .2ex}% 38 | {\normalfont\small\bfseries}} 39 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 40 | {-1ex plus -.5ex minus -.2ex}% 41 | {1ex plus .2ex}% 42 | {\normalfont\footnotessize\bfseries}} 43 | \renewcommand\small{\@setfontsize\small{10}{11}} 44 | \makeatother 45 | 46 | % Define BibTeX command 47 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 48 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 49 | 50 | % Don't print section numbers 51 | \setcounter{secnumdepth}{0} 52 | 53 | \setlength{\parindent}{0pt} 54 | \setlength{\parskip}{1pt plus 0.5ex} 55 | 56 | \newcommand{\tab}{\hspace{.02\textwidth}} 57 | \newcommand{\ds}{\displaystyle} 58 | 59 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 60 | 61 | % Redefine some commands for newtxmath boldness 62 | \renewcommand{\grad}{\nabla} 63 | \renewcommand{\curl}[1]{\nabla\times#1} 64 | \renewcommand{\div}[1]{\nabla\cdot#1} 65 | \renewcommand{\cross}{\times} 66 | 67 | % ----------------------------------------------------------------------- 68 | 69 | \begin{document} 70 | \raggedright 71 | \footnotesize 72 | \begin{multicols}{3} 73 | % multicol parameters 74 | % These lengths are set only within the two main columns 75 | %\setlength{\columnseprule}{0.25pt} 76 | \setlength{\premulticols}{1pt} 77 | \setlength{\postmulticols}{1pt} 78 | \setlength{\multicolsep}{1pt} 79 | \setlength{\columnsep}{2pt} 80 | 81 | \begin{center} 82 | \Large{\underline{PHYS 403 Formula Sheet}} 83 | \end{center} 84 | 85 | \section{Thermodynamics} 86 | \begin{tabular}{lll} 87 | \toprule 88 | Potential & Function & Differential\\ 89 | \midrule 90 | Internal & $U$ & $dU = T\,dS - p\,dV$\\ 91 | Enthalpy & $H = U + pV$ & $dH = T\,dS + V\,dp$\\ 92 | Helmoltz & $F = U -TS$ & $dF = -S\,dT$\\ 93 | Gibbs & $G = U -TS + pV$ & $dG = -S\,dT + V\,dp$\\ 94 | \bottomrule 95 | \end{tabular} 96 | 97 | Internal Energy:\\ 98 | \tab $\Delta U = Q + W$ 99 | 100 | First Law of Thermodynamics:\\ 101 | \tab $dU = \delta Q - p\,dV$ 102 | 103 | Reversible Process:\\ 104 | \tab $dU = T\,dS$\\ 105 | \tab $\ds dS = \frac{\delta Q}{T}$ 106 | 107 | Heat Capacities:\\ 108 | \tab $\ds C_V = \left(\pdv{U}{T}\right)_V \qquad C_p = \left(\pdv{H}{T}\right)_p$ 109 | 110 | Entropy:\\ 111 | \tab $S = k_B \ln(W)$ 112 | 113 | \section{The Canonical Distribution} 114 | Boltzmann Factor:\\ 115 | \tab $e^{-\beta E_i}$ 116 | 117 | The Canonical Distribution:\\ 118 | \tab $P_i = Z^{-1}e^{-\beta E_i}$ 119 | 120 | Partition Function with degeneracy $g_j$:\\ 121 | \tab $\ds Z = \sum_i g_ie^{-\beta E_i}$ 122 | 123 | Continuous Partition function for degeneracy per unit volume $g(E)$:\\ 124 | \tab $\ds Z = V\int_{0}^{\infty}g(E)e^{-\beta E}\mathop{dE}$ 125 | 126 | Mean energy of a microsystem:\\ 127 | \tab $\ds \expval{E} = \sum_i P_i E_i = -\frac{1}{Z}\pdv{Z}{\beta}$ 128 | 129 | Mean energy of a macrosystem with $N$ identical and weakly interacting microsystems:\\ 130 | \tab $U = N\hspace{-0.75mm}\expval{E}$ 131 | 132 | Many-particle partition function:\\ 133 | \tab $Z_N = Z_1^N$ 134 | 135 | Free Energy:\\ 136 | \tab $Z_N = e^{-\beta F}$ 137 | 138 | Entropy for probability $P_i = Z_N^{-1}e^{-\beta E_i}$ that the system is in the $i$-th macrostate:\\ 139 | \tab $\ds S = -k_B \sum_i P_i\ln(P_i)$ 140 | 141 | \subsection{$N$ Distinguishable Particles in a Box} 142 | Degeneracies for spin $s$:\\ 143 | \tab $\ds g_\mathit{1D}(E) = (2s+1)\frac{2}{\hbar}\sqrt{\frac{m}{2E}}$\\ 144 | \tab $\ds g_\mathit{2D}(E) = (2s+1)\frac{m}{2\pi\hbar^2}$\\ 145 | \tab $\ds g_\mathit{3D}(E) = (2s+1)\frac{1}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$ 146 | 147 | \section{The Grand Canonical Distribution} 148 | Chemical Potential:\\ 149 | \tab $\ds \mu = -T\left(\pdv{S}{N}\right)_{U,V} = \left(\pdv{F}{N}\right)_{T,V}$ 150 | 151 | Gibbs Factor:\\ 152 | \tab $e^{\beta(n\mu - E_i)}$ 153 | 154 | The Grand Canonical Distribution:\\ 155 | \tab $P(n, E_i) = \Xi^{-1}e^{\beta(n\mu - E_i)}$ 156 | 157 | Grand Partition Function:\\ 158 | \tab $\ds \Xi = \sum_{i,n}e^{\beta(n\mu - E_i)}$ 159 | 160 | Absolute Activity:\\ 161 | \tab $\alpha = e^{\beta\mu}$ 162 | 163 | Mean number of particles:\\ 164 | \tab $\ds \expval{n} = \Xi^{-1} \sum_{i,n}ne^{\beta(n\mu - E_i)} = \frac{1}{\beta\Xi}\left(\pdv{\Xi}{\mu}\right)_T = \frac{\alpha}{\Xi}\left(\pdv{\Xi}{\alpha}\right)_T$ 165 | 166 | Mean energy:\\ 167 | \tab $\ds \expval{E} = -\frac{1}{\Xi}\left(\pdv{\Xi}{\beta}\right)_T + \mu\hspace{-0.75mm}\expval{n}$ 168 | 169 | \section{Quantum and Classical Gasses} 170 | One-particle distribution function over energy:\\ 171 | \tab $f(E) = \expval{n(E)}$ 172 | 173 | Fermi-Dirac Distribution:\\ 174 | \tab $\ds f(E) = \frac{1}{e^{\beta(E - \mu)}- 1}$ 175 | 176 | Total number of particles:\\ 177 | \tab $\ds N = \int_{0}^{\infty}Vg(E)f(E)\mathop{dE}$ 178 | 179 | Fermi Energy ($\mu \approx E_F$ at very low temperatures):\\ 180 | \tab $\ds n = \int_{0}^{E_F}g(E)\mathop{dE}$ 181 | 182 | Bose-Einstein Distribution:\\ 183 | \tab $\ds f(E) = \frac{1}{e^{\beta(E - \mu)}-1} = \frac{1}{\alpha^{-1}e^{\beta E}-1}$ 184 | 185 | Maxwell-Boltzmann Distribution ($f(E) \ll 1$):\\ 186 | \tab $f(E) = e^{\beta(\mu - E)}$ 187 | 188 | % Footer content 189 | \rule{0.3\linewidth}{0.25pt} 190 | \scriptsize\\ 191 | Updated \today\\ 192 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 193 | \end{multicols} 194 | \end{document} 195 | -------------------------------------------------------------------------------- /ELEC 401 - Analog CMOS Integrated Circuit Design/elec401fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{mdframed} 16 | \usepackage{amsbsy} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \makeatother 47 | 48 | % Define BibTeX command 49 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 50 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 51 | 52 | % Don't print section numbers 53 | \setcounter{secnumdepth}{0} 54 | 55 | 56 | \setlength{\parindent}{0pt} 57 | \setlength{\parskip}{1pt plus 0.5ex} 58 | 59 | \newcommand{\tab}{\hspace*{1em}} 60 | \newcommand{\ds}{\displaystyle} 61 | 62 | % Redefine some commands for newtxmath boldness 63 | \renewcommand{\grad}{\nabla} 64 | \renewcommand{\curl}[1]{\nabla\times#1} 65 | \renewcommand{\div}[1]{\nabla\cdot#1} 66 | \renewcommand{\cross}{\times} 67 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 68 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 69 | 70 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 71 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 72 | 73 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 74 | \theoremstyle{definition} 75 | \newmdtheoremenv{theorem}{Theorem} 76 | 77 | % ----------------------------------------------------------------------- 78 | 79 | \begin{document} 80 | 81 | \raggedright 82 | \footnotesize 83 | \begin{multicols}{3} 84 | 85 | \raggedcolumns 86 | 87 | % multicol parameters 88 | % These lengths are set only within the two main columns 89 | %\setlength{\columnseprule}{0.25pt} 90 | \setlength{\premulticols}{1pt} 91 | \setlength{\postmulticols}{1pt} 92 | \setlength{\multicolsep}{1pt} 93 | \setlength{\columnsep}{2pt} 94 | 95 | \begin{center} 96 | \Large{\underline{ELEC 401 Formula Sheet}} 97 | \end{center} 98 | 99 | \section{MOS Transistors} 100 | Regions of Operation:\\ 101 | \tab NMOS: 102 | \begin{equation*} 103 | \begin{cases} 104 | V_{GS} < V_{TH}& \text{Cut-off}\\ 105 | V_{GS} > V_{TH}, V_{DS} \ll 2(V_{GS} - V_{TH}) & \text{Deep Triode}\\ 106 | V_{GS} > V_{TH}, V_{DS} < V_{GS} - V_{TH} & \text{Triode}\\ 107 | V_{GS} > V_{TH}, V_{DS} > V_{GS} - V_{TH} & \text{Saturation} 108 | \end{cases} 109 | \end{equation*} 110 | \tab PMOS: 111 | \begin{equation*} 112 | \begin{cases} 113 | V_{SG} < \abs{V_{TH}}& \text{Cut-off}\\ 114 | V_{SG} > \abs{V_{TH}}, V_{SD} \ll 2(V_{SG} - \abs{V_{TH}}) & \text{Deep Triode}\\ 115 | V_{SG} >\abs{V_{TH}}, V_{SD} < V_{SG} - \abs{V_{TH}} & \text{Triode}\\ 116 | V_{SG} > \abs{V_{TH}}, V_{SD} > V_{SG} - \abs{V_{TH}} & \text{Saturation} 117 | \end{cases} 118 | \end{equation*} 119 | 120 | Long Channel Current Equations:\\ 121 | \tab NMOS ($I_{DS}$): 122 | \begin{equation*} 123 | \begin{cases} 124 | 0 & \text{Cut-off}\\ 125 | \ds \mu_n C_\text{ox} \frac{W}{L} (V_{GS} - V_{TH}) V_{DS} & \text{Deep Triode}\\[0.5em] 126 | \ds \mu_n C_\text{ox} \frac{W}{L} \left[(V_{GS} - V_{TH}) V_{DS} - \frac{V_{DS}^2}{2}\right] & \text{Triode}\\[1em] 127 | \ds \frac{1}{2} \mu_n C_\text{ox} \frac{W}{L} (V_{GS} - V_{TH})^2 & \text{Saturation} 128 | \end{cases} 129 | \end{equation*} 130 | 131 | \tab PMOS ($I_{SD}$): 132 | \begin{equation*} 133 | \begin{cases} 134 | 0 & \text{Cut-off}\\ 135 | \ds \mu_p C_\text{ox} \frac{W}{L} (V_{SG} - \abs{V_{TH}}) V_{SD} & \text{Deep Triode}\\[0.5em] 136 | \ds \mu_p C_\text{ox} \frac{W}{L} \left[(V_{SG} - \abs{V_{TH}}) V_{SD} - \frac{V_{SD}^2}{2}\right] & \text{Triode}\\[1em] 137 | \ds \frac{1}{2} \mu_p C_\text{ox} \frac{W}{L} (V_{SG} - \abs{V_{TH}})^2 & \text{Saturation} 138 | \end{cases} 139 | \end{equation*} 140 | 141 | Transconductance (NMOS):\\ 142 | \tab $\ds g_m = \eval{\pdv{I_D}{V_{GS}}}_{V_{DS}}$\\ 143 | \tab $\ds g_m = \mu_n C_\text{ox} \frac{W}{L} (V_{GS} - V_{TH})$\\ 144 | \tab $\ds g_m = \sqrt{2\mu_n C_\text{ox} \frac{W}{L} I_D} = \frac{2I_D}{V_{GS} - V_{TH}}$ 145 | 146 | Body Effect:\\ 147 | \tab $V_\text{TH} = V_\text{TH0} + \gamma\left(\sqrt{\abs{2\Phi_F + V_{SB}}} - \sqrt{\abs{2\Phi_F}}\right)$ 148 | \tab $\ds \gamma = \frac{\sqrt{2q\varepsilon_\text{si}N_\text{sub}}}{C_\text{ox}}$ 149 | 150 | Channel Length Modulation:\\ 151 | \tab $\ds I_D = \frac{1}{2} \mu_n C_\text{ox} \frac{W}{L} (V_{GS} - V_{TH})^2 (1 + \lambda V_{DS})$ 152 | 153 | Sub-threshold Conduction:\\ 154 | \tab $I_D = I_0 e^{\frac{V_{GS}}{\zeta V_T}}$ 155 | 156 | Device Capacitances:\\ 157 | \bgroup 158 | \def\arraystretch{1.2}% 159 | \tab \begin{tabular}{|l|c | c |c |} 160 | \hline 161 | & Cut-off & Triode & Saturation\\ 162 | \hline 163 | $C_{GS}$ & $C_\text{ov}$ & $C_\text{ov} + \frac{C_1}{2}$ & $C_\text{ov} + \frac{2}{3}C_1$\\ 164 | \hline 165 | $C_{GD}$ & $C_\text{ov}$ & $C_\text{ov} + \frac{C_1}{2}$ & $C_\text{ov}$\\ 166 | \hline 167 | $C_{GB}$ & $\frac{C_1 C_2}{C_1 + C_2} \leq C_{GB} \leq C_1$ & 0 & 0\\ 168 | \hline 169 | $C_{SB}$ & $C_5$ & $C_5 + \frac{C_2}{2}$ & $C_5 + \frac{2}{3}C_2$\\ 170 | \hline 171 | $C_{DB}$ & $C_6$ & $C_6 + \frac{C_2}{2}$ & $C_6$\\ 172 | \hline 173 | \end{tabular} 174 | \egroup 175 | 176 | Small-Signal Model:\\ 177 | \tab $\ds i_D = g_m v_{GS} + \frac{v_{DS}}{r_o} + g_{mb}v_{BS}$ 178 | \tab $\ds g_{mb} = \eta g_m = \frac{\gamma}{2\sqrt{\abs{2\Phi_F + V_{SB}}}} g_m$ 179 | 180 | % Footer content 181 | \rule{0.3\linewidth}{0.25pt} 182 | \scriptsize\\ 183 | Updated \today\\ 184 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 185 | \end{multicols}% 186 | 187 | \end{document} 188 | -------------------------------------------------------------------------------- /MECH 280 - Introduction to Fluid Mechanics/mech280fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | 14 | % This sets page margins to .5 inch if using letter paper, and to 1cm 15 | % if using A4 paper. (This probably isn't strictly necessary.) 16 | % If using another size paper, use default 1cm margins. 17 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 18 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 19 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | } 23 | 24 | % Turn off header and footer 25 | \pagestyle{empty} 26 | 27 | 28 | % Redefine section commands to use less space 29 | \makeatletter 30 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 31 | {-1ex plus -.5ex minus -.2ex}% 32 | {0.5ex plus .2ex}%x 33 | {\normalfont\normalsize\bfseries}} 34 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 35 | {-1explus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}% 37 | {\normalfont\small\bfseries}} 38 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 39 | {-1ex plus -.5ex minus -.2ex}% 40 | {1ex plus .2ex}% 41 | {\normalfont\footnotessize\bfseries}} 42 | \makeatother 43 | 44 | % Define BibTeX command 45 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 46 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 47 | 48 | % Don't print section numbers 49 | \setcounter{secnumdepth}{0} 50 | 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 58 | \newcommand{\mdv}[2]{\frac{\mathrm{D}#1}{\mathrm{D}#2}} 59 | 60 | % Redefine some commands for newtxmath boldness 61 | \renewcommand{\grad}{\nabla} 62 | \renewcommand{\curl}[1]{\nabla\times#1} 63 | \renewcommand{\div}[1]{\nabla\cdot#1} 64 | \renewcommand{\cross}{\times} 65 | 66 | 67 | % ----------------------------------------------------------------------- 68 | 69 | \begin{document} 70 | 71 | \raggedright 72 | \footnotesize 73 | \begin{multicols}{3} 74 | \raggedcolumns 75 | 76 | % multicol parameters 77 | % These lengths are set only within the two main columns 78 | %\setlength{\columnseprule}{0.25pt} 79 | \setlength{\premulticols}{1pt} 80 | \setlength{\postmulticols}{1pt} 81 | \setlength{\multicolsep}{1pt} 82 | \setlength{\columnsep}{2pt} 83 | 84 | \begin{center} 85 | \Large{\underline{MECH 280 Formula Sheet}} 86 | \end{center} 87 | 88 | \section{Vectors \& Matrices} 89 | Inner Product (aka scalar or dot product):\\ 90 | \tab $\vb{a}\cdot\vb{b} = \abs{\vb{a}}\abs{\vb{b}}\cos\theta$ 91 | 92 | Cross Product (aka vector product):\\ 93 | \tab $\vb{a}\cross\vb{b} = \abs{\vb{a}}\abs{\vb{b}}\sin\theta\,\vu{n}$ 94 | 95 | Triple Product:\\ 96 | \tab $\vb{a} \times (\vb{b} \times \vb{c}) = (\vb{a} \cdot \vb{c})\vb{b} - (\vb{a} \cdot \vb{b})\vb{c}$\\ 97 | \tab $(\vb{a} \times \vb{b}) \cdot \vb{c} = (\vb{c} \times \vb{a}) \cdot \vb{b} = (\vb{b} \times \vb{c}) \cdot \vb{a}$ 98 | 99 | Outer Product (aka tensor product):\\ 100 | \tab $\vb{v} \otimes \vb{w} \begin{bmatrix}v_{1}w_{1}&&v_{1}w_{2}&&\cdots&&v_{1}w_{m}\\ 101 | v_{2}w_{1}&&v_{2}w_{2}&&\cdots &&v_{2}w_{m}\\ 102 | \vdots &&\vdots &&\ddots &&\vdots \\ 103 | v_{n}w_{1}&&v_{n}w_{2}&&\cdots &&v_{n}w_{m}\end{bmatrix}$ 104 | 105 | \section{Vector Derivatives} 106 | \subsection{Cartesian} 107 | Gradient:\\ 108 | \tab $\ds \grad{f} = \pdv{f}{x}\vu{x} + \pdv{f}{y}\vu{y} + \pdv{f}{z}\vu{z}$ 109 | 110 | Divergence:\\ 111 | \tab $\ds \div{\vb{v}} = \pdv{v_x}{x} + \pdv{v_y}{y} + \pdv{v_z}{z}$ 112 | 113 | Curl:\\ 114 | \vspace{-3mm} 115 | \tab $\ds \curl{\vb{v}} = \left(\pdv{v_z}{y} - \pdv{v_y}{z}\right)\vu{x} + \left(\pdv{v_x}{z} - \pdv{v_z}{x}\right)\vu{y} + \left(\pdv{v_y}{x} - \pdv{v_x}{y}\right)\vu{z}$ 116 | 117 | \subsection{Spherical} 118 | ($\theta$ is the polar angle.) 119 | 120 | Gradient:\\ 121 | \tab $\ds \grad{f} = \pdv{f}{r}\vu{r} + \frac{1}{r}\pdv{f}{\theta}\vu*{\theta} + \frac{1}{r\sin\theta}\pdv{f}{\phi}\vu*{\phi}$ 122 | 123 | Divergence:\\ 124 | \tab $\ds \div{\vb{v}} = \frac{1}{r^2}\pdv{r}(r^2v_r) + \frac{1}{r\sin\theta}\pdv{\theta}(\sin\theta v_\theta) + \frac{1}{r\sin\theta}\pdv{v_\theta}{\phi}$ 125 | 126 | Curl:\\ 127 | \tab $\ds \curl{\vb{v}} = \frac{1}{r\sin\theta}\left[\pdv{\theta}(\sin\theta\, v_\phi)- \pdv{v_\theta}{\phi}\right]\vu{r}\,+$\\ 128 | \tab \tab $\ds \frac{1}{r}\left[\frac{1}{\sin\theta}\pdv{v_r}{\phi}-\pdv{r}(r v_\phi)\right]\vu*{\theta} + \frac{1}{r}\left[\pdv{r}(r v_\theta)-\pdv{v_r}{\theta}\right]\vu*{\phi}$ 129 | 130 | Tensor:\\ 131 | \tab $\grad\vb{u} = \begin{bmatrix} 132 | \vspace{2mm} 133 | \ds \pdv{u_x}{x} &&\ds \pdv{u_y}{x}&&\ds \pdv{u_z}{x}\\ 134 | \vspace{2mm} 135 | \ds \pdv{u_x}{y} &&\ds \pdv{u_y}{y}&&\ds \pdv{u_z}{y} \\ 136 | \ds \pdv{u_x}{z} &&\ds \pdv{u_y}{z}&&\ds \pdv{u_z}{z}\\ 137 | \end{bmatrix}$ 138 | 139 | \subsection{Cylindrical} 140 | Gradient:\\ 141 | \tab $\ds \grad{f} = \pdv{f}{s}\vu{s}+\frac{1}{s}\pdv{f}{\phi}\vu*{\phi}+\pdv{f}{z}\vu{z}$ 142 | 143 | Divergence:\\ 144 | \tab $\ds \div{\vb{v}} = \frac{1}{s}\pdv{s}(sv_s)+\frac{1}{s}\pdv{v_\phi}{\phi} + \pdv{v_z}{z}$ 145 | 146 | Curl:\\ 147 | \tab $\ds \curl{\vb{v}} = \left[\frac{1}{s}\pdv{v_z}{\phi}-\pdv{v_\phi}{z}\right]\vu{s}+\left[\pdv{v_s}{z}-\pdv{v_z}{s}\right]\vu*{\phi}+\frac{1}{s}\left[\pdv{s}(sv_\phi)-\pdv{v_s}{\phi}\right]\vu{z}$ 148 | 149 | \section{Vector Calculus} 150 | Divergence Theorem:\\ 151 | \tab $\ds \int (\div \vb{A})\, dV = \oint \vb{A}\cdot d\vb{S}$ 152 | 153 | Stoke's Theorem:\\ 154 | \tab $\ds \int (\curl{\vb{A}})\cdot d\vb{S} = \oint \vb{A}\cdot d\vb{l}$ 155 | 156 | Directional Derivative (slope of a scalar field in the direction of the vector $\vb{u}$):\\ 157 | \tab $\grad f \cdot \vb{u}$ 158 | 159 | Material Derivative:\\ 160 | \tab $\ds \mdv{\rho}{t} = \pdv{\rho}{t} + \vb{u}\cdot \grad\rho$ 161 | \tab $\ds \mdv{\vb{A}}{t} = \pdv{\vb{A}}{t} + \vb{u}\cdot \grad\vb{A}$ 162 | 163 | Liebniz Rule:\\ 164 | \tab $\ds \frac{d}{dt} \int_\text{CV(t)}\rho dV = \int_\text{CV}\pdv{\rho}{t}dt + \rho\int_\text{CS} \vb{u}_\text{CS}\cdot d\vb{S}$ 165 | 166 | \section{Integral Equations of Motion} 167 | Flux:\\ 168 | \tab $\ds \phi = \int_\text{CS} \rho \vb{u}\cdot d\vb{S}$ 169 | 170 | Conservation of Mass:\\ 171 | \tab $\ds \int_\text{CV}\pdv{\rho}{t} = - \int_\text{CS} \rho \vb{u}\cdot d\vb{S}$ 172 | 173 | Conservation of Momentum:\\ 174 | $\ds \frac{d}{dt} \int_\text{CV}\rho \vb{u}dV = - \int_\text{CS}(\rho\vb{u})\vu{u}\cdot d\vb{S} + \int_\text{CV}\vb{f}_b\,dV + \int_\text{CS}\vb{t}\,dS$ 175 | 176 | \section{Differential Equations of Motion} 177 | Conservation of Mass:\\ 178 | \tab $\ds \mdv{\rho}{t} + \rho\div{\vb{u}} = 0$ 179 | 180 | Conservation of Momentum:\\ 181 | \tab $\ds \rho \mdv{\vb{u}}{t} = \vb{f}_b + \vb{f}_c$ 182 | 183 | % Footer content 184 | \rule{0.3\linewidth}{0.25pt} 185 | \scriptsize\\ 186 | Updated \today\\ 187 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 188 | \end{multicols} 189 | \end{document} 190 | -------------------------------------------------------------------------------- /PHYS 350 - Applications of Classical Mechanics/phys350fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{booktabs} 16 | \usepackage{mdframed} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \makeatother 47 | 48 | % Define BibTeX command 49 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 50 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 51 | 52 | % Don't print section numbers 53 | \setcounter{secnumdepth}{0} 54 | 55 | 56 | \setlength{\parindent}{0pt} 57 | \setlength{\parskip}{1pt plus 0.5ex} 58 | 59 | \newcommand{\tab}{\hspace{.02\textwidth}} 60 | \newcommand{\ds}{\displaystyle} 61 | \newcommand{\Lagr}{\mathcal{L}} 62 | 63 | % Redefine some commands for newtxmath boldness 64 | \renewcommand{\grad}{\nabla} 65 | \renewcommand{\curl}[1]{\nabla\times#1} 66 | \renewcommand{\div}[1]{\nabla\cdot#1} 67 | \renewcommand{\cross}{\times} 68 | 69 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 70 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 71 | 72 | \def\rcurs{{\mbox{$\resizebox{.09in}{.08in}{\includegraphics[trim= 1em 0 14em 0,clip]{ScriptR}}$}}} 73 | \def\brcurs{{\mbox{$\resizebox{.09in}{.08in}{\includegraphics[trim= 1em 0 14em 0,clip]{BoldR}}$}}} 74 | 75 | % ----------------------------------------------------------------------- 76 | 77 | \begin{document} 78 | 79 | \raggedright 80 | \footnotesize 81 | \begin{multicols}{3} 82 | 83 | 84 | % multicol parameters 85 | % These lengths are set only within the two main columns 86 | %\setlength{\columnseprule}{0.25pt} 87 | \setlength{\premulticols}{1pt} 88 | \setlength{\postmulticols}{1pt} 89 | \setlength{\multicolsep}{1pt} 90 | \setlength{\columnsep}{2pt} 91 | 92 | \raggedcolumns 93 | 94 | \begin{center} 95 | \Large{\underline{PHYS 350 Formula Sheet}} 96 | \end{center} 97 | 98 | \section{Lagrangian Mechanics} 99 | Hamilton's Principle: $\Lagr(\mathbf{q}, \mathbf{\dot{q}}, t)$ minimizes\\ 100 | \tab $\Lagr(\mathbf{q}, \dot{\mathbf{q}}, t) \rightarrow S[t] = \int_{t_1}^{t_2} \Lagr(\mathbf{q}, \dot{\mathbf{q}}, t)\,dt$ 101 | 102 | Euler-Lagrangian Equation:\\ 103 | \tab $\ds \dv{t}\left(\pdv{\Lagr}{\dot{q}_i}\right) - \pdv{\Lagr}{q_i} = 0 \qquad i = 1,2,\ldots,s$ 104 | 105 | Conservation of Energy:\\ 106 | \tab $\ds \pdv{\Lagr}{t} = 0$\\ 107 | \tab $\ds E = T + U = \sum_{i=1}^{s}\dot{q}_i \pdv{\Lagr}{q_i} - \Lagr$ 108 | 109 | Conservation of Momentum:\\ 110 | \tab $\ds \pdv{\Lagr}{q_i} = 0$\\ 111 | \tab $\ds p_i = \pdv{\Lagr}{\dot{q}_i}$ 112 | 113 | Systems with $s=1$, $\pdv{L}{t} = 0$:\\ 114 | \begin{itemize} 115 | \item $\ds \Lagr(q, \dot{q}) = \frac{\alpha(q)\dot{q}^2}{2} - U(q)$ 116 | \item $E \geq U(q)$ 117 | \item $U(q_o) = E$ are turning points 118 | \item $\ds \int_{0}^{T} dt = \int_{0}^{Q} \frac{1}{\sqrt{\frac{2}{\alpha(q)}(E - U(q))}}dq$ 119 | \end{itemize} 120 | 121 | \section{Two Body Problem} 122 | Generalized Coordinates:\\ 123 | \tab $\brcurs = \vb{r_1} - \vb{r_2}$\\ 124 | \tab $\ds \vb{R_\text{CM}} = \frac{m_1\vb{r_1} + m_2 \vb{r_2}}{m_1 + m_2}$\\ 125 | \tab $\ds \dot{\vb{r_1}} = \frac{m_2}{m_1 + m_2}\dot{\brcurs} + \vb{R_\text{CM}} \qquad \dot{\vb{r_2}} = -\frac{m_1}{m_1 + m_2}\dot{\brcurs} + \vb{R_\text{CM}}$\\ 126 | 127 | Lagrangian:\\ 128 | \tab $\ds \Lagr = \frac{\mu}{2}\abs{\dot{\brcurs}}^2 + \frac{M}{2}\abs{\dot{\vb{R_\text{CM}}}}^2 - U(\brcurs)$\\ 129 | \tab $\ds \mu = \frac{m_1m_2}{m_1 + m_2} \qquad M = m_1 + m_2$ 130 | 131 | Reduction to independent problems:\\ 132 | \tab $\Lagr = \Lagr_\text{CM} + \Lagr_\text{rel}$\\ 133 | \tab $\ds \Lagr_\text{CM} = \frac{M}{2}\abs{\dot{\vb{R_\text{CM}}}}^2$\\ 134 | \tab $\ds \Lagr_\text{rel} = \frac{\mu}{2}\abs{\dot{\brcurs}}^2 - U(\brcurs)$ 135 | 136 | Angular Momentum in Polar Coordinates:\\ 137 | \tab $\ell = \mu \rcurs^2 \dot{\phi}$ 138 | 139 | \section{Planetary Motion} 140 | \tab $\ds U(\rcurs) = - \frac{Gm_1m_2}{\rcurs} = -\frac{\alpha}{\rcurs}$ 141 | 142 | Eccentricity when $\ell \leq 0$:\\ 143 | \tab $\ds e = \sqrt{1 + \frac{E_\text{rel}}{U_o}} \qquad U_o = \abs{\min\{U_\text{eff}(\rcurs)\})}$\\ 144 | \tab Trajectory = $\begin{cases} 145 | \text{Constant radius orbit} & e = 0\\ 146 | \text{Ellipse} & 0 < e < 1\\ 147 | \text{Parabola} & e = 1\\ 148 | \text{Hyperbola} & e > 1 149 | \end{cases}$ 150 | 151 | % Column break 152 | \vfill\null 153 | \section{Small Oscillations} 154 | Equilibrium Point:\\ 155 | \tab $\ds \dv{U}{q}\bigg\rvert_{q = q_0} = 0$ 156 | 157 | Stability Criterion of Equilibrium Point ($s=1$):\\ 158 | \tab $\ds \dv[2]{U}{q} > 0$ 159 | 160 | Stability Criterion of Equilibrium Point ($s \geq 2$):\\ 161 | \tab $\ds \pdv{U}{q_i}{q_j} = K_{ij} = K_{ji} \geq 0$ 162 | 163 | Small Angle Approximation:\\ 164 | \tab Taylor expand around equilibrium point of the Lagrangian $q_0$ and keep up to first term that contributes to the Lagrangian. 165 | 166 | General Solution to $\ddot{x} = -\omega^2 (x - x_0) + f(t)$:\\ 167 | \tab Let $z(t) = \dot{x}(t) + i\omega x(t) \qquad x(t) = x_0 + \Im{z(t)}/\omega$\\ 168 | \tab $\ds z(t) = e^{i\omega t}\left[z(0) + \int_{0}^{t} e^{-i\omega \tau} f(\tau)\,d\tau\right]$ 169 | 170 | 171 | \section{Rigid Body Motion} 172 | Kinetic Energy:\\ 173 | \tab $T = \frac{M}{2}\abs{\vb{V_0}}^2 + M\vb{\rcurs}_\text{CM} \cdot (\vb{V_0} \cross \vb{\Omega}) + \frac{1}{2}\vb{\Omega}\hat{I_0}\vb{\Omega}$\\ 174 | where $\vb{V_o}$ is the velocity of the chosen reference point $\mathcal{O}$, $\vb{\rcurs}_\text{CM} = \vb{R}_\text{CM} - \vb{\rcurs}_\text{O}$ is the position of the CM with respect to $\mathcal{O}$, and $\vb{\Omega}$ is the angular velocity of the body. 175 | 176 | Moment of Inertia:\\ 177 | \tab $\ds I_{xx}^{(0)} = \int_\mathcal{V} \rho(\vb{r}) (y^2 + z^2)\,dV \qquad I_{xy}^{(0)} = \int_\mathcal{V} \rho(\vb{r}) xy\,dV$ 178 | 179 | Parallel Axis Theorem:\\ 180 | \tab $I_{xx}^{(0)} = I_{xx}^{\text{CM}} + M(d_y^2 + d_z^2)$ 181 | 182 | % Footer content 183 | \rule{0.3\linewidth}{0.25pt} 184 | \scriptsize\\ 185 | Updated \today\\ 186 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 187 | \end{multicols} 188 | \end{document} 189 | -------------------------------------------------------------------------------- /ELEC 404 - RF Integrated Circuits/elec404fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{mdframed} 16 | \usepackage{amsbsy} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \renewcommand\small{\@setfontsize\small{10}{11}} 47 | \makeatother 48 | 49 | % Define BibTeX command 50 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 51 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 52 | 53 | % Don't print section numbers 54 | \setcounter{secnumdepth}{0} 55 | 56 | 57 | \setlength{\parindent}{0pt} 58 | \setlength{\parskip}{1pt plus 0.5ex} 59 | 60 | \newcommand{\tab}{\hspace*{1em}} 61 | \newcommand{\ds}{\displaystyle} 62 | 63 | % Redefine some commands for newtxmath boldness 64 | \renewcommand{\grad}{\nabla} 65 | \renewcommand{\curl}[1]{\nabla\times#1} 66 | \renewcommand{\div}[1]{\nabla\cdot#1} 67 | \renewcommand{\cross}{\times} 68 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 69 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 70 | 71 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 72 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 73 | 74 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 75 | \theoremstyle{definition} 76 | \newmdtheoremenv{theorem}{Theorem} 77 | 78 | % ----------------------------------------------------------------------- 79 | 80 | \begin{document} 81 | 82 | \raggedright 83 | \footnotesize 84 | \begin{multicols}{3} 85 | 86 | \raggedcolumns 87 | 88 | % multicol parameters 89 | % These lengths are set only within the two main columns 90 | %\setlength{\columnseprule}{0.25pt} 91 | \setlength{\premulticols}{1pt} 92 | \setlength{\postmulticols}{1pt} 93 | \setlength{\multicolsep}{1pt} 94 | \setlength{\columnsep}{2pt} 95 | 96 | \begin{center} 97 | \Large{\underline{ELEC 404 Formula Sheet}} 98 | \end{center} 99 | 100 | \section{Basic Concepts in RF Design} 101 | Signal power in dBm:\\ 102 | \tab $\ds P_{\mathit{sig}\rvert\text{dBm}} = 10\log\left(\frac{P_\mathit{sig}}{1 \text{ mW}}\right)$ 103 | 104 | \subsection{Non-Linearity} 105 | Peak 1-dB compression point:\\ 106 | \tab $\ds P_\text{1dB} = \sqrt{0.145\abs{\frac{\alpha_1}{\alpha_3}}}$ 107 | 108 | Input Third Intercept Point:\\ 109 | \tab $\ds A_\mathit{IIP3} = \sqrt{\frac{4}{3}\abs{\frac{\alpha_1}{\alpha_3}}}$\\ 110 | \tab $\ds \frac{A_\mathit{IIP3}}{P_\text{1dB}} \approx 9.6 \text{dB}$ 111 | 112 | Cascaded Non-Linear Stages:\\ 113 | \tab $\ds A_\mathit{IIP3} = \sqrt{\frac{4}{3}\left\vert\frac{\alpha_1\beta_1}{\alpha_3\beta_1 + 2 \alpha_1\alpha_2\beta_2 + \alpha_1^3 \beta_3}\right\vert}$\\ 114 | \tab $\ds \frac{1}{ A_\mathit{IIP3}^2} = \left\vert\frac{1}{ A_{\mathit{IIP3},1}^2} + \frac{3\alpha_2\beta_2}{2\beta_1} + \frac{\alpha_1^2}{A_{\mathit{IIP3},2}^2}\right\vert$\\ 115 | \tab $\ds \frac{1}{A_\mathit{IIP3}^2} \approx \frac{1}{A_{\mathit{IIP3},1}^2} + \frac{\alpha_1^2}{A_{\mathit{IIP3},2}^2} + \frac{\alpha_1^2\beta_1^2}{A_{\mathit{IIP3},3}^2} + \cdots$ 116 | 117 | \subsection{Noise} 118 | Output Spectrum of a linear, time invariant system:\\ 119 | \tab $S_y(f) = S_x(f)\abs{H(f)}^2$ 120 | 121 | Resistor Noise Power Spectral Density:\\ 122 | \tab $\overline{V_n^2} = 4kTR \qquad \overline{I_n^2} = 4kTR$ 123 | 124 | MOSFET Noise Power Spectral Density ($\gamma \approx 2/3$):\\ 125 | \tab $\overline{V_n^2} = 4kT\gamma/g_m \qquad \overline{V_n^2} = 4kT\gamma g_m$ 126 | 127 | MOSFET $1/f$ noise corner frequency for some process constant $K$:\\ 128 | \tab $\ds f_c = \frac{K}{WLC_\mathit{ox}}\frac{g_m}{4kT\gamma}$ 129 | 130 | Noise Figure:\\ 131 | \tab $\ds \text{NF} = \frac{\mathit{SNR}_\mathit{in}}{\mathit{SNR}_\mathit{out}} \qquad \text{NF}|_\text{dB} = 10\log\left(\frac{\mathit{SNR}_\mathit{in}}{\mathit{SNR}_\mathit{out}}\right)$ 132 | 133 | Attenuation Factor:\\ 134 | \tab $\ds \alpha = \frac{Z_\mathit{in}}{Z_\mathit{in} + R_S}$ 135 | 136 | Noise figure as the total noise at the output divided by the noise at the output due to the source impedance ($\overline{V_n^2}$ is the output noise of device):\\ 137 | \tab $\ds \text{NF} = \frac{1}{\overline{V_\mathit{RS}^2}}\cdot\frac{\overline{V_\mathit{RS}^2}\abs{\alpha}^2A_v^2+\overline{V_n^2}}{\abs{\alpha}^2A_v^2} = 1 + \frac{\overline{V_n^2}}{\abs{\alpha}^2A_v^2}\cdot\frac{1}{\overline{V_\mathit{RS}^2}}$ 138 | 139 | Available power gain:\\ 140 | \tab $\ds A_P = \frac{\text{Power delivered to a matched load by stage}}{\text{Power delivered to a matched load by source}}$ 141 | 142 | Noise figure of cascaded stages (Friis' Equation):\\ 143 | \tab $\ds \text{NF}_\mathit{tot} = 1 + (\text{NF}_1 - 1) + \frac{\text{NF}_2 - 1}{A_{P1}} + \dots + \frac{\text{NF}_m - 1}{A_{P1}\cdots A_{P(m-1)}}$ 144 | 145 | \subsection{Passive Impedance Transformation} 146 | Capacitor Quality Factor:\\ 147 | \tab $\ds Q_S = \frac{1}{R_SC\omega} \qquad Q_P = \frac{R_PC\omega}{1}$ 148 | 149 | Inductor Quality Factor:\\ 150 | \tab $\ds Q_S = \frac{L\omega}{R_S} \qquad Q_P = \frac{R_P}{L\omega}$ 151 | 152 | Series to Parallel Conversion $(Q_P = Q_S)$:\\ 153 | \tab $R_P = (Q_S^2 + 1)R_S$ 154 | \tab $\ds C_P = \left(\frac{Q_S^2}{Q_S^2 + 1}\right) C_S \qquad L_P = \left(\frac{Q_S^2 + 1}{Q_S^2}\right) L_S$ 155 | 156 | \subsection{Scattering Parameters} 157 | \includegraphics[width=\linewidth]{scattering_parameters.png} 158 | \tab $V_1^- = S_{11}V_1^+ + S_{12}V_2^+$\\ 159 | \tab $V_2^- = S_{21}V_1^+ + S_{22}V_2^+$ 160 | 161 | Accuracy of input matching:\\ 162 | \tab $\ds S_{11} = \frac{V_1^-}{V_1^+}\rvert_{V_2^+ = 0} \approx \frac{Z_\mathit{in} - R_S}{Z_\mathit{in} + R_S}$ 163 | 164 | Reverse isolation:\\ 165 | \tab $\ds S_{12} = \frac{V_1^-}{V_2^+}\rvert_{V_1^+ = 0}$ 166 | 167 | Accuracy of output matching:\\ 168 | \tab $\ds S_{22} = \frac{V_2^-}{V_2^+}\rvert_{V_1^+ = 0} \approx \frac{Z_\mathit{out} - R_S}{Z_\mathit{out} + R_S}$ 169 | 170 | Circuit gain:\\ 171 | \tab $\ds S_{21} = \frac{V_2^-}{V_2^+}\rvert_{V_2^+ = 0}$ 172 | 173 | Scattering parameters in dB:\\ 174 | \tab $S_{mn\vert\text{dB}} = 20\log\abs{S_{mn}}$ 175 | 176 | 177 | % Footer content 178 | \rule{0.3\linewidth}{0.25pt} 179 | \scriptsize\\ 180 | Updated \today\\ 181 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 182 | \end{multicols}% 183 | 184 | \end{document} 185 | -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250mt1fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{physics} 11 | \usepackage{newtxtext,newtxmath} 12 | 13 | % This sets page margins to .5 inch if using letter paper, and to 1cm 14 | % if using A4 paper. (This probably isn't strictly necessary.) 15 | % If using another size paper, use default 1cm margins. 16 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 17 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 18 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 19 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | } 22 | 23 | % Turn off header and footer 24 | \pagestyle{empty} 25 | 26 | % Redefine section commands to use less space 27 | \makeatletter 28 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 29 | {-1ex plus -.5ex minus -.2ex}% 30 | {0.5ex plus .2ex}%x 31 | {\normalfont\large\bfseries}} 32 | \renewcommand{\subsubsection}{\@startsection{subsection}{2}{0mm}% 33 | {-1explus -.5ex minus -.2ex}% 34 | {0.5ex plus .2ex}% 35 | {\normalfont\normalsize\bfseries}} 36 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 37 | {-1ex plus -.5ex minus -.2ex}% 38 | {1ex plus .2ex}% 39 | {\normalfont\small\bfseries}} 40 | \makeatother 41 | 42 | % Define BibTeX command 43 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 44 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 45 | 46 | % Don't print section numbers 47 | \setcounter{secnumdepth}{0} 48 | 49 | \setlength{\parindent}{0pt} 50 | \setlength{\parskip}{1pt plus 0.5ex} 51 | 52 | \newcommand{\tab}{\hspace{.02\textwidth}} 53 | \newcommand{\ds}{\displaystyle} 54 | 55 | % ----------------------------------------------------------------------- 56 | 57 | \begin{document} 58 | \raggedright 59 | \footnotesize 60 | \begin{multicols}{3} 61 | 62 | 63 | % multicol parameters 64 | % These lengths are set only within the two main columns 65 | %\setlength{\columnseprule}{0.25pt} 66 | \setlength{\premulticols}{1pt} 67 | \setlength{\postmulticols}{1pt} 68 | \setlength{\multicolsep}{1pt} 69 | \setlength{\columnsep}{2pt} 70 | 71 | \begin{center} 72 | \Large{\underline{PHYS 250 MT1 Formula Sheet}} 73 | \end{center} 74 | 75 | \section{Energy \& Waves} 76 | \tab $E_{\text{max}}^2 \approx (\text{amplitude})^2$\\ 77 | $\bullet$ Does not depend on frequency or matter. 78 | 79 | Energy of a photon:\\ 80 | \tab $\ds E = hf = \frac{hc}{\lambda} $\\ 81 | $\bullet$ Interference is evidence that light is a wave. 82 | 83 | \section{Photoelectric Effect} 84 | \begin{itemize}[leftmargin=0.5cm] 85 | \itemsep0em 86 | \item Current is linearly proportional to intensity. 87 | \item Current appears without delay. 88 | \item Photoelectrons are only emitted if the light frequency $f$ exceeds a threshold frequency $f_0$ 89 | \item The value of $f_0$ depends on cathode material. 90 | \item Current becomes independent of $V$ for large $V$. 91 | \item If the voltage is made negative, then the current decreases until some stopping potential. 92 | \item $V_{\text{stop}}$ is independent of light intensity. 93 | \item Electron immediately has enough energy to escape. 94 | \item Number of electrons $\propto$ Intensity 95 | \item Maximum $E_k \propto$ Frequency 96 | \end{itemize} 97 | 98 | Classical Interpretation: 99 | \begin{itemize}[leftmargin=0.5cm] 100 | \itemsep0em 101 | \item Metal is heated to high temperature to allow Thermal Emission 102 | \item Can possibly raise electron temperature to significantly higher than the metal, so that the electron can emit without the metal melting. 103 | \item Nothing to suggest threshold frequency 104 | \item Does not explain instantaneous current 105 | \item Does not explain why $V_{\text{stop}}$ is constant. 106 | \end{itemize} 107 | 108 | Stopping Potential:\\ 109 | \tab $\ds V_{\text{stop}} = \frac{hf - E_0}{\text{e}}$ 110 | 111 | The Photon Rate:\\ 112 | \tab $\ds P = \dv{N}{t} hf$ 113 | 114 | \subsection{Emission and Absorption} 115 | \begin{itemize}[leftmargin=0.5cm] 116 | \itemsep0em 117 | \item Atom jumps from lower energy to higher energy state by absorbing a photon. It can emit a photon of the same frequency as it jumps back. (Spontaneous Transmission) 118 | \item Stimulated Emission: Production of two identical photons by one photon interacting with an excited atom. Only occurs if the first photon's frequency matches the energy difference. 119 | \item A laser uses a chain reaction of stimulated emission in many excited atoms. The number of excited atoms must out number the non-excited atoms to be stable. 120 | \item Population Inversion: Having an amount of atoms N such that the number of excited atoms is proportionally larger than the number of non-excited atoms. 121 | \end{itemize} 122 | 123 | Balmer's Formula ($\lambda$ in hydrogen spectrum):\\ 124 | \tab $\ds \frac{91.18\text{nm}}{\frac{1}{m^2} - \frac{1}{n^2}} \text{ for m = 1,2,3... \& } n > m$ 125 | 126 | \subsection{Wave Function} 127 | We want to relate the probability functions with electrons, but there are no waves for electrons. We assume there is some continuous, wave-like function for matter that is analogous for light. 128 | 129 | Probability Density:\\ 130 | \tab $P(x) = \abs{\psi (x)}^2$ 131 | 132 | Normalization:\\ 133 | \tab $\ds \int_{-\infty}^{\infty} \abs{\psi (x)}^2 dx = 1$ 134 | 135 | \subsection{Quantum Models} 136 | Bohr Model can't explain 137 | \begin{itemize}[leftmargin=0.5cm] 138 | \itemsep0em 139 | \item Why angular momentum is quantized 140 | \item Why electrons don't radiate energy when in orbits 141 | \item How does electron know what orbit to jump to? 142 | \item Can't be generalized 143 | \item Shapes of molecular orbits 144 | \item Molecular bonds 145 | \item Very closely spaced spectral lines 146 | \end{itemize} 147 | 148 | \subsection{Schroedinger Equation} 149 | $$\dv[2]{\psi}{x}+\frac{2m}{\hbar^2}[E - U(x)]\psi (x) = 0$$ 150 | $$\hbar = h/2\pi$$ 151 | de Broglie wavelength:\\ 152 | \tab $ \ds \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m E_k}}$ 153 | 154 | Restrictions 155 | \begin{itemize}[leftmargin=0.5cm] 156 | \itemsep0em 157 | \item $\psi (x)$ is continuous 158 | \item $\psi (x)$ = 0 if $x$ is in a region where te particle is impossible to be in 159 | \item $\psi (x) \rightarrow 0$ as $x \rightarrow \infty$ 160 | \item $\psi (x)$ is a normalized function 161 | \end{itemize} 162 | 163 | \subsection{Potential Wells} 164 | \begin{itemize}[leftmargin=0.5cm] 165 | \itemsep0em 166 | \item A particle with energy $E > U_0$ an escape into the classically forbidden region. 167 | \item Particle's energy is quantized 168 | \item There area finite number of bound states 169 | \item $\psi (x)$ extends into the classically forbidden region 170 | \item Node spacing is smaller when kinetic energy is larger 171 | \item Classical particle is more likely to be found where it is moving slowly 172 | \item Wave function amplitude is larger where the kinetic energy is smaller 173 | \end{itemize} 174 | Wave Function in the classically forbidden region:\\ 175 | \tab $\psi (x) = \psi_{\text{edge}} e^{-(x-L)/\eta}$ 176 | 177 | Penetration distance:\\ 178 | \tab $\ds \eta = \frac{\hbar}{\sqrt{2m(U_0 - E)}}$ 179 | 180 | \begin{itemize}[leftmargin=0.5cm] 181 | \itemsep0em 182 | \item Quantum tunneling requires no energy 183 | \item Tunneling requires oscillatory solutions on the other side 184 | \end{itemize} 185 | 186 | Tunneling Probability:\\ 187 | \tab $P_{\text{tunnel}} = e^{-2w/\eta}$ for some edge $x=w$ 188 | 189 | % Footer content 190 | \rule{0.3\linewidth}{0.25pt} 191 | \scriptsize\\ 192 | Updated \today\\ 193 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 194 | \end{multicols} 195 | \end{document} -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250mt2fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{physics} 11 | \usepackage{newtxtext,newtxmath} 12 | 13 | % This sets page margins to .5 inch if using letter paper, and to 1cm 14 | % if using A4 paper. (This probably isn't strictly necessary.) 15 | % If using another size paper, use default 1cm margins. 16 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 17 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 18 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 19 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | } 22 | 23 | % Turn off header and footer 24 | \pagestyle{empty} 25 | 26 | % Redefine section commands to use less space 27 | \makeatletter 28 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 29 | {-1ex plus -.5ex minus -.2ex}% 30 | {0.5ex plus .2ex}%x 31 | {\normalfont\large\bfseries}} 32 | \renewcommand{\subsubsection}{\@startsection{subsection}{2}{0mm}% 33 | {-1explus -.5ex minus -.2ex}% 34 | {0.5ex plus .2ex}% 35 | {\normalfont\normalsize\bfseries}} 36 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 37 | {-1ex plus -.5ex minus -.2ex}% 38 | {1ex plus .2ex}% 39 | {\normalfont\small\bfseries}} 40 | \makeatother 41 | 42 | % Define BibTeX command 43 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 44 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 45 | 46 | % Don't print section numbers 47 | \setcounter{secnumdepth}{0} 48 | 49 | \setlength{\parindent}{0pt} 50 | \setlength{\parskip}{1pt plus 0.5ex} 51 | 52 | \newcommand{\tab}{\hspace{.02\textwidth}} 53 | \newcommand{\ds}{\displaystyle} 54 | 55 | % ----------------------------------------------------------------------- 56 | 57 | \begin{document} 58 | \raggedright 59 | \footnotesize 60 | \begin{multicols}{3} 61 | % multicol parameters 62 | % These lengths are set only within the two main columns 63 | %\setlength{\columnseprule}{0.25pt} 64 | \setlength{\premulticols}{1pt} 65 | \setlength{\postmulticols}{1pt} 66 | \setlength{\multicolsep}{1pt} 67 | \setlength{\columnsep}{2pt} 68 | 69 | \begin{center} 70 | \Large{\underline{PHYS 250 MT2 Cheat Sheet}} 71 | \end{center} 72 | 73 | \section{Energy \& Waves} 74 | \tab $E_{\text{max}}^2 \approx (\text{amplitude})^2$\\ 75 | $\bullet$ Does not depend on frequency or matter. 76 | 77 | Energy of a photon:\\ 78 | \tab $\ds E = hf = \frac{hc}{\lambda} $\\ 79 | $\bullet$ Interference is evidence that light is a wave. 80 | 81 | \section{Emission and Absorption} 82 | \begin{itemize}[leftmargin=0.5cm] 83 | \itemsep0em 84 | \item Atom jumps from lower energy to higher energy state by absorbing a photon. It can emit a photon of the same frequency as it jumps back. (Spontaneous Transmission) 85 | \item Stimulated Emission: Production of two identical photons by one photon interacting with an excited atom. Only occurs if the first photon's frequency matches the energy difference. 86 | \item A laser uses a chain reaction of stimulated emission in many excited atoms. The number of excited atoms must out number the non-excited atoms to be stable. 87 | \item Population Inversion: Having an amount of atoms N such that the number of excited atoms is proportionally larger than the number of non-excited atoms. 88 | \end{itemize} 89 | 90 | Balmer's Formula ($\lambda$ in hydrogen spectrum):\\ 91 | \tab $\ds \frac{91.18\text{nm}}{\frac{1}{m^2} - \frac{1}{n^2}} \text{ for m = 1,2,3... \& } n > m$ 92 | 93 | \section{Bohr Model} 94 | \begin{itemize}[leftmargin=0.5cm] 95 | \itemsep0em 96 | \item Electrons can exist only in certain orbits. A particular arrangement of electrons is a stationary state. 97 | \item Each stationary state has a discrete energy. 98 | \end{itemize} 99 | 100 | Hydrogen radius:\\ 101 | \tab $r_n = n^2 a_B$ 102 | 103 | Hydrogen Energy:\\ 104 | \tab $E_n = -13.60 \text{ eV} / n^2$ 105 | 106 | Bohr Model can't explain 107 | \begin{itemize}[leftmargin=0.5cm] 108 | \itemsep0em 109 | \item Why angular momentum is quantized 110 | \item Why electrons don't radiate energy when in orbits 111 | \item How does electron know what orbit to jump to? 112 | \item Can't be generalized 113 | \item Shapes of molecular orbits 114 | \item Molecular bonds 115 | \item Very closely spaced spectral lines 116 | \end{itemize} 117 | 118 | \section{Schroedinger Equation} 119 | $$\dv[2]{\psi}{x}+\frac{2m}{\hbar^2}[E - U(x)]\psi (x) = 0$$ 120 | $$\hbar = h/2\pi$$ 121 | de Broglie wavelength:\\ 122 | \tab $ \ds \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m E_k}}$ 123 | 124 | Restrictions 125 | \begin{itemize}[leftmargin=0.5cm] 126 | \itemsep0em 127 | \item $\psi (x)$ is continuous 128 | \item $\psi (x)$ = 0 if $x$ is in a region where te particle is impossible to be in 129 | \item $\psi (x) \rightarrow 0$ as $x \rightarrow \infty$ 130 | \item $\psi (x)$ is a normalized function 131 | \end{itemize} 132 | 133 | \section{Potential Wells} 134 | \begin{itemize}[leftmargin=0.5cm] 135 | \itemsep0em 136 | \item A particle with energy $E > U_0$ an escape into the classically forbidden region. 137 | \item Particle's energy is quantized 138 | \item There area finite number of bound states 139 | \item $\psi (x)$ extends into the classically forbidden region 140 | \item Node spacing is smaller when kinetic energy is larger 141 | \item Classical particle is more likely to be found where it is moving slowly 142 | \item Wave function amplitude is larger where the kinetic energy is smaller 143 | \end{itemize} 144 | Wave Function in the classically forbidden region:\\ 145 | \tab $\psi (x) = \psi_{\text{edge}} e^{-(x-L)/\eta}$ 146 | 147 | Penetration distance:\\ 148 | \tab $\ds \eta = \frac{\hbar}{\sqrt{2m(U_0 - E)}}$ 149 | 150 | \begin{itemize}[leftmargin=0.5cm] 151 | \itemsep0em 152 | \item Quantum tunneling requires no energy 153 | \item Tunneling requires oscillatory solutions on the other side 154 | \item $U_0 - E$ can be the metal's work function 155 | \end{itemize} 156 | 157 | Infinite well energy:\\ 158 | \tab $\ds E_n =\frac{n^2 \pi^2 \hbar^2}{2mL^2}$ 159 | 160 | Tunneling Probability:\\ 161 | \tab $P_{\text{tunnel}} = e^{-2w/\eta}$ for potential well width of $w$ 162 | 163 | \section{Wave Packets} 164 | \begin{itemize}[leftmargin=0.5cm] 165 | \itemsep0em 166 | \item A localized particle 167 | \item Travels with constant speed 168 | \item For any wave packet $\Delta f \Delta t \geq 1$ 169 | \end{itemize} 170 | 171 | Uncertainty:\\ 172 | \tab $\ds \Delta x = v_x \Delta t = \frac{p_x}{m}\Delta t$ 173 | 174 | Uncertainty Principle:\\ 175 | \tab $\Delta x\, \Delta p_x \geq h / 2$ 176 | 177 | \section{Measurement} 178 | \begin{itemize}[leftmargin=0.5cm] 179 | \itemsep0em 180 | \item Measuring changes the system 181 | \item Measuring collapses wavefunciton to a specific eigenstate 182 | \item Cannot know both position and energy. 183 | \item Measuring position $\rightarrow$ Probability density changes with time 184 | \item Measuring energy $\rightarrow$ Probability density does not change 185 | \end{itemize} 186 | 187 | \section{Hydrogen Atom} 188 | Bohr Radius:\\ 189 | \tab $\ds a_B = \frac{4 \pi \varepsilon_0 \hbar^2}{me^2}$ 190 | 191 | Energy:\\ 192 | \tab $E_n = -13.60 \text{ eV} / n^2$, $n = 1, 2, 3...$ 193 | 194 | Momentum:\\ 195 | \tab $L = \sqrt{l(l+1)}\,\hbar$, $l = 0,1,2...n-1$\\ 196 | \tab $L_z = m\hbar$, $m = -l, -l+1,...0,...l-1,l$ 197 | 198 | Symbols for $l$:\\ 199 | \tab 200 | $0 \rightarrow s$, 201 | $1 \rightarrow p$, 202 | $2 \rightarrow d$, 203 | $3 \rightarrow f$ 204 | 205 | % Footer content 206 | \rule{0.3\linewidth}{0.25pt} 207 | \scriptsize\\ 208 | Updated \today\\ 209 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 210 | \end{multicols} 211 | \end{document} -------------------------------------------------------------------------------- /APSC 278 - Engineering Materials/apsc278fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{mdframed} 16 | \usepackage{amsbsy} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \makeatother 47 | 48 | % Define BibTeX command 49 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 50 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 51 | 52 | % Don't print section numbers 53 | \setcounter{secnumdepth}{0} 54 | 55 | 56 | \setlength{\parindent}{0pt} 57 | \setlength{\parskip}{1pt plus 0.5ex} 58 | 59 | \newcommand{\tab}{\hspace*{1em}} 60 | \newcommand{\ds}{\displaystyle} 61 | 62 | % Redefine some commands for newtxmath boldness 63 | \renewcommand{\grad}{\nabla} 64 | \renewcommand{\curl}[1]{\nabla\times#1} 65 | \renewcommand{\div}[1]{\nabla\cdot#1} 66 | \renewcommand{\cross}{\times} 67 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 68 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 69 | 70 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 71 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 72 | 73 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 74 | \theoremstyle{definition} 75 | \newmdtheoremenv{theorem}{Theorem} 76 | 77 | % ----------------------------------------------------------------------- 78 | 79 | \begin{document} 80 | 81 | \raggedright 82 | \footnotesize 83 | \begin{multicols*}{3} 84 | 85 | \raggedcolumns 86 | 87 | % multicol parameters 88 | % These lengths are set only within the two main columns 89 | %\setlength{\columnseprule}{0.25pt} 90 | \setlength{\premulticols}{1pt} 91 | \setlength{\postmulticols}{1pt} 92 | \setlength{\multicolsep}{1pt} 93 | \setlength{\columnsep}{2pt} 94 | 95 | \begin{center} 96 | \Large{\underline{APSC 278 Formula Sheet}} 97 | \end{center} 98 | 99 | \section{Bonding and Properties} 100 | Equilibrium at $r_0$:\\ 101 | \tab $F_\text{net} = \dv{E}{r} = 0$ 102 | 103 | Thermal Expansion:\\ 104 | \tab $\Delta L = L_0 \alpha \Delta T$ 105 | 106 | \section{Mechanical Properties} 107 | Hooke's Law:\\ 108 | \tab $\sigma = E \epsilon$ 109 | 110 | Poisson's Ratio:\\ 111 | \tab $\ds \nu = - \frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\epsilon_z}$ 112 | 113 | Shear Modulus:\\ 114 | \tab $\ds G = \frac{\tau}{\gamma} = \frac{F}{A_0 \tan\theta}$ 115 | 116 | \tab $E = 2G(1+\nu)$ 117 | 118 | Toughness:\\ 119 | \tab $\ds \text{Toughness} = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon}\approx \frac{\sigma_\text{YS} + \sigma_\text{UTS}}{2}\epsilon_f $ 120 | 121 | Modulus of Resilience:\\ 122 | \tab $\ds U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\sigma_\text{YS}^2}{2E}$ 123 | 124 | Work Hardening:\\ 125 | \tab $\sigma_T = K \epsilon_T^n$ 126 | 127 | True Stress and Strain:\\ 128 | \tab $\ds \epsilon_\text{true} = \int_{L_0}^{L_i}\frac{dL}{L} = \ln(\epsilon_\text{Eng} + 1)$ 129 | 130 | \tab $\sigma_\text{true} = \sigma_\text{eng}(\epsilon_\text{eng} + 1)$ 131 | 132 | \section{Crystal Structures} 133 | \bgroup 134 | \def\arraystretch{1.2}% 135 | \tab \begin{tabular}{|l|c | c | c |} 136 | \hline 137 | & \begin{tabular}[x]{@{}c@{}}Lattice\\Parameter ($a$)\end{tabular} & \begin{tabular}[x]{@{}c@{}}Atoms per\\unit cell\end{tabular} & APF\\ 138 | \hline 139 | BCC & $\frac{4\sqrt{3}}{3} R$ & 2 & 0.68\\ 140 | \hline 141 | FCC & $2\sqrt{2}R$ & 4 & 0.74\\ 142 | \hline 143 | HCP & - & 6 & 0.74\\ 144 | \hline 145 | \end{tabular} 146 | \egroup 147 | 148 | Density (g/cm$^3$):\\ 149 | \tab $\ds \rho_\text{Th} = \frac{NMW_i}{V_c N_A}$\\ 150 | \vspace{2mm} 151 | \tab $n$ = number of atoms per unit cell\\ 152 | \tab $MW_i$ = Atomic weight (g/mol)\\ 153 | \tab $V_c$ = unit cell volume\\ 154 | \tab $N_A$ = Avogadro's Number ($6.023 \cdot 10^{23}$ atoms/mol) 155 | 156 | \section{Effect of Temperature on Deformation} 157 | Homologous Temperature:\\ 158 | \tab $\ds T_H = \frac{T_\text{deformation}}{T_\text{melt}}$ 159 | 160 | Homologous Temperature for pure metals:\\ 161 | \tab $\begin{cases} 162 | \text{Cold working} & T_H < 0.4\\ 163 | \text{Hot working} & T_H > 0.4 164 | \end{cases}$ 165 | 166 | Homologous Temperature for alloys:\\ 167 | \tab $\begin{cases} 168 | \text{Cold working} & T_H < 0.6\\ 169 | \text{Hot working} & T_H > 0.6 170 | \end{cases}$ 171 | 172 | Cold Working (rolling):\\ 173 | \tab \% cold work = $\ds \frac{t_0 - t_f}{t_0} \times 100\%$ 174 | 175 | Recrystallization rate:\\ 176 | \tab rate $\ds = A\exp(-\frac{Q}{RT})$\\ 177 | \tab $t_\text{recrx} = 1/\text{rate}$\\ 178 | \tab $Q = $ thermal activation energy 179 | 180 | Grain Growth:\\ 181 | \tab $D(t) = D_0t^m$ 182 | 183 | Hall-Petch Equation for grain size $d$:\\ 184 | \tab $\sigma_\text{YS} = \sigma_0 + k_y d^{-1/2}$ 185 | 186 | Creep rate:\\ 187 | \tab $\ds \dot{\epsilon}_{c,ss} = K_2\sigma^n\exp(-\frac{Q_c}{RT})$ 188 | 189 | Larson Miller parameter ($C \approx 20$):\\ 190 | \tab $m = T(C + \log_{10}(t_r))$ 191 | 192 | \section{Fracture} 193 | 194 | Stress Concentration Factor for an elliptical notch:\\ 195 | \tab $\ds \sigma_m = \sigma_\text{net} \left(1 + 2 \sqrt{\frac{a}{r_t}}\right) = \sigma_\text{net}K_t$ 196 | 197 | Griffith Equation for surface energy $\gamma_s$:\\ 198 | \tab $\ds \sigma_\text{critical} = \sqrt{\frac{2E\gamma_s}{\pi a_c}}$ 199 | 200 | Griffith Equation extension to ductile materials:\\ 201 | \tab $\ds \sigma_\text{critical} = \sqrt{\frac{2EG_c}{\pi a_c}}$\\ 202 | \tab $G_c$ = critical strain energy release rate 203 | 204 | Stress Intensity Factor:\\ 205 | \tab $K = Y\sigma\sqrt{\pi a_c}$ 206 | 207 | Paris Equation (crack growth in steady state creep):\\ 208 | \tab $\ds \dv{a}{N} = C(\Delta K)^N$ 209 | 210 | \section{Composites} 211 | Isostrain:\\ 212 | \tab $\epsilon_c = \epsilon_m = \epsilon_f$\\ 213 | \tab $F_c = F_f + F_m$\\ 214 | \tab $E_c = E_ff_f + E_mf_m$\\ 215 | \tab $\ds \frac{F_f}{F_c} = \frac{f_f}{f_f + \frac{E_m}{E_f}f_m}$ 216 | 217 | Isostress:\\ 218 | \tab $\sigma_c = \sigma_m = \sigma_f$\\ 219 | \tab $\epsilon_c = \epsilon_m f_m + \epsilon_f f_f$\\ 220 | \tab $\ds E_c = \frac{E_m E_f}{E_mf_f + E_f f_m}$\\ 221 | 222 | \section{Electrical Properties} 223 | Resistance in a straight conductor:\\ 224 | \tab $\ds R = \frac{\rho l}{A}$ 225 | 226 | Transmission Line Sag:\\ 227 | \tab $\ds \delta \approx \frac{wl^2}{8H}$\\ 228 | \tab $L = l + \frac{8\delta^2}{3l}$\\ 229 | \tab $L_\text{total} = L(1 + \epsilon_\sigma + \epsilon_T)$ 230 | 231 | Conductivity in metals and semiconductors:\\ 232 | \tab $\sigma = n\abs{e}\mu_e + p\abs{e}\mu_p$\\ 233 | \tab For metals, $p = 0$. For intrinsic semiconductors, $n = p$. 234 | 235 | % Footer content 236 | \rule{0.3\linewidth}{0.25pt} 237 | \scriptsize\\ 238 | Updated \today\\ 239 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 240 | \end{multicols*}% 241 | 242 | \end{document} 243 | -------------------------------------------------------------------------------- /PHYS 250 - Introduction to Modern Physics/phys250finalfs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{physics} 11 | \usepackage{newtxtext,newtxmath} 12 | 13 | % This sets page margins to .5 inch if using letter paper, and to 1cm 14 | % if using A4 paper. (This probably isn't strictly necessary.) 15 | % If using another size paper, use default 1cm margins. 16 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 17 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 18 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 19 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | } 22 | 23 | % Turn off header and footer 24 | \pagestyle{empty} 25 | 26 | % Redefine section commands to use less space 27 | \makeatletter 28 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 29 | {-1ex plus -.5ex minus -.2ex}% 30 | {0.5ex plus .2ex}%x 31 | {\normalfont\large\bfseries}} 32 | \renewcommand{\subsubsection}{\@startsection{subsection}{2}{0mm}% 33 | {-1explus -.5ex minus -.2ex}% 34 | {0.5ex plus .2ex}% 35 | {\normalfont\normalsize\bfseries}} 36 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 37 | {-1ex plus -.5ex minus -.2ex}% 38 | {1ex plus .2ex}% 39 | {\normalfont\small\bfseries}} 40 | \makeatother 41 | 42 | % Define BibTeX command 43 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 44 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 45 | 46 | % Don't print section numbers 47 | \setcounter{secnumdepth}{0} 48 | 49 | \setlength{\parindent}{0pt} 50 | \setlength{\parskip}{1pt plus 0.5ex} 51 | 52 | \newcommand{\tab}{\hspace{.02\textwidth}} 53 | \newcommand{\ds}{\displaystyle} 54 | 55 | % ----------------------------------------------------------------------- 56 | 57 | \begin{document} 58 | \raggedright 59 | \footnotesize 60 | \begin{multicols}{3} 61 | 62 | 63 | % multicol parameters 64 | % These lengths are set only within the two main columns 65 | %\setlength{\columnseprule}{0.25pt} 66 | \setlength{\premulticols}{1pt} 67 | \setlength{\postmulticols}{1pt} 68 | \setlength{\multicolsep}{1pt} 69 | \setlength{\columnsep}{2pt} 70 | 71 | \begin{center} 72 | \Large{\underline{PHYS 250 Final Formula Sheet}} 73 | \end{center} 74 | 75 | \section{Energy \& Waves} 76 | \tab $E_{\text{max}}^2 \approx (\text{amplitude})^2$\\ 77 | $\bullet$ Does not depend on frequency or matter. 78 | 79 | Energy of a photon:\\ 80 | \tab $\ds E = hf = \frac{hc}{\lambda} $\\ 81 | $\bullet$ Interference is evidence that light is a wave. 82 | 83 | \section{Photoelectric Effect} 84 | \begin{itemize}[leftmargin=0.5cm] 85 | \itemsep0em 86 | \item The value of $f_0$ depends on cathode material. 87 | \item $V_{\text{stop}}$ is independent of light intensity. 88 | \item Number of electrons $\propto$ Intensity 89 | \item Maximum $E_k \propto$ Frequency 90 | \end{itemize} 91 | Stopping Potential: \tab $\ds V_{\text{stop}} = \frac{hf - E_0}{e}$\\ 92 | The Photon Rate: \tab $\ds P = \dv{N}{t} hf$ 93 | 94 | \section{Emission and Absorption} 95 | \begin{itemize}[leftmargin=0.5cm] 96 | \itemsep0em 97 | \item Atom transitions to higher energy state by absorbing a photon. Emits a photon of the same frequency if it jumps back. 98 | \item Stimulated Emission: Production of two identical photons by one photon interacting with an excited atom. Only occurs if the first photon's frequency matches the energy difference. 99 | \item Population Inversion: Having proportionally larger excited atoms than the number of non-excited atoms. 100 | \end{itemize} 101 | 102 | Balmer's Formula ($\lambda$ in hydrogen spectrum):\\ 103 | \tab $\ds \frac{91.18\text{nm}}{\frac{1}{m^2} - \frac{1}{n^2}} \text{ for m = 1,2,3... \& } n > m$ 104 | 105 | \section{Bohr Model} 106 | \begin{itemize}[leftmargin=0.5cm] 107 | \itemsep0em 108 | \item Electrons exist only in certain orbits. A particular arrangement of electrons is a stationary state. 109 | \item Each stationary state has a discrete energy. 110 | \end{itemize} 111 | 112 | Hydrogen radius: \tab $r_n = n^2 a_B$ 113 | 114 | Hydrogen Energy: \tab $E_n = -13.60 \text{ eV} / n^2$ 115 | 116 | \section{Schroedinger Equation} 117 | $$\dv[2]{\psi}{x}+\frac{2m}{\hbar^2}[E - U(x)]\psi (x) = 0$$ 118 | $$\hbar = h/2\pi$$ 119 | de Broglie wavelength:\\ 120 | \tab $ \ds \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m E_k}}$ 121 | 122 | \section{Potential Wells} 123 | \begin{itemize}[leftmargin=0.5cm] 124 | \itemsep0em 125 | \item A particle with energy $E > U_0$ an escape into the classically forbidden region. 126 | \item Node spacing is smaller when $E_K$ is larger 127 | \item Classical particle is more likely to be found where it is moving slowly 128 | \item $\psi (x)$ amplitude is larger where $E_K$ is smaller 129 | \end{itemize} 130 | Wave Function in the classically forbidden region:\\ 131 | \tab $\psi (x) = \psi_{\text{edge}} e^{-(x-L)/\eta}$ 132 | 133 | Penetration distance:\\ 134 | \tab $\ds \eta = \frac{\hbar}{\sqrt{2m(U_0 - E)}}$ 135 | \begin{itemize}[leftmargin=0.5cm] 136 | \itemsep0em 137 | \item Quantum tunneling requires no energy and has oscillatory solutions on the other side 138 | \item $U_0 - E$ can be the metal's work function 139 | \end{itemize} 140 | 141 | Infinite well energy:\\ 142 | \tab $\ds E_n =\frac{n^2 \pi^2 \hbar^2}{2mL^2}$ 143 | 144 | Tunneling Probability:\\ 145 | \tab $P_{\text{tunnel}} = e^{-2w/\eta}$ for potential well width of $w$ 146 | 147 | 148 | \section{Measurement} 149 | \begin{itemize}[leftmargin=0.5cm] 150 | \itemsep0em 151 | \item Measuring collapses wave function to a specific eigenstate 152 | \item Cannot know both position and energy. 153 | \item Measuring position $\rightarrow$ $\abs{\psi(x)}^2$ changes in time 154 | \item Measuring energy $\rightarrow$ $\abs{\psi(x)}^2$ no change in time 155 | \end{itemize} 156 | 157 | \section{Wave Packets} 158 | \begin{itemize}[leftmargin=0.5cm] 159 | \itemsep0em 160 | \item A localized particle with constant speed 161 | \item For any wave packet $\Delta f \Delta t \geq 1$ 162 | \end{itemize} 163 | Uncertainty: \tab $\Delta x = v_x \Delta t = \frac{p_x}{m}\Delta t$\\ 164 | Uncertainty Principle: \tab $\Delta x \Delta p_x \geq h / 2$ 165 | \section{Hydrogen Atom} 166 | Bohr Radius: \tab $\ds a_B = \frac{4 \pi \varepsilon_0 \hbar^2}{me^2}$ 167 | 168 | Energy:\\ 169 | \tab $E_n = -13.60 eV / n^2$, $n = 1, 2, 3...$ 170 | 171 | Momentum:\\ 172 | \tab $L = \sqrt{l(l+1)}\hbar$, $l = 0,1,2...n-1$\\ 173 | \tab $L_z = m\hbar$, $m = -l, -l+1,...0,...l-1,l$ 174 | 175 | Symbols for $l$:\\ 176 | \tab 177 | $0 \rightarrow s$, 178 | $1 \rightarrow p$, 179 | $2 \rightarrow d$, 180 | $3 \rightarrow f$ 181 | 182 | Radial probability: \tab $P_r(r) = 4\pi r^2 \abs{R_{nl}(r)}$ 183 | 184 | Spin: \tab $S_z = m_s \hbar$, $m_s = \pm 1/2$ 185 | 186 | Spin Angular Momentum: \tab $S=\sqrt{3}/2\hbar$ 187 | 188 | Pauli Exclusion Principle: No two electrons can have the same set of quantum numbers. If one electron is present in a state, it excludes all others. 189 | 190 | High l $\rightarrow$ circular orbit 191 | 192 | \section{Special Relativity} 193 | \begin{itemize}[leftmargin=0.5cm] 194 | \itemsep0em 195 | \item Laws of physics are the same in all inertial frames 196 | \item Any two events occurring simultaneously in one reference frame are not simultaneous in any reference frame moving relative to the original. 197 | \item Proper time: The time interval between two events occurring in the same position. 198 | \end{itemize} 199 | \tab $\ds \gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}$ \hspace{0.5cm} $\ds \gamma_p = \frac{1}{\sqrt{1-(\frac{u}{c})^2}}$ 200 | 201 | Time Dilation: 202 | \tab $\Delta t = \gamma \Delta \tau$ 203 | 204 | Length Contraction: 205 | \tab $L' = \frac{L}{\gamma}$ 206 | 207 | Spacetime Interval: 208 | \tab $s^2 = c^2(\Delta t)^2 - (\Delta x)^2$ 209 | 210 | Relativistic Momentum: 211 | \tab $p = \gamma_p m u$ 212 | 213 | Relativistic Energy:\\ 214 | \tab $E = \gamma_p mc^2 = E_0 + K = mc^2 + (\gamma_p - 1)mc^2$\\ 215 | \tab $pc = \frac{u}{c}E$ 216 | 217 | % Footer content 218 | \rule{0.3\linewidth}{0.25pt} 219 | \scriptsize\\ 220 | Updated \today\\ 221 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 222 | \end{multicols} 223 | \end{document} -------------------------------------------------------------------------------- /MECH 360 - Mechanics of Materials/mech360fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{gensymb} 12 | \usepackage{newtxtext,newtxmath} 13 | 14 | % This sets page margins to .5 inch if using letter paper, and to 1cm 15 | % if using A4 paper. (This probably isn't strictly necessary.) 16 | % If using another size paper, use default 1cm margins. 17 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 18 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 19 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | } 23 | 24 | % Turn off header and footer 25 | \pagestyle{empty} 26 | 27 | 28 | % Redefine section commands to use less space 29 | \makeatletter 30 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 31 | {-1ex plus -.5ex minus -.2ex}% 32 | {0.5ex plus .2ex}%x 33 | {\normalfont\large\bfseries}} 34 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 35 | {-1explus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}% 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 39 | {-1ex plus -.5ex minus -.2ex}% 40 | {1ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \makeatother 43 | 44 | % Define BibTeX command 45 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 46 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 47 | 48 | % Don't print section numbers 49 | \setcounter{secnumdepth}{0} 50 | 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | 58 | 59 | % ----------------------------------------------------------------------- 60 | 61 | \begin{document} 62 | 63 | \raggedright 64 | \footnotesize 65 | \begin{multicols}{3} 66 | 67 | 68 | % multicol parameters 69 | % These lengths are set only within the two main columns 70 | %\setlength{\columnseprule}{0.25pt} 71 | \setlength{\premulticols}{1pt} 72 | \setlength{\postmulticols}{1pt} 73 | \setlength{\multicolsep}{1pt} 74 | \setlength{\columnsep}{2pt} 75 | 76 | \begin{center} 77 | \Large{\underline{MECH 360 Formula Sheet}} 78 | \end{center} 79 | 80 | \section{Stress \& Strain} 81 | Average normal stress:\\ 82 | \tab $\ds \sigma = \frac{P}{A}$ 83 | 84 | Average shear stress:\\ 85 | \tab $\ds \tau = \frac{V}{A}$ 86 | 87 | Double shear:\\ 88 | \tab $\ds \tau = \frac{P}{2A}$ 89 | 90 | Bearing stress:\\ 91 | \tab $\ds \sigma_b = \frac{P}{A} = \frac{P}{td}$ 92 | 93 | Stresses on a 2-force member\\($\theta$ measured from vertical): \\ 94 | \tab $\ds \sigma = \frac{P}{A_\perp}\cos^2\theta \hspace{1cm} \tau = \frac{P}{A_\perp}\sin\theta\cos\theta$ 95 | 96 | Factor of safety:\\ 97 | \tab $\ds \text{Factor of safety} = \frac{\text{Ultimate Load}}{\text{Allowable Load}}$ 98 | 99 | Normal strain:\\ 100 | \tab $\ds \epsilon = \frac{\delta}{L} = \frac{d\delta}{dx}$ 101 | 102 | Local shear strain (Change of $\pi/2$):\\ 103 | \tab $\gamma = \pi/2 - \theta$ 104 | 105 | \section{Axial Load} 106 | Hooke's Law and Modulus of Elasticity:\\ 107 | \tab $\sigma = E\epsilon$ 108 | 109 | Elastic deformation under axial loading:\\ 110 | \tab $\ds \delta = \frac{FL}{AE} = \sum_{i}\frac{F_i L_i}{A_i E_i}$ 111 | 112 | Temperature change:\\ 113 | \tab $\delta_T = L_o\alpha\Delta T \hspace{1cm}$ 114 | 115 | Poisson's Ration:\\ 116 | \vspace{1mm} 117 | \tab $\ds \nu = -\frac{\epsilon_{\text{lat}}}{\epsilon_\text{long}}$ 118 | 119 | Shear Stress-Strain Diagrams:\\ 120 | \tab $\ds G = \frac{E}{2(1 + \nu)}$\\ 121 | \tab $\tau = G\gamma \hspace{1cm} \text{(elastic region)}$ 122 | 123 | Elastic Strain Energy:\\ 124 | \tab $\ds u = \int_{0}^{\sigma} \sigma\,d\epsilon = \frac{1}{2}\frac{\sigma^2}{E}$ 125 | 126 | \section{Torsion} 127 | Polar Moment of Inertia:\\ 128 | \tab $J = \int r^2\,dA$\\ 129 | \tab $\ds J = \frac{\pi c^4}{2}$ \hspace{1cm} (full tube)\\ 130 | \vspace{1mm} 131 | \tab $\ds J = \frac{\pi}{2}(c^4-a^4)$ \hspace{1cm} (hollow tube) 132 | 133 | Shear Stress:\\ 134 | \tab $\ds \tau = \frac{T\rho}{J} \hspace{1cm } \tau_{\text{max}} = \frac{Tc}{J}$ 135 | 136 | Power:\\ 137 | \tab $P = T\omega$ 138 | 139 | Angle of Twist:\\ 140 | \tab $\ds \phi = \frac{TL}{JG} = \int_{0}^{L}\frac{T(x)}{J(x)G(x)}\,dx$ 141 | 142 | Stress Concentrations:\\ 143 | \tab $\ds \tau_{\text{max}} = K\frac{Tc}{J}$ 144 | 145 | \section{Bending} 146 | Distributed Load Intensity at each point:\\ 147 | \tab $\ds w = \frac{dV}{dx}$ 148 | 149 | Shear at each point:\\ 150 | \tab $\ds V = \frac{dM}{dx}$ 151 | 152 | Normal Strain:\\ 153 | \vspace{1mm} 154 | \tab $\ds \epsilon_{x} = -\frac{y}{p} = -\frac{y}{c}\epsilon_{\text{max}}$ 155 | 156 | Normal Stress:\\ 157 | \vspace{1mm} 158 | \tab $\ds \sigma = -\frac{y}{c}\sigma_{\text{max}}$\\ 159 | \vspace{1mm} 160 | \tab $\ds \sigma = \frac{My}{I}$ 161 | \tab $\ds \sigma_{\text{max}} = \frac{Mc}{I}$ 162 | 163 | Second Moment of Inertia:\\ 164 | \tab $I = \int y^2\,dA$\\ 165 | \tab Circle: $\ds I = \frac{\pi}{4}r^4$\\ 166 | \tab Rectangle: $\ds I = \frac{1}{12}bh^3$ 167 | 168 | Neutral Axis:\\ 169 | \tab $\int y\,dA = 0$ 170 | \tab $\overline{\text{Y}}=\frac{\sum_i^n\overline{y}A}{\sum_i^nA}$ 171 | 172 | Section Modulus:\\ 173 | \tab $S = I/c$ 174 | 175 | Parallel Axis Theorem:\\ 176 | \tab $I_{\parallel} = I_G + Md^2$ 177 | 178 | Composite Beams:\\ 179 | \tab $n = \dfrac{E_2}{E_1}$ for $E_2 > E_1$\\ 180 | \vspace{1mm} 181 | \tab $\sigma_2 = n \sigma_1$ 182 | 183 | Product of Inertia:\\ 184 | \tab $I_{xy} = \int xy\,dA$\\ 185 | \tab $I_{x'} = I_x \cos^2\theta + I_y\sin^2\theta - I_{xy}\sin\theta\cos\theta$\\ 186 | \tab $I_{y'} = I_x \sin^2\theta + I_y\cos^2\theta - I_{xy}\sin\theta\cos\theta$\\ 187 | \tab $I_{x'y'} = I_x\sin\theta\cos\theta-I_y\sin\theta\cos\theta+I_{xy}(\cos^2\theta-\sin^2\theta)$ 188 | 189 | Asymmetric Bending:\\ 190 | \tab $\ds \sigma_x = -\frac{M_z y}{I_z}+\frac{M_y z}{I_y}$\\ 191 | \tab $\ds \tan \phi = \frac{y}{z} = \frac{I_z}{I_y}\tan\theta$ 192 | 193 | \section{Stress Transformations} 194 | \tab $\ds \sigma_x' = \frac{\sigma_x + \sigma_y}{2} + \frac{\sigma_x - \sigma_y}{2}\cos2\theta + \tau_{xy}\sin2\theta$ 195 | \\[2mm] 196 | \tab $\ds \sigma_y' = \frac{\sigma_x + \sigma_y}{2} - \frac{\sigma_x - \sigma_y}{2}\cos2\theta - \tau_{xy}\sin2\theta$ 197 | \\[2mm] 198 | \tab $\ds \tau_{x'y'} = -\frac{\sigma_x - \sigma_y}{2}\sin2\theta + \tau_{xy}\cos2\theta$ 199 | 200 | Principal Stresses:\\ 201 | \tab $\ds \sigma_{1,2} = \frac{\sigma_x + \sigma _y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ 202 | 203 | Maximum In-Plane Shear Stress:\\ 204 | \vspace{1mm} 205 | \tab $\ds \tau_{\text{max}} = R = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2}$ 206 | 207 | Angle of Principal In-Plane Stresses:\\ 208 | \tab $\ds \tan 2\theta_p = \frac{2\tau_{xy}}{(\sigma_x - \sigma_y)}$ 209 | 210 | Angle of Maximum In-Plane Stresses:\\ 211 | \tab $\ds \tan 2\theta_s = -\frac{(\sigma_x - \sigma_y)}{2\tau_{xy}}$ 212 | 213 | 3D Analysis:\\ 214 | \tab $\tau_{\text{max}} = \frac{1}{2}|\sigma_{\text{max}} - \sigma_{\text{min}}|$ 215 | 216 | \section{Theories of Failure} 217 | Maximum-Shearing-Stress Criterion for a yield strength $\sigma_Y$ and principal stresses $\sigma_a, \sigma_b$:\\ 218 | \tab Same sign: $|\sigma_a| < \sigma_Y$ \,and\, $|\sigma_b| < \sigma_Y$\\ 219 | \tab Opposite sign: $|\sigma_a - \sigma_b| < \sigma_Y$ 220 | 221 | Distortion Energy per unit volume:\\ 222 | \tab $\ds u_d = \frac{1}{6G}(\sigma^2_a - \sigma_a \sigma_b + \sigma_b^2)$ 223 | 224 | Maximum Distortion Energy Criterion:\\ 225 | \tab $\sigma^2_a - \sigma_a \sigma_b + \sigma_b^2 < \sigma_Y^2$ 226 | 227 | 228 | \section{Thin-Walled Pressure Vessel Stress} 229 | \subsubsection{Cylindrical} 230 | Hoop Stress:\\ 231 | \tab $\ds \sigma_1 = \frac{pr}{t}$ 232 | 233 | Longitudinal Stress:\\ 234 | \tab $\ds \sigma_2 = \frac{pr}{2t}$ 235 | 236 | Maximum In-Plane Shearing Stress:\\ 237 | \tab $\ds \tau_{\text{max}} = \frac{1}{2}\sigma_2 = \frac{pr}{4t}$ 238 | 239 | Maximum Out-of-Plane Shearing Stress (45\degree rotation around a longitudinal axis):\\ 240 | \tab $\ds \tau_{\text{max}} = \frac{pr}{2t}$ 241 | 242 | \subsubsection{Circle} 243 | \tab $\ds \sigma_1 = \sigma_2 \frac{pr}{2t}$ 244 | 245 | Maximum In-Plane Shearing Stress:\\ 246 | \tab $\ds \tau_{\text{max}} = 0$ (reduces to a point) 247 | 248 | Maximum Out-of-Plane Shearing Stress (45\degree rotation around a longitudinal axis):\\ 249 | \tab $\ds \tau_{\text{max}} = \frac{1}{2}\sigma_1 = \frac{pr}{4t}$ 250 | 251 | % Footer content 252 | \rule{0.3\linewidth}{0.25pt} 253 | \scriptsize\\ 254 | Updated \today\\ 255 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 256 | 257 | \end{multicols} 258 | \end{document} 259 | -------------------------------------------------------------------------------- /CPSC 340 - Machine Learning and Data Mining/cpsc340fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{mdframed} 16 | \usepackage{amsbsy} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \renewcommand\small{\@setfontsize\small{10}{11}} 47 | \makeatother 48 | 49 | % Define BibTeX command 50 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 51 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 52 | 53 | % Don't print section numbers 54 | \setcounter{secnumdepth}{0} 55 | 56 | 57 | 58 | \setlength{\parindent}{0pt} 59 | \setlength{\parskip}{1pt plus 0.5ex} 60 | 61 | \newcommand{\tab}{\hspace*{1em}} 62 | \newcommand{\ds}{\displaystyle} 63 | 64 | % Redefine some commands for newtxmath boldness 65 | \renewcommand{\grad}{\nabla} 66 | \renewcommand{\curl}[1]{\nabla\times#1} 67 | \renewcommand{\div}[1]{\nabla\cdot#1} 68 | \renewcommand{\cross}{\times} 69 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 70 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 71 | 72 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 73 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 74 | 75 | \DeclareMathOperator*{\argmax}{argmax} 76 | \DeclareMathOperator*{\argmin}{argmin} 77 | 78 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 79 | \theoremstyle{definition} 80 | \newmdtheoremenv{theorem}{Theorem} 81 | 82 | % ----------------------------------------------------------------------- 83 | 84 | \begin{document} 85 | 86 | \raggedright 87 | \footnotesize 88 | \begin{multicols}{3} 89 | 90 | \raggedcolumns 91 | 92 | % multicol parameters 93 | % These lengths are set only within the two main columns 94 | %\setlength{\columnseprule}{0.25pt} 95 | \setlength{\premulticols}{1pt} 96 | \setlength{\postmulticols}{1pt} 97 | \setlength{\multicolsep}{1pt} 98 | \setlength{\columnsep}{2pt} 99 | 100 | \begin{center} 101 | \Large{\underline{CPSC 340 Formula Sheet}} 102 | \end{center} 103 | 104 | Norms:\\ 105 | \tab $\ds \norm{\vb{y}}_2 = \sqrt{\sum_{i=1}^{n}y_i^2} \qquad\qquad \norm{\vb{y}}_1 = \sum_{i=1}^{n}\abs{y_i}$\\ 106 | \tab $\ds \norm{\vb{y}}_0 = \sum_{i=1}^{n} (1 \text{ if } y_i \neq 0)$\\ 107 | \tab $\ds \norm{\vb{y}}_\infty = \max(\abs{y_1}, \abs{y_2}, \ldots, \abs{y_n})$ 108 | \tab $\ds \norm{W}_F = \sqrt{\sum_{j=1}^{d}\sum_{c=1}^{k}w_{jc}^2}$ 109 | 110 | \section{Supervised Learning} 111 | \subsection{K-Nearest Neighbors} 112 | \begin{itemize} 113 | \itemsep 0em 114 | \item Find $k$ nearest values of $\vb{x}_i$ that are most similar to $\tilde{\vb{x}}_i$ 115 | \item Use mode of corresponding $y_i$ 116 | \end{itemize} 117 | 118 | \subsection{Naive Bayes} 119 | \tab $\ds P(y_i \mid \vb{x_i}) = \frac{P(\vb{x}_i \mid y_i)P(y_i)}{P(\vb{x}_i)} \propto P(\vb{x}_i \mid y_i)P(y_i)$ 120 | 121 | Conditional Independence Assumption:\\ 122 | \tab $\ds P(\vb{x}_i \mid y_i) \approx \prod_{j=1}^{d}P(x_{ij} \mid y_i)$ 123 | 124 | Probability Assumption:\\ 125 | \tab $\ds P(x_{ij} = k \mid y_i = c) = \frac{\text{\# times } (y_i = c, x_{ij} = k)}{n}$ 126 | 127 | Laplace Smoothing:\\ 128 | \tab $\bullet$ Add $\beta$ to numerator, and add 1 for each possible label to denominator. 129 | 130 | \section{Regression} 131 | Linear Regression:\\ 132 | \tab $\hat{y}_i = \vb{w}^T\vb{x}$ 133 | 134 | Least Squares Objective:\\ 135 | \tab $f(\vb{w}) = \frac{1}{2}\sum_{i=1}^{n}(\vb{w}^T\vb{x}_i - y_i)^2$ 136 | 137 | Normal Equations:\\ 138 | \tab $X^TX\vb{w} = X^T\vb{y}$ 139 | 140 | Huber Loss Approximation:\\ 141 | \tab $h(z) = \begin{cases} 142 | \frac{1}{2}z^2 & \abs{z} < 1\\ 143 | \abs{z} - \frac{1}{2} & \abs{z} > 1 144 | \end{cases}$ 145 | 146 | Log-sum-exp Approximation:\\ 147 | \tab $\max_i\{z_i\} \approx \log\left(\sum_{i}\exp(z_i)\right)$ 148 | 149 | \columnbreak 150 | Gradient Descent:\\ 151 | \tab $\vb{w}^{t+1} = \vb{w}^t - \alpha^t\grad{f(\vb{w}^t)}$ 152 | 153 | General Polynomial Features ($d = 1$):\\ 154 | \tab $\ds Z = \begin{bmatrix} 155 | 1 & x_1 & x_1^2 & \dots & x_1^p \\ 156 | 1 & x_2 & x_2^2 & \dots & x_2^p \\ 157 | \vdots & \vdots & \vdots & \ddots & \vdots \\ 158 | 1 & x_n & x_n^2 & \dots & x_n^p \\ 159 | \end{bmatrix}$ 160 | 161 | Gaussian Radial Basis Functions:\\ 162 | \tab $\ds g(\varepsilon) = \exp(-\frac{\varepsilon^2}{2\sigma^2})$ 163 | 164 | Gaussian Radial Basis Transform:\\ 165 | \tab $\ds Z = \begin{bmatrix} 166 | g(\norm{\vb{x}_1 - \vb{x}_1}) & \dots & g(\norm{\vb{x}_1 - \vb{x}_n}) \\ 167 | g(\norm{\vb{x}_2 - \vb{x}_1}) & \dots & g(\norm{\vb{x}_2 - \vb{x}_n}) \\ 168 | \vdots & \ddots & \vdots \\ 169 | g(\norm{\vb{x}_n - \vb{x}_1}) & \dots & g(\norm{\vb{x}_n - \vb{x}_n}) \\ 170 | \end{bmatrix}$ 171 | 172 | Gram Matrix:\\ 173 | \tab $K = XX^T$ 174 | 175 | Kernel Trick:\\ 176 | \tab $\hat{\vb{y}} = \tilde{Z}\vb{v} = \tilde{Z}Z^T(ZZ^T+\lambda I)\vb{y} = \tilde{K}(K+\lambda I)\vb{y}$ 177 | 178 | Kernel Trick with Polynomials:\\ 179 | \tab $K_{ij} = (1 + \vb{x}_i^T\vb{x}_j)^p \qquad \tilde{K}_{ij} = (1 + \tilde{\vb{x}_i}^T\vb{x}_j)^p$ 180 | 181 | \section{Linear Classifiers} 182 | Binary Classification:\\ 183 | \begin{itemize}[noitemsep, topsep=0pt] 184 | \itemsep 0em 185 | \item Encode using $y_i \in \{-1, 1\}$ 186 | \item Use $\text{sign}(\vb{w}^T\vb{x}_i)$ as prediction. 187 | \end{itemize} 188 | 189 | 0-1 Loss Function (\# of classification errors):\\ 190 | \tab $f(\vb{w}) = \norm{\text{sign}(X\vb{w}) - \vb{y}}_0$ 191 | 192 | Hinge Loss (convex upper bound on 0-1 loss):\\ 193 | \tab $f(\vb{w}) = \sum_{i=1}^{n}\max\{0, 1 - y_i\vb{w}^T\vb{x}_i\}$ 194 | 195 | Support Vector Machine:\\ 196 | \tab $f(\vb{w}) = \sum_{i=1}^{n}\max\{0, 1 - y_i\vb{w}^T\vb{x}_i\} + \frac{\lambda}{2}\norm{\vb{w}}_2^2$ 197 | 198 | Logistic Loss:\\ 199 | \tab $f(\vb{w}) = \sum_{i=1}^{n}\log(1 + \exp(-y_i\vb{w}^T\vb{x}_i))$ 200 | 201 | Softmax for class $c$:\\ 202 | \tab $\ds P(y_i = c) = \frac{\exp(\vb{w}_c^T\vb{x}_i)}{\sum_{c=1}^{k}\exp(\vb{w}_c^T\vb{x}_i)}$ 203 | 204 | Softmax Loss for classes $y_i$:\\ 205 | \tab $f(\vb{w}) = - \vb{w}_{y_i}^T\vb{x}_i + \log(\sum_{c=1}^{k}\exp(\vb{w}_c^T\vb{x}_i))$ 206 | 207 | \section{MLE and MAP} 208 | Maximum Likelihood Estimation:\\ 209 | \tab $\ds \hat{\vb{w}} \in \argmax_{\vb{w}} \{P(D \mid \vb{w})\}$ 210 | 211 | Minimizing Negative Log Likelihood to maximize likelihood:\\ 212 | \tab $\ds \hat{\vb{w}} \in \argmin_{\vb{w}} \{-\log(P(D \mid \vb{w}))\}$ 213 | 214 | Maximum a Posteriori Estimation:\\ 215 | \tab $\ds \hat{\vb{w}} \in \argmax_{\vb{w}} \{P(\vb{w} \mid D)\}$\\ 216 | 217 | Minimizing Negative Log Likelihood to maximize a posteriori:\\ 218 | \tab $\ds \vb{w} \in \argmin_{\vb{w}} \left\{-\sum_{i=1}^{n}\log(P(\vb{D}_i \mid \vb{w})) - \log(P(\vb{w}))\right\}$ 219 | 220 | \section{Matrix Factorization} 221 | \tab $X \approx ZW \qquad \vb{x}_i \approx W^T\vb{z}_i \qquad x_{ij} \approx (\vb{w}^j)^T\vb{z}_i$ 222 | \subsection{Principal Component Analysis} 223 | PCA Objective Function:\\ 224 | \tab $f(W,Z) = \sum_{i=1}^{n}\norm{W^T\vb{z}_i - \vb{x}_i}_2^2 = \norm{ZW-X}_F^2$ 225 | 226 | Prediction:\\ 227 | \begin{itemize}[noitemsep, topsep=0pt] 228 | \item Center: replace each $\tilde{x}_{ij}$ with $(\tilde{x}_{ij} - \mu_j)$ 229 | \item Find $\tilde{Z}$ minimizing squared error: $\tilde{Z} = \tilde{X}W^T(WW^T)^{-1}$ 230 | \end{itemize} 231 | 232 | Gradients:\\ 233 | \tab $\grad{f(W, Z_0)} = Z^TZW - Z^TX$\\ 234 | \tab $\grad{f(W_0, Z)} = ZWW^T-XW^T$ 235 | 236 | Variance Explained:\\ 237 | \tab $\ds 1 - \frac{\norm{ZW-X}_F^2}{\norm{X}_F^2}$ 238 | 239 | \subsection{Non-Negative Matrix Factorization} 240 | \tab Require $Z, W$ to have non-negative values. 241 | 242 | Projected Gradient Algorithm:\\ 243 | \tab $\vb{w}^{t+1/2} = \vb{w}^t - \alpha^t\grad{f(\vb{w}^t)}$\\ 244 | \tab $\vb{w}_j^{t+1} = \max\{0, \vb{w}_j\}$ 245 | 246 | \subsection{Multi-Dimensional Scaling} 247 | Traditional MDS cost function:\\ 248 | \tab $f(Z) = \sum_{i=1}^{n}\sum_{j=i+1}^{n}(\norm{\vb{z}_i - \vb{z}_j} - \norm{\vb{x}_i - \vb{x}_j})^2$ 249 | 250 | \section{Neural Networks} 251 | Objective function for one hidden layer:\\ 252 | \tab $f(\vb{v},W) = \frac{1}{2}\sum_{i=1}^{n}(\vb{v}^T \vb{h}(W\vb{x}_i)-y_i)^2$ 253 | 254 | % Footer content 255 | \rule{0.3\linewidth}{0.25pt} 256 | \scriptsize\\ 257 | Updated \today\\ 258 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 259 | \end{multicols}% 260 | 261 | \end{document} 262 | -------------------------------------------------------------------------------- /MATH 305 - Applied Complex Analysis/math305fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | \usepackage{mdframed} 14 | 15 | % This sets page margins to .5 inch if using letter paper, and to 1cm 16 | % if using A4 paper. (This probably isn't strictly necessary.) 17 | % If using another size paper, use default 1cm margins. 18 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 19 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 20 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 23 | } 24 | 25 | % Turn off header and footer 26 | \pagestyle{empty} 27 | 28 | 29 | % Redefine section commands to use less space 30 | \makeatletter 31 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 32 | {-1ex plus -.5ex minus -.2ex}% 33 | {0.5ex plus .2ex}%x 34 | {\normalfont\normalsize\bfseries}} 35 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 36 | {-1explus -.5ex minus -.2ex}% 37 | {0.5ex plus .2ex}% 38 | {\normalfont\small\bfseries}} 39 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 40 | {-1ex plus -.5ex minus -.2ex}% 41 | {1ex plus .2ex}% 42 | {\normalfont\footnotessize\bfseries}} 43 | \makeatother 44 | 45 | % Define BibTeX command 46 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 47 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 48 | 49 | % Don't print section numbers 50 | \setcounter{secnumdepth}{0} 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | \newcommand{\conj}[1]{\overline{#1}} 58 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 59 | \newcommand{\set}[1]{\left\{ #1 \right\}} 60 | \DeclareMathOperator{\Arg}{Arg} 61 | 62 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 63 | 64 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 65 | \theoremstyle{definition} 66 | \newmdtheoremenv{theorem}{Theorem} 67 | 68 | 69 | % ----------------------------------------------------------------------- 70 | 71 | \begin{document} 72 | 73 | \raggedright 74 | \footnotesize 75 | \raggedcolumns 76 | \begin{multicols}{3} 77 | 78 | 79 | % multicol parameters 80 | % These lengths are set only within the two main columns 81 | %\setlength{\columnseprule}{0.25pt} 82 | \setlength{\premulticols}{1pt} 83 | \setlength{\postmulticols}{1pt} 84 | \setlength{\multicolsep}{1pt} 85 | \setlength{\columnsep}{2pt} 86 | 87 | \begin{center} 88 | \Large{\underline{MATH 305 Formula Sheet}} 89 | \end{center} 90 | 91 | \section{Complex Numbers} 92 | Operators:\\ 93 | \tab $\ds \Re(z) = \frac{1}{2}(z + \conj{z})$ \qquad $\ds \Im(z) = \frac{1}{2i}(z - \conj{z})$ 94 | 95 | Conjugation:\\ 96 | \tab $\conj{z_1z_1} = \conj{z_1}\cdot\conj{z_2}$ \qquad $\conj{z_1/z_1} = \conj{z_1}/\conj{z_2}$ 97 | 98 | Modulus:\\ 99 | \tab $\abs{z} = \sqrt{x^2 + y^2} = \sqrt{z\cdot\conj{z}}$ \qquad $\abs{z} = \abs{\conj{z}}$\\ 100 | \tab $\abs{z_1z_1} = \abs{z_1}\abs{z_2}$\\ 101 | \tab $\abs{z_1 + z_2} \leq \abs{z_1} + \abs{z_2}$ 102 | 103 | Argument Function:\\ 104 | \tab $\arg(z) = \arg(\abs{z}e^{i\varphi}) = \set{\varphi + 2\pi k \,\vert\, k\in \mathbb{Z}}$\\ 105 | \tab $\Arg(z) = \set{\arg(z) + 2\pi k \,\vert\, k \in \mathbb{Z} \land -\pi < \Arg(z) \leq \pi}$ 106 | 107 | De Moivre's Formula:\\ 108 | \tab $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ 109 | 110 | Roots of Unity:\\ 111 | For $z = re^{i\theta}$, the $n$-th root of $z$ is\\ 112 | \tab $\ds z^{1/n} = r^{1/n}\exp\left(i\frac{\theta + 2\pi k}{n}\right)$ \qquad $k = 0, 1, 2, \ldots , n-1$\\ 113 | \tab $1^{1/n} = e^{i2\pi k/m}$ 114 | 115 | \section{Complex Functions} 116 | \defn{Continuity}: Suppose $f(z)$ is defined on a domain $D$ and $z_0 \in D$. Then $f(z)$ is continuous at $z_0$ if:\\ 117 | \tab $\ds \lim_{z\rightarrow z_0}f(z) = f(z_0)$ 118 | 119 | \defn{Differentiability}: Let $f$ be defined in a neighborhood of $z_0$. Then $f$ is differentiable at $z_0$ if the following limit exists:\\ 120 | \tab $\ds \dv{f}{z}(z_0) \equiv f'(z_0) = \lim_{\Delta z\rightarrow 0} \frac{f(z_0 + \Delta z) - f(z_0)}{\Delta z}$ 121 | 122 | \defn{Analyticity}: A complex valued function $f(z)$ is said to be analytic on an open domain $D$ if it has a derivative at every point in $D$. 123 | 124 | Cauchy-Riemann Equations:\\ 125 | For $f(z) = U(x,y) + iV(x,y)$, the CR equations are\\ 126 | \tab $\ds \pdv{U}{x} =\pdv{V}{y} \qquad \pdv{U}{y} = -\pdv{V}{x}$ 127 | 128 | \begin{theorem} 129 | Let $f(z) = U(x,y) + iV(x,y)$ be defined in some open set $G$ containing the point $z_0$. If the first partial derivatives of $U$ and $V$ exist in $G$, are continuous at $z_0$, and satisfy the Cauchy-Riemann equations at $z_0$, then $f$ is differentiable at $z_0$. 130 | \end{theorem} 131 | 132 | \begin{theorem} 133 | If $\ds \pdv{f}{\conj{z}} = 0$, then $f$ is differentiable. 134 | \end{theorem} 135 | 136 | \defn{Harmonic}: A function $u(x,y)$ is harmonic if $\Delta{u} = 0$. 137 | 138 | \begin{theorem} 139 | If $f(z) = U(x,y) + iV(x,y)$ is analytic, then $U$ and $V$ are harmonic functions. 140 | \end{theorem} 141 | 142 | Harmonic Conjugate:\\ 143 | \tab Let $f(z) = U + Vi$ be an analytic function. Then $V$ is a harmonic conjugate of $U$. 144 | 145 | \section{Elementary Functions} 146 | \tab $e^z = e^x(\cos y + i\sin y)$\\ 147 | \tab $\ds \sin z = \frac{e^{iz} - e^{-iz}}{2i} \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2}$\\ 148 | \tab $\ds \sinh z = \frac{e^{z} - e^{-z}}{2} \qquad \cosh z = \frac{e^{z} + e^{-z}}{2}$\\ 149 | \tab $\log z = \ln|z| + i\arg(z)$\\ 150 | \tab $\sin^{-1}z = -i\log(iz \pm \sqrt{1-z^2})$\\ 151 | \tab $\cos^{-1}z = -i\log(z + \sqrt{z^2-1})$\\ 152 | \tab $\ds \tan^{-1}z = \frac{i}{2}\log\left(\frac{1-iz}{1+iz}\right)$ 153 | 154 | \begin{theorem} 155 | $\Arg(z)$ is harmonic in $\mathbb{C} - \set{x < 0 \,\vert\, x\in\mathbb{R}}$. 156 | \end{theorem} 157 | \defn{Branch}: A function $F(z)$ is a branch of a multiple-valued function $f(z)$ if $F(z)$ is single-valued, continuous in some domain, and $F(z) \in f(z)$. 158 | \defn{Branch Cut}: Discontinuous points of an argument function. 159 | 160 | \section{Complex Integration} 161 | Fundamental Theorem of Calculus:\\ 162 | \tab $\ds \int_{a}^{b} f(z)\,dz = F(b) - F(a)$ 163 | 164 | Contour Integral:\\ 165 | \tab $\ds \int_{\Gamma}f(z) = \int_{a}^{b} f(r(t))r'(t)\,dt$ 166 | 167 | \begin{theorem} 168 | If $f$ is continuous on the contour $\Gamma$ and if $\abs{f(z)} < M$ for all $z$ on $\Gamma$, then $\abs{\int_\Gamma f(z)\,dz} \leq M\ell(\Gamma)$, where $\ell(\Gamma)$ is the length of $\Gamma$. 169 | \end{theorem} 170 | 171 | Path Independence:\\ 172 | \tab If $f(z)$ is continuous in an domain $D$ and has an anti-derivative $F(z)$ throughout $D$, then with initial point $z_I$ and terminal point $z_T$, for any $\Gamma \in D$, $\int_\Gamma f(z)\,dz = F(z_T) - F(z_I)$ 173 | 174 | Deformation Invariance:\\ 175 | \tab Let $f$ be analytic in a domain $D$ containing the loops $\Gamma_0$ and $\Gamma_1$. If the loops can be deformed continuously to one-another, then $\int_{\Gamma_0} f(z)\,dz = \int_{\Gamma_1}f(z)\,dz$ 176 | 177 | Cauchy's Theorem:\\ 178 | \tab If $f$ is analytic in a simply connected domain $D$ and $\Gamma$ is any closed contour, then $\int_\Gamma f(z)\,dz = 0$. 179 | 180 | Cauchy's Integral Formula:\\ 181 | \tab Let $\Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in the domain enclosed by $\Gamma$, and $z_0$ is any point inside $\Gamma$, then\\ 182 | \tab $\ds f(z_0) = \frac{1}{2\pi i}\int_{\Gamma}\frac{f(z)}{z-z_0}dz$\\ 183 | \tab $\ds f^{(n)}(z) = \frac{n!}{2\pi i}\int_{\Gamma}\frac{f(\sigma)}{(\sigma-z)^{n+1}}\,d\sigma$ 184 | 185 | \section{Complex Analysis} 186 | \begin{theorem} 187 | A bounded entire function must be constant. 188 | \end{theorem} 189 | 190 | Maximum Modulus Principle:\\ 191 | \tab If $f$ is analytic in a domain $D$ and $\abs{f(z)}$ achieves its maximum value at a point $z_0$ in $D$, then $f$ is constant in $D$. 192 | 193 | \begin{theorem} 194 | A function analytic in a bounded domain and continuous up to and including its boundary attains its maximum modulus on the boundary. 195 | \end{theorem} 196 | 197 | \begin{theorem} 198 | Suppose at each point in some closed domain enclosed by $\Gamma$ $f$ is analytic or is a pole. Then $\ds N_0(f) - N_p(f)= \frac{1}{2\pi i}\int_{\Gamma}\frac{f'(z)}{f(z)}\,dz$ 199 | \end{theorem} 200 | 201 | Argument Principle:\\ 202 | \tab $\ds \int_{\Gamma}\frac{f'(z)}{f(z)}\,dz = i(\text{Total change of }\arg(f(z))\text{ along }\Gamma)$ 203 | 204 | Nyquist Stability Criterion:\\ 205 | \tab Let $\Gamma_+$ be the countour from $(0, \infty)$ to $(0,0)$. For a polynomial function $p(z)$ of order $n$, the number of zeroes in the right-half plane is given by:\\ 206 | \tab $\ds N_0(p(z)) = \frac{1}{2\pi}\left(n\pi + 2\Delta_{\Gamma_+}(\arg(p(z)))\right)$ 207 | 208 | Rouche's Theorem:\\ 209 | \tab Suppose $f$ and $h$ are analytic functions on a domain enclosed by $\Gamma$ and that $\abs{h(z)} < \abs{f(z)} \, \forall z \in \Gamma$. Then $N_0(f) = N_0(f + h)$. 210 | 211 | Laurent Series:\\ 212 | \tab Assume $f$ is analytic in some annulus $r < \abs{z - z_0} < R$. Then we can write $\ds \sum_{j=-\infty}^{\infty}a_j(z-z_0)^j$. 213 | 214 | Singularities:\\ 215 | \tab Let $f$ have an isolated singularity at $z_0$, and let $f$ have a Laurent series expansion in $r < \abs{z - z_0} < R$. Then 216 | \begin{itemize}[leftmargin=2em] 217 | \itemsep0em 218 | \item If $a_j = 0$ for all $j < 0$, we say $z_0$ is a removable singularity. 219 | \item If $a_{-m}\neq0 $ for some positive integer $m$, but $a_j = 0$ for all $j < -m$, we say that $z_0$ is a pole of order $m$ for $f$. 220 | \item If $a_j \neq 0$ for all $j < 0$, then we say $z_0$ is an essential singularity of $f$. 221 | \end{itemize} 222 | 223 | \section{Residue Theory} 224 | \defn{Residue}: Suppose $f$ has a Laurent series expansion around a point $z_0$. Then $\Res(f, z_0) = a_{-1}$. 225 | 226 | \begin{theorem} 227 | Suppose $f(z) = P(z)/Q(z)$ and $Q(z)$ has a simple zero at $z_0$. Then $\ds \Res(f, z_0)= \frac{P(z_0)}{Q'(z_0)}$. 228 | \end{theorem} 229 | \begin{theorem} 230 | If $f$ has a pole of order $m$ at $z_0$, then $\ds \Res(f, z_0) = \lim_{ z\rightarrow z_0}\frac{1}{(m-1)!} \frac{d^{m-1}}{dz^{m-1}}[f(z)(z-z_0)^m]$ 231 | \end{theorem} 232 | 233 | Cauchy's Residue Theorem:\\ 234 | \tab If $\Gamma$ is a simple closed positively oriented contour and $f$ is analytic inside and on $\Gamma$ except at the points $z_1, z_2, \ldots , z_n$, then $\ds \int_{\Gamma}f(z)\,dz = 2\pi i \sum_{j=1}^{n}\Res(f, z_j)$ 235 | 236 | % Footer content 237 | \rule{0.3\linewidth}{0.25pt} 238 | \scriptsize\\ 239 | Updated \today\\ 240 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 241 | \end{multicols} 242 | \end{document} 243 | -------------------------------------------------------------------------------- /MECH 466 - Automatic Control/mech466fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{booktabs} 16 | \usepackage{mdframed} 17 | 18 | % This sets page margins to .5 inch if using letter paper, and to 1cm 19 | % if using A4 paper. (This probably isn't strictly necessary.) 20 | % If using another size paper, use default 1cm margins. 21 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 22 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 23 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 24 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | } 27 | 28 | % Turn off header and footer 29 | \pagestyle{empty} 30 | 31 | 32 | % Redefine section commands to use less space 33 | \makeatletter 34 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 35 | {-1ex plus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}%x 37 | {\normalfont\normalsize\bfseries}} 38 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 39 | {-1explus -.5ex minus -.2ex}% 40 | {0.5ex plus .2ex}% 41 | {\normalfont\small\bfseries}} 42 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 43 | {-1ex plus -.5ex minus -.2ex}% 44 | {1ex plus .2ex}% 45 | {\normalfont\footnotessize\bfseries}} 46 | \makeatother 47 | 48 | % Define BibTeX command 49 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 50 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 51 | 52 | % Don't print section numbers 53 | \setcounter{secnumdepth}{0} 54 | 55 | 56 | \setlength{\parindent}{0pt} 57 | \setlength{\parskip}{1pt plus 0.5ex} 58 | 59 | \newcommand{\tab}{\hspace*{1em}} 60 | \newcommand{\ds}{\displaystyle} 61 | 62 | % Change itemize indent 63 | \setlist[itemize]{leftmargin=1.5em, itemsep=0em} 64 | 65 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 66 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 67 | 68 | \newenvironment{Figure} 69 | {\par\medskip\noindent\minipage{\linewidth}} 70 | {\endminipage\par\medskip} 71 | 72 | % ----------------------------------------------------------------------- 73 | 74 | \begin{document} 75 | 76 | \raggedright 77 | \footnotesize 78 | \begin{multicols}{3} 79 | 80 | 81 | % multicol parameters 82 | % These lengths are set only within the two main columns 83 | %\setlength{\columnseprule}{0.25pt} 84 | \setlength{\premulticols}{1pt} 85 | \setlength{\postmulticols}{1pt} 86 | \setlength{\multicolsep}{1pt} 87 | \setlength{\columnsep}{2pt} 88 | 89 | \raggedcolumns 90 | 91 | \begin{center} 92 | \Large{\underline{MECH 466 Formula Sheet}} 93 | \end{center} 94 | 95 | \section{Continuous Time Signals} 96 | Linearity:\\ 97 | \tab $\mathcal{S}[\alpha x(t) + \beta y(t)] = \alpha\mathcal{S}[x(t)] + \beta\mathcal{S}[y(t)]$ 98 | 99 | Time Invariance:\\ 100 | \tab If $\mathcal{S}[x(t)] = y(t)$ then $\mathcal{S}[x(t \pm \tau)] = y(t \pm \tau)$. 101 | 102 | \section{Laplace Transform} 103 | Initial Value Theorem:\\ 104 | \tab $\ds f(0+) \Leftrightarrow \lim_{s\rightarrow\infty} sF(s)$ 105 | 106 | Final Value Theorem:\\ 107 | \tab $\ds \lim_{t\rightarrow\infty} f(t) \Leftrightarrow \lim_{s\rightarrow 0} sF(s)$ 108 | 109 | \section{Stability} 110 | \begin{itemize} 111 | \item BIBO stability: any bounded input provides a bounded output 112 | \item Asymptotic stability: Initial conditions generates $y(t)$ converges to zero. 113 | \end{itemize} 114 | 115 | Characteristic Equation:\\ 116 | \tab For $G(s) = N(s)/D(s)$, the characteristic equation is 117 | $D(s) = 0$. 118 | 119 | Stability Condition in $s$-Domain:\\ 120 | \tab All poles in open left hand plane $\iff$ System is BIBO and asymptotically stable\\ 121 | 122 | Marginal Stability:\\ 123 | \begin{itemize} 124 | \item $G(s)$ has no pole in the open RHP 125 | \item $G(s)$ has at least one simple pole on the imaginary axis 126 | \item $G(s)$ has no repeated pole on the imaginary axis 127 | \item BIBO stable except for sinusoidal inputs. 128 | \item For any non-zero initial condition, the output neither converges to zero nor diverge. 129 | \end{itemize} 130 | 131 | Polynomials:\\ 132 | \begin{itemize} 133 | \item For 1st and 2nd order polynomials, all roots are in LHP $\iff$ coefficients have the same sign 134 | \item For 3rd and higher orders, all roots are in LHP $\implies$ coefficients have the same sign 135 | \end{itemize} 136 | 137 | \subsection{Routh-Hurwitz Criterion} 138 | \begin{itemize} 139 | \item The number of roots in the open right half-plane is equal to the number of sign changes in the first column of Routh array. 140 | \item If zero row appears in the Routh array, roots are in imaginary axis or RHP. 141 | \item If zero row appears, replace the zero with the coefficients of derivative of auxiliary polynomial. 142 | \item Auxiliary polynomial: The polynomial above the zero row. 143 | \end{itemize} 144 | 145 | \section{Steady State Error} 146 | Step Function:\\ 147 | \tab $\ds r(t) = Ru(t) \implies e_\text{ss} = \frac{R}{1+K_p} = \frac{R}{1+L(0)}$ 148 | 149 | Ramp Function:\\ 150 | \tab $\ds r(t) = Rtu(t) \implies e_\text{ss} = \frac{R}{K_v} = \frac{R}{\lim\limits_{s\to 0}sL(s)}$ 151 | 152 | Parabolic Function:\\ 153 | \tab $\ds r(t) = \frac{Rt}{2}u(t) \implies e_\text{ss} = \frac{R}{K_a} = \frac{R}{\lim\limits_{s\to 0}s^2L(s)}$ 154 | 155 | \section{First Order System} 156 | \tab $\ds G(s)=\frac{K}{Ts+1}$ 157 | 158 | Step Response:\\ 159 | \tab $y(t) = K(1-e^{t/T})u(t)$ 160 | 161 | Time Constant:\\ 162 | \tab $\ds T = \frac{1}{|\text{Real part of pole}|}$\\ 163 | \tab Response is slower the closer the pole is to the imaginary axis 164 | 165 | Settling Time:\\ 166 | \tab 5\%: $\approx 3T$ 167 | \tab 2\%: $\approx 4T$ 168 | 169 | \section{Second Order System} 170 | \tab $\ds G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$ 171 | 172 | \subsection{Underdamped Case ($0 < \zeta < 1$)} 173 | Step Response:\\ 174 | \tab $\ds y(t) = 1-\frac{e^{-\zeta\omega_n t}}{\sqrt{1-\zeta^2}}\sin\left(\omega_d t + \arccos\zeta\right)$\\ 175 | \tab $\omega_d = \omega_n\sqrt{1-\zeta^2}$ 176 | 177 | Poles:\\ 178 | \tab $s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}$ 179 | 180 | Settling Time:\\ 181 | \tab 5\%: $\approx 3/\zeta\omega_n$ 182 | \tab 2\%: $\approx 4/\zeta\omega_n$ 183 | 184 | Peak Time:\\ 185 | \tab $T_p = \pi/\omega_d$ 186 | 187 | Overshoot:\\ 188 | \tab 16\%: $\zeta = 0.5$ 189 | \tab 5\%: $\zeta = \sqrt{2}/2$\\ 190 | \tab $\ds y_\text{max} = 1+e^{-\zeta\pi/\sqrt{1-\zeta^2}}$\\ 191 | \tab Percent overshoot = $100 e^{-\zeta\pi/\sqrt{1-\zeta^2}}$ 192 | 193 | \section{Root Locus Method} 194 | A graphical method to show how the poles move of a feedback system when K varies from 0 to $\infty$ in the form\\ 195 | \tab $\ds H(S) = \frac{KL(s)}{1+KL(s)}$ \hspace{1cm} $\ds L(s) = \frac{N(s)}{D(s)}$ 196 | \begin{itemize} 197 | \item RL includes all points on real axis to the left of an odd number of real poles/zeros 198 | \item RL originates from the poles of $L$ and terminates at the zeros of $L$, including infinity zeros. 199 | \item Number of asymptotes: $r = \text{deg}(D(S)) - \text{deg}(N(s))$ 200 | \item Intersection of asymptotes: $\ds \frac{\sum \text{poles} - \sum \text{zeroes}}{r}$ 201 | \item Angle of asymptotes: $\ds \frac{\pi(2k+1)}{r}\quad k = 0,1,\ldots r-1$ 202 | \item Breakaway points: Each root of $L'(s^{\star}) = 0$, where $K = -1/L(s^\star) > 0$ 203 | \item Angle condition: $s_0$ is on the root locus $\iff$ $\angle L(s_0) = 180^\circ$ 204 | \end{itemize} 205 | 206 | \section{Lead \& Lag Compensators} 207 | \tab $\ds C(s) = K\frac{s + z}{s + p} \quad \text{or} \quad C(s) = K\frac{\frac{s}{z} +1 }{\frac{s}{p} + 1}$ 208 | 209 | \subsection{Lead ($z < p$)} 210 | \begin{itemize} 211 | \item Moves intersection of asymptotes to the left 212 | \item Improves transient response (faster) 213 | \end{itemize} 214 | \subsection{Lag ($z > p$)} 215 | \begin{itemize} 216 | \item Moves intersection of asymptotes to the right 217 | \item Reduces steady state error 218 | \end{itemize} 219 | 220 | Phase-lag design: 221 | \begin{enumerate} 222 | \item Adjust DC gain of OL system by a constant gain $K$ to satisfy low-frequency requirement. 223 | \item On Bode plot, find the frequency $\omega_g$ where $\angle G(j\omega_g) = -180^\circ + \phi_m + 5^\circ$ for the required PM $\phi_m$. 224 | \item Select $z$ and $p$: $z = 0.1 \omega_g$, $p = z/\abs{KG(j\omega_g)}$. 225 | \end{enumerate} 226 | 227 | Phase-lead design: 228 | \begin{enumerate} 229 | \item Select $z$ near uncompensated $\omega_g$. 230 | \item Select $p > z$ by trial and error. 231 | \item Check PM and setting time. 232 | \end{enumerate} 233 | 234 | \section{Frequency Response} 235 | For a sinusoidal input $A\sin(\omega t)$ and system $G(s)$, the output is $A\abs{G(j\omega)}\sin(\omega t + \angle G(j\omega))$\\ 236 | 237 | First Order System with corner frequency $1/T$:\\ 238 | \tab $\ds G(j\omega) = \frac{K}{j\omega T + 1}\approx \begin{cases} 239 | K & 1 \gg \omega T \\ 240 | \frac{K}{j\omega T} & 1 \ll \omega T 241 | \end{cases}$ 242 | 243 | \subsection{Basic Functions} 244 | Constant Gain:\\ 245 | \tab $\abs{G(j\omega)} = K, \; \angle G(j\omega) = 0^\circ$ 246 | 247 | Differentiator:\\ 248 | \tab $\abs{G(j\omega)} = \omega, \; \angle G(j\omega) = \angle j\omega = 90^\circ$ 249 | 250 | Integrator:\\ 251 | \tab $\abs{G(j\omega)} = \frac{1}{\omega}, \; \angle G(j\omega) = \angle\frac{1}{j\omega} = -90^\circ$ 252 | 253 | Double Integrator:\\ 254 | \tab $\abs{G(j\omega)} = \frac{1}{\omega^2}, \; \angle G(j\omega) = \angle\frac{1}{(j\omega)^2} = -180^\circ$ 255 | 256 | Time Delay:\\ 257 | \tab $\abs{G(j\omega)} = 1, \; \angle G(j\omega) = -\omega T$ 258 | 259 | Second Order System:\\ 260 | \begin{Figure} 261 | \centering 262 | \includegraphics[width=\linewidth]{SecondOrderFreqResponse.jpg} 263 | \end{Figure} 264 | Resonant Frequency: $\omega_n \sqrt{1 - 2 \zeta^2} \approx \omega_n$ 265 | 266 | Peak Gain: \tab $\ds \frac{1}{2\zeta\sqrt{1-\zeta^2}} \approx \frac{1}{2\zeta}$ 267 | 268 | At the resonant frequency, the gain is unity when $\zeta = 1/\sqrt{2}$. 269 | 270 | \subsection{Nyquist Stability} 271 | For a open-loop transfer function $L(s)$: 272 | \begin{itemize} 273 | \item CL system is stable $\iff Z := P + N = 0$ 274 | \item $Z$: \# of CL poles in open RHP. 275 | \item $P$: \# of OL poles in open RHP. 276 | \item $N$: \# of clockwise encirclement of -1 277 | \item $N = -1$ is a counter-clockwise encirclement. 278 | \end{itemize} 279 | 280 | \subsection{Relative Stability} 281 | Gain crossover frequency $\omega_g$:\\ 282 | \tab $\abs{L(j\omega_g)} = 1$ 283 | 284 | Phase crossover frequency $\omega_p$\\ 285 | \tab $\angle L(j\omega_p) = -180^\circ$ 286 | 287 | Gain Margin:\\ 288 | \tab $\ds \text{GM} = 20\log_{10}\frac{1}{\abs{L(j\omega_p)}}$ 289 | 290 | Phase Margin:\\ 291 | \tab $\ds \text{PM} = \angle L(j\omega_g) + 180^\circ$ 292 | 293 | % Footer content 294 | \rule{0.3\linewidth}{0.25pt} 295 | \scriptsize\\ 296 | Updated \today\\ 297 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 298 | 299 | \end{multicols} 300 | \end{document} 301 | -------------------------------------------------------------------------------- /MATH 318 - Probability with Physical Applications/math318fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage[landscape]{geometry} 8 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 9 | \usepackage{color,graphicx,overpic} 10 | \usepackage{hyperref} 11 | \usepackage{enumitem} 12 | \usepackage{upgreek} 13 | \usepackage[italicdiff]{physics} 14 | \usepackage{newtxtext,newtxmath} 15 | \usepackage{booktabs} 16 | \usepackage{mdframed} 17 | \usepackage{amsbsy} 18 | 19 | % This sets page margins to .5 inch if using letter paper, and to 1cm 20 | % if using A4 paper. (This probably isn't strictly necessary.) 21 | % If using another size paper, use default 1cm margins. 22 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 23 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 24 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 27 | } 28 | 29 | % Turn off header and footer 30 | \pagestyle{empty} 31 | 32 | 33 | % Redefine section commands to use less space 34 | \makeatletter 35 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 36 | {-1ex plus -.5ex minus -.2ex}% 37 | {0.5ex plus .2ex}%x 38 | {\normalfont\normalsize\bfseries}} 39 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 40 | {-1explus -.5ex minus -.2ex}% 41 | {0.5ex plus .2ex}% 42 | {\normalfont\small\bfseries}} 43 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 44 | {-1ex plus -.5ex minus -.2ex}% 45 | {1ex plus .2ex}% 46 | {\normalfont\footnotessize\bfseries}} 47 | \makeatother 48 | 49 | % Define BibTeX command 50 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 51 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 52 | 53 | % Don't print section numbers 54 | \setcounter{secnumdepth}{0} 55 | 56 | 57 | \setlength{\parindent}{0pt} 58 | \setlength{\parskip}{1pt plus 0.5ex} 59 | 60 | \newcommand{\tab}{\hspace*{1em}} 61 | \newcommand{\ds}{\displaystyle} 62 | 63 | % Redefine some commands for newtxmath boldness 64 | \renewcommand{\grad}{\nabla} 65 | \renewcommand{\curl}[1]{\nabla\times#1} 66 | \renewcommand{\div}[1]{\nabla\cdot#1} 67 | \renewcommand{\cross}{\times} 68 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 69 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 70 | 71 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 72 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 73 | 74 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 75 | \theoremstyle{definition} 76 | \newmdtheoremenv{theorem}{Theorem} 77 | 78 | % ----------------------------------------------------------------------- 79 | 80 | \begin{document} 81 | 82 | \raggedright 83 | \footnotesize 84 | \begin{multicols}{3} 85 | 86 | 87 | % multicol parameters 88 | % These lengths are set only within the two main columns 89 | %\setlength{\columnseprule}{0.25pt} 90 | \setlength{\premulticols}{1pt} 91 | \setlength{\postmulticols}{1pt} 92 | \setlength{\multicolsep}{1pt} 93 | \setlength{\columnsep}{2pt} 94 | 95 | \raggedcolumns 96 | 97 | \begin{center} 98 | \Large{\underline{MATH 318 Formula Sheet}} 99 | \end{center} 100 | 101 | \section{Probability Theory} 102 | Probability Function:\\ 103 | \begin{itemize} 104 | \itemsep 0em 105 | \item $0 \leq P \leq 1$ 106 | \item $P(S) = 1$ 107 | \item $E_1 \cap E_2 = \emptyset \implies P(E_1 \cup E_2) = P(E_1) + P(E_2)$ 108 | \item $P(E_1) + P(E_2) = P(E_1 \cup E_2) + P(E_1 \cap E_2)$ 109 | \end{itemize} 110 | 111 | Conditional Probability:\\ 112 | \tab $\ds P(E|F) = \frac{P(E \cap F)}{P(F)}$ 113 | 114 | Two events are said to be independent if:\\ 115 | \tab $P(E \cap F) = P(E)P(F)$ 116 | 117 | \begin{theorem} 118 | Let $F_1, F_2 \ldots F_n$ be a partition of the sample space $S$. Assume $F_i \cap F_j = \emptyset$ for any $i \neq j$. Then for any event $E \subset S$, 119 | \begin{enumerate} 120 | \itemsep 0em 121 | \item $P(E) = \sum_{i}^{n} P(E|F_i)P(F_i)$ 122 | \item $\ds P(F_j|E) = \frac{P(E|F_j)P(F_j)}{\sum_{i}^{n}P(E|F_i)P(F_i)}$ (Bayes' 123 | Formula) 124 | \end{enumerate} 125 | \end{theorem} 126 | 127 | \section{Random Variables} 128 | Memory-less Property:\\ 129 | \tab $P(X > m + n | X > n) = P(X > m)$ 130 | 131 | Expectation Value:\\ 132 | \tab $\expval{X} = \sum_{i=0}^{\infty} x_i p(X = x_i) = \sum_{i=0}^{\infty} x_i p(x_i)$\\ 133 | \tab $\expval{X} = \int_{-\infty}^\infty xf(x)\,dx$ 134 | 135 | Cumulative Distribution Function:\\ 136 | \tab $F(x) = \int_{-\infty}^{x} f(t)\,dt$ 137 | 138 | Law of the Unconscious Statistician:\\ 139 | \tab $\expval{g(X)} = \int_{-\infty}^{\infty}g(x)f(x)\,dx$ 140 | 141 | Linearity of Expectation:\\ 142 | \tab $\expval{aX+b} = a\expval{X} + b$ 143 | 144 | Moments:\\ 145 | \tab $n$-th moment of $X = \begin{cases} 146 | \int_{-\infty}^{\infty}x^n f(x)\,dx & \,\\ 147 | \sum_{i}^{\infty} x_i^n p(x_i) & \, 148 | \end{cases}$ 149 | 150 | Variance:\\ 151 | \tab $\Var{X} = \expval{(X-\expval{X})^2} = \expval{X^2} - \expval{X}^2$ 152 | 153 | Joint Continuity:\\ 154 | \tab $P((X,Y)\in C) = \iint_C f(x,y)\,dx\,dy$ 155 | 156 | Marginal Distribution:\\ 157 | \tab $P(X\in A) = P(X \in A, Y \in \mathbb{R}) = \int_A \int_{-\infty}^{\infty}f(x,y)\,dy\,dx$ 158 | 159 | Independence:\\ 160 | If $X,Y$ are independent, then\\ 161 | \tab $P(X\leq a, Y\leq b) = P(X\leq a)P(Y\leq b)$\\ 162 | \tab $\expval{g(X)h(Y)} = \expval{g(X)}\expval{h(Y)}$ 163 | 164 | Covariance:\\ 165 | \tab $\Cov{X,Y} = \expval{(X-\expval{X})(Y-\expval{Y})} = \expval{XY} - \expval{X}\expval{Y}$ 166 | 167 | Correlation Coefficient:\\ 168 | \tab $\ds \rho(X,Y) = \frac{\Cov{X,Y}}{\sqrt{\Var{X}\Var{Y}}} \in [-1,1]$ 169 | 170 | Cauchy-Swartz Inequality:\\ 171 | \tab $\abs{\expval{XY}}^2 \leq \expval{X^2}\expval{Y^2}$ 172 | 173 | Sum of Random Variables:\\ 174 | \tab $\Var{X+Y} = \Var{X} + \Var{Y} + 2\Cov{X,Y}$\\ 175 | \tab $\ds F_{X+Y}(a)=\int_{-\infty}^\infty\int_{-\infty}^{a-y}f_X(x)f_Y(y)\,dx\,dy$\\ 176 | \tab $\ds f_{X+Y}(a)=\int_{-\infty}^\infty f_X(a-y)f_Y(y)\,dy$ 177 | 178 | Conditional Probability Distribution:\\ 179 | \tab $\ds f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$ 180 | 181 | Conditional Expectation:\\ 182 | \tab $\expval{X|Y} = \sum_x xp_{X|Y}(x,y)$\\ 183 | \tab $\expval{X|Y} = \int_{-\infty}^{\infty}xf_{X|Y}(x,y)\,dx$\\ 184 | \tab $\expval{X} = \expval{\expval{X|Y}} = \sum_y\expval{X|Y=y}P(Y=y)$\\ 185 | \tab $\expval{X} = \expval{\expval{X|Y}} = \int_{-\infty}^\infty \expval{X|Y=y}f_Y(y)\,dy$ 186 | 187 | \subsection{Characteristic Functions} 188 | \tab $\phi_X(t) = \expval{e^{itX}}$ 189 | \tab $M(t) = \expval{e^{tX}}$ 190 | 191 | Extracting Moments: 192 | \tab $\ds \dv[n]{}{t}\rvert_{t=0} \phi(t) = \expval{i^nX^n}$ 193 | 194 | Inversion Theorem:\\ 195 | \tab $f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\phi(t) e^{-itx}\,dt$ 196 | 197 | Shifting:\\ 198 | \tab $\ds \phi_{aX+b}(t) = e^{itb}\phi_X(at)$ 199 | 200 | \subsection{Convergence of Random Variables} 201 | Convergence in Distribution:\\ 202 | \tab $\lim_{n\rightarrow\infty}F_n(x) = F(x) \hspace{0.5em}\forall x \hspace{0.5em}\text{cont.} \iff X_n \xrightarrow{D} X$ 203 | 204 | \thm{Continuity Theorem}:\\ 205 | \tab Let $X_1, X_2, \ldots$ be random variables with CDFs $F_1, F_2, \ldots$ and characteristics functions $\phi_1, \phi_2, \ldots$. Then 206 | \tab $\cdot$ If $F_n \rightarrow F$, where $F$ is the CDF of some random variable $X$, then $\lim\limits_{n\rightarrow\infty} \phi_n(t) = \phi(t)$.\\ 207 | \tab $\cdot$ If $\lim\limits_{n\rightarrow\infty} \phi_n(t) = \phi(t)$ and $\phi(t)$ is continuous at $t = 0$, then $\phi$ is the characteristic function of some random variable $X$ and $F_n \rightarrow X$ and $X_n \xrightarrow{D} X$. 208 | 209 | \thm{Weak Law of Large Numbers}: Let $X_1, X_2, \ldots X_n$ be iid random variables. Assume $\expval{X} = \mu < \infty$. Let $S_n = \sum_{i=1}^{n}X_i$. Then $\frac{S}{n} \xrightarrow{D} \mu$. 210 | 211 | \thm{Central Limit Theorem}: Let $X_i$ be iid random variables. Let $\expval{X_i} = \mu < \infty$, $\Var{X_i} = \sigma^2 < \infty$, and $S_n = \sum_{i=1}^{n}X_i$. Then\\ 212 | \tab $\ds \frac{S_n - n\mu}{\sigma\sqrt{n}} \xrightarrow{D} N(0,1)$. 213 | 214 | \section{Statistical Estimation} 215 | \defn{Estimator}: A function of the data that is used to estimate the unknown parameter. 216 | 217 | Estimator of the mean $\mu$:\\ 218 | \tab Sample Mean $\ds \bar{X} = \frac{1}{n}\sum_{1}^{n}X_i$ 219 | 220 | Estimator of the variance $\sigma^2$:\\ 221 | \tab Sample Variance $\ds S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2$ 222 | 223 | \defn{Confidence Interval}: For some interval $A \subset \mathbb{R}$, an estimator is in the $a\%$ confidence level if $P(\bar{X} \in A) = a\%$. 224 | 225 | \section{Random Walks} 226 | \textbf{Symmetric random walks in $\mathbb{Z}^d$}:\\ 227 | \tab Number of visits to the origin $M$: $\expval{M} = (1-u)^{-1}$.\\ 228 | \tab Probability walk returns to the origin $u = 1 - 1/\expval{M}$.\\ 229 | \tab If $u = 1$, the walk is recurrent, otherwise transient. 230 | 231 | \section{Markov Chains} 232 | Transition Matrix $P$: Rows add to 1.\\ 233 | One-Step Transition Probability:\\ 234 | \tab $P_{ij} = P(X_{n+1} = j | X_n = i)$ \quad $\sum_j P_{ij} = 1$ 235 | 236 | N-Step Transition Probability:\\ 237 | \tab $P^n = P(X_{l+n} = j | X_l = i)$ 238 | 239 | Chapman-Kolmogorov Equation:\\ 240 | \tab $P_{ij}^{n+m} = \sum_{k} P_{ik}^n P_{kj}^n$\\ 241 | \tab $P^{n+m} = P^nP^m$ 242 | 243 | 244 | \end{multicols} 245 | 246 | \newpage 247 | 248 | \begin{minipage}[t]{0.33\textwidth} 249 | Classification of States: 250 | \begin{itemize} 251 | \item A state $i$ is absorbing if $P_{ii} = 1$ 252 | \item $j$ is accessible from $i$ if $P_{ij}^n > 0$ for some $n$. 253 | \item $i$ and $j$ communicate ($i \leftrightarrow j$) if $j$ is accessible from $i$ and $i$ is accessible from $j$. 254 | \item If $i$ is recurrent and $j$ is accessible from $i$, $i \leftrightarrow j$. 255 | \item If $i$ is recurrent and $i \leftrightarrow j$, then $j$ is also recurrent. 256 | \end{itemize} 257 | 258 | Irreducibility:\\ 259 | \tab A Markov chain is irreducible if there is only one class (all states communicate). 260 | 261 | Periodicity of state $i$:\\ 262 | \tab Period $d = \gcd\{n \geq 1 : P_{ii}^n > 0\}$\\ 263 | \tab $d = 1 \implies i$ is aperiodic 264 | 265 | Transience and Recurrence:\\ 266 | \tab $f_i = P(\exists n \text{ s.t. } X_n = i | X_0 = i)$\\ 267 | \tab $f_i = 1 \implies i$ is recurrent (every path leads to $i$)\\ 268 | \tab $f_i < 1 \implies i$ is transient 269 | 270 | Recurrent State for $T_i$ = time of first return to $i$:\\ 271 | \tab $\expval{T_i | X_0 = i} < \infty \implies$ positive recurrent\\ 272 | \tab $\expval{T_i | X_0 = i} = \infty \implies$ null recurrent 273 | 274 | \defn{Ergodic}: A aperiodic, positive recurrent state is called ergodic. A Markov chain is ergodic if all its states are ergodic. 275 | 276 | \thm{Existence of Equilibrium Distribution}: For an irreducible, ergodic Markov chain, the limit\\ 277 | \tab $\pi_j = \lim\limits_{n\rightarrow\infty} P_{ij}^n$\\ 278 | exists for all $j$ and is independent of state $i$. 279 | \begin{enumerate} 280 | \item $\boldsymbol{\pi}$ is the unique solution of $\boldsymbol{\pi} = \boldsymbol{\pi}P$ and $\sum_{j} \pi_j = 1$ 281 | \item Let $N_j(n)$ be the number of visits to state $j$ after $n$ steps. Then $\ds \pi_j = \lim_{n\rightarrow\infty} \frac{N_j(n)}{n}$ 282 | \item $\pi_j = 1/m_j$ where $m_j = \expval{T_j | X_0 = j}$ 283 | \end{enumerate} 284 | 285 | Time Reversal:\\ 286 | \tab Given a Markov chain $(X_n)^N_{n=0}$ with stationary distribution $\boldsymbol{\pi}$ and with $P(X_0 = j) = \pi_j$, let $Y_n = X_{N-n}$. Then $(Y_n)^N_{n=0}$ is a Markov chain with transition probabilities $Q_{ij} = P_{ji}\frac{\pi_j}{\pi_i}$ and stationary distribution $\boldsymbol{\pi}$. 287 | 288 | \defn{Time Reversibility}: A markov chain is time reversible if $Q_{ij} = P_{ij}$ $\forall i,j$. In this case, $\pi_i P_{ij} = \pi_j P_{ji}$. 289 | 290 | % Footer content 291 | \rule{0.3\linewidth}{0.25pt} 292 | \scriptsize\\ 293 | Updated \today\\ 294 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 295 | \end{minipage}% 296 | \hspace{0.5em} 297 | \begin{minipage}[t]{0.35\textwidth} 298 | \subsection{Random Variables} 299 | \begin{tabular}{lllll} 300 | \toprule 301 | Distribution & Mass/Density Function & Mean & Variance & Characteristic Function\\ 302 | \midrule\vspace{1mm} 303 | 304 | Binomial$(n,p)$ & $p(i) = \binom{n}{i}p^i (1-p)^{n-i}$ & $np$ & $np(1-p)$ & $(1-p+e^{it})^n$\\ 305 | 306 | Geometric$(p)$ & $p(k) = (1-p)^{k-1}p$ & $1/p$ & $\ds \frac{1-p}{p^2}$ & $\ds \frac{pe^{it}}{1-(1-p)e^{it}}$\\ 307 | 308 | Poisson$(\lambda)$ & $\ds p(i) = \frac{\lambda^i}{i!}e^{-\lambda}$ & $\lambda$ & $\lambda$ & $e^{\lambda(e^{it} - 1)}$\\ 309 | 310 | Uniform$(a,b)$ & 311 | $\ds f(x) = \begin{cases} 312 | \frac{1}{b-a} & x \in [a,b]\\ 313 | 0 & \text{otherwise} 314 | \end{cases}$ & $\ds \frac{a+b}{2}$ & $\ds \frac{(b-a)^2}{12}$ & $\ds \frac{e^{ita}-e^{itb}}{it(b-a)}$\\ 315 | 316 | Exponential$(\lambda)$ & 317 | $\ds f(x) = \begin{cases} 318 | \lambda e^{-\lambda x} & x \geq 0\\ 319 | 0 & x < 0 320 | \end{cases}$ & $1/\lambda$ & $1/\lambda^2$ & $\ds \frac{\lambda}{\lambda - it}$\\ 321 | 322 | Normal$(\mu, \sigma^2)$ & $\ds f(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ & $\mu$ & $\sigma^2$ & $e^{i\mu t-\sigma^2 t^2/2}$\\ 323 | \bottomrule 324 | \end{tabular} 325 | \vspace{0.5em} 326 | \section{Reserved} 327 | 328 | 329 | \end{minipage} 330 | \begin{minipage}[t]{0.30\textwidth} 331 | \vspace{19em} 332 | \subsection{Reserved} 333 | \end{minipage} 334 | 335 | \end{document} 336 | -------------------------------------------------------------------------------- /PHYS 304 - Introduction to Quantum Mechanics/phys304fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | 14 | % This sets page margins to .5 inch if using letter paper, and to 1cm 15 | % if using A4 paper. (This probably isn't strictly necessary.) 16 | % If using another size paper, use default 1cm margins. 17 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 18 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 19 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | } 23 | 24 | % Turn off header and footer 25 | \pagestyle{empty} 26 | 27 | 28 | % Redefine section commands to use less space 29 | \makeatletter 30 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 31 | {-1ex plus -.5ex minus -.2ex}% 32 | {0.5ex plus .2ex}%x 33 | {\normalfont\normalsize\bfseries}} 34 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 35 | {-1explus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}% 37 | {\normalfont\small\bfseries}} 38 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 39 | {-1ex plus -.5ex minus -.2ex}% 40 | {1ex plus .2ex}% 41 | {\normalfont\footnotessize\bfseries}} 42 | \makeatother 43 | 44 | % Define BibTeX command 45 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 46 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 47 | 48 | % Don't print section numbers 49 | \setcounter{secnumdepth}{0} 50 | 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | \newcommand{\ap}{a_+} 58 | \newcommand{\am}{a_-} 59 | 60 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 61 | 62 | % Redefine some commands for newtxmath boldness 63 | \renewcommand{\grad}{\nabla} 64 | \renewcommand{\curl}[1]{\nabla\times#1} 65 | \renewcommand{\div}[1]{\nabla\cdot#1} 66 | \renewcommand{\cross}{\times} 67 | 68 | % ----------------------------------------------------------------------- 69 | 70 | \begin{document} 71 | 72 | \raggedright 73 | \footnotesize 74 | \begin{multicols}{3} 75 | 76 | 77 | % multicol parameters 78 | % These lengths are set only within the two main columns 79 | %\setlength{\columnseprule}{0.25pt} 80 | \setlength{\premulticols}{1pt} 81 | \setlength{\postmulticols}{1pt} 82 | \setlength{\multicolsep}{1pt} 83 | \setlength{\columnsep}{2pt} 84 | 85 | \raggedcolumns 86 | 87 | \begin{center} 88 | \Large{\underline{PHYS 304 Formula Sheet}} 89 | \end{center} 90 | 91 | \section{The Wave Function} 92 | Time dependent Schrodinger Equation:\\ 93 | \tab $\ds i\hbar \pdv{\Psi}{t} = -\frac{\hbar^2}{2m}\pdv[2]{\Psi}{x}+ V\Psi$ 94 | 95 | Standard Deviation:\\ 96 | \tab $\sigma = \sqrt{\expval{j^2} - \expval{j}^2}$ 97 | 98 | Momentum: \\ 99 | \tab $\ds \expval{p} = m \dv{\expval{x}}{t} = \int\Psi^* \left(\frac{\hbar}{i}\pdv{}{x}\right)\Psi\,dx$ 100 | 101 | \section{Infinite Square Well} 102 | Time Independent Schrodinger Equation:\\ 103 | \tab $\ds -\frac{\hbar^2}{2m}\pdv[2]{\psi}{x}+ V\psi = E\psi$ 104 | 105 | Eigenstate Expansion:\\ 106 | \tab $\ds \Psi(x,t) = \sum_{n=1}^{\infty} c_n \psi_n(x)e^{-iE_nt/\hbar} = \sum_{n=1}^{\infty} c_n \Psi_n(x,t)$ 107 | 108 | Energy In Infinite Square Well:\\ 109 | \tab $\ds E_n = \frac{n^2\pi^2\hbar^2}{2ma^2}$ 110 | 111 | Stationary States:\\ 112 | \tab $\ds \psi_n(x) = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi}{a}x\right)$ 113 | 114 | Determining Coefficients:\\ 115 | \tab $\ds c_n = \sqrt{\frac{2}{a}}\int_{0}^{a}\sin\left(\frac{n\pi}{a}x\right)\Psi(x,0)\,dx$ 116 | 117 | Expectation Value of Energy:\\ 118 | \tab $\expval{H} = \sum_{n=1}^{\infty}\abs{c_n}^2E_n$ 119 | 120 | \section{Harmonic Oscillator} 121 | \tab $k = \omega^2m$ 122 | 123 | Ladder Operators:\\ 124 | \tab $a_\pm = \frac{1}{\sqrt{2\hbar m\omega}}(\mp ip + m\omega x)$\\ 125 | \tab $\ap \psi_n = \sqrt{n+1}\,\psi_{n+1}$\\ 126 | \tab $\am \psi_n = \sqrt{n}\,\psi_{n-1}$\\ 127 | 128 | Operators:\\ 129 | \tab $\ds x = \sqrt{\frac{\hbar}{2m\omega}}(\ap + \am)$\\ 130 | \tab $\ds p = i\sqrt{\frac{\hbar m\omega}{2}}(\ap - \am)$\\ 131 | \tab $\ds x^2 = \frac{\hbar}{2m\omega}\left[(\ap)^2 + (\ap\am) + (\am\ap) + (\am)^2\right]$ 132 | 133 | Commutation:\\ 134 | \tab $\comm{x}{p} = i\hbar$\\ 135 | \tab $\comm{\am}{\ap} = 1$\\ 136 | \tab $\comm{AB}{C} = A\comm{B}{C} + \comm{A}{C}B$ 137 | 138 | Hamiltonian:\\ 139 | \tab $\ds H = \hbar \omega \left(\ap \am + \frac{1}{2}\right)$\\ 140 | \tab $\ds \ap \am + \am\ap = 2 \left(\frac{H}{\hbar \omega}\right)$\\ 141 | 142 | States:\\ 143 | \tab $\ds \psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-\frac{m\omega}{2\hbar}x^2}$\\ 144 | \tab $\ds \psi_n = \frac{1}{\sqrt{n!}}(\ap)^n\psi_0$ 145 | 146 | Energy:\\ 147 | \tab $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ 148 | 149 | \section{Generalized Statistical Interpretation} 150 | Momentum Expansion:\\ 151 | \tab $\ds \Phi(p,t) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{-ipx/\hbar}\,\Psi(x,t)\,dx$\\ 152 | \tab $\ds \Psi(x,t) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}e^{ipx/\hbar}\,\Phi(p,t)\,dp$ 153 | 154 | \section{Angular Momentum} 155 | $L_x = yp_z -zp_y$ \quad $L_y = zp_x - xp_z$ \quad $L_z = xp_y - yp_x$ 156 | 157 | Commutators:\\ 158 | \tab $\comm{L_x}{L_y} = i\hbar L_z$ \quad $\comm{L_y}{L_z} = i\hbar L_x$ \quad $\comm{L_z}{L_x} = i\hbar L_y$\\ 159 | \tab $\comm{L_z}{x} = i\hbar y$ \quad $\comm{L_z}{y} = -i\hbar x$ \quad $\comm{L_z}{z} = 0$\\ 160 | \tab $\comm{L_z}{p_x} = i\hbar p_y$ \quad $\comm{L_z}{p_y} = -i\hbar p_x$ \quad $\comm{L_z}{p_z} = 0$ 161 | 162 | Square of Angular Momentum:\\ 163 | \tab $L^2 \equiv L_x^2 + L_y^2 + L_z^2$ \qquad $\comm{L^2}{\vb{L}} = 0$ 164 | 165 | Ladder Operators:\\ 166 | \tab $L_\pm = L_x \pm iL_y$ \qquad $\comm{L_z}{L_\pm} = \pm i\hbar L_\pm$\\ 167 | \tab $L^2 = L_\pm L_\mp + L_z^2 \mp \hbar L_z$\\ 168 | \tab $\ds L_x = \frac{L_+ + L_-}{2}$ \qquad $\ds L_y = \frac{L_+ - L_-}{2i}$\\ 169 | \tab $L_+ \ket{l,l} = 0$ \qquad $L_- \ket{l,-l} =0$ 170 | 171 | Eigenvalues:\\ 172 | \tab $L^2 \ket{l,m} = \hbar^2 l(l+1)\ket{l,m}$\\ 173 | \tab $L_z \ket{l,m} = \hbar m \ket{l,m}$\\ 174 | \tab $L_\pm \ket{l,m} = \hbar \sqrt{l(l+1) - m(m\pm 1)}\ket{l,m\pm1}$\\ 175 | 176 | Expectation Value when $L_x\ket{\psi} = L_y\ket{\psi}$:\\ 177 | \tab $\ds \bra{\psi}L_x^2\ket{\psi} = \bra{\psi}L_y^2\ket{\psi} = \frac{1}{2}\bra{\psi}L^2 - L_z^2\ket{\psi}$ 178 | 179 | Spherical Coordinate Representation:\\ 180 | \tab $L_x = \ds \frac{\hbar}{i}\left(-\sin\phi \pdv{\theta} - \cos\phi\cot\theta\pdv{\phi}\right)$\\ 181 | \tab $L_y = \ds \frac{\hbar}{i}\left(+\cos\phi \pdv{\theta} - \sin\phi\cot\theta\pdv{\phi}\right)$\\ 182 | \tab $\ds L_z = \frac{\hbar}{i}\pdv{\theta}$\\ 183 | \tab $\ds L_\pm = \pm\hbar e^{\pm i \theta}\left(\pdv{\theta} \pm i\cot\theta\pdv{\phi}\right)$\\ 184 | \tab $\ds L_+L_- = -\hbar^2\left(\pdv[2]{\theta}+\cot\theta\pdv{\theta} + \cot^2\theta\pdv[2]{\phi} + i\pdv{\phi}\right)$\\ 185 | \tab $\ds L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\pdv{\theta}\left(\sin\theta\pdv{\theta}\right) + \frac{1}{\sin^2\theta}\pdv[2]{\phi}\right]$ 186 | 187 | \section*{Spin} 188 | Commutators:\\ 189 | \tab $\comm{S_x}{S_y} = i\hbar S_z$ \quad $\comm{S_y}{S_z} = i\hbar S_x$ \quad $\comm{S_z}{S_x} = i\hbar S_y$\\ 190 | 191 | Eigenvalues:\\ 192 | \tab $S^2 \ket{s,m} = \hbar^2 s(s+1)\ket{s,m}$\\ 193 | \tab $S_z \ket{s,m} = \hbar m \ket{s,m}$\\ 194 | \tab $S_\pm \ket{s,m} = \hbar \sqrt{s(s+1) - m(m\pm 1)}\ket{s,m\pm1}$\\ 195 | 196 | Square of Angular Momentum:\\ 197 | \tab $S^2 \equiv S_x^2 + S_y^2 + S_z^2$ \qquad $\comm{S^2}{\vb{S}} = 0$ 198 | 199 | \section*{Spin 1/2} 200 | Spinors \& Eigenspinors:\\ 201 | \tab $\ds \chi = \begin{bmatrix}a \\ b\end{bmatrix},\, \abs{a}^2 + \abs{b}^2 = 1 \qquad \chi^z_+ = \begin{bmatrix}1 \\ 0\end{bmatrix} \qquad \chi^z_- = \begin{bmatrix}0 \\ 1\end{bmatrix}$\\ 202 | \tab $\ds \chi^x_+ = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ 1\end{bmatrix} \qquad \chi^x_- = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ -1\end{bmatrix}$\\ 203 | \tab $\ds \chi^y_+ = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ i\end{bmatrix} \qquad \chi^y_- = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ -i\end{bmatrix}$\\ 204 | 205 | Spin Operators:\\ 206 | \tab $\ds \vb{S}_x = \frac{\hbar}{2}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} \quad \vb{S}_y = \frac{\hbar}{2}\begin{bmatrix}0 & -i \\ i & 0 \end{bmatrix} \quad \vb{S}_z = \frac{\hbar}{2}\begin{bmatrix}1 & 0 \\ 0 & -1 \end{bmatrix}$\\ 207 | \tab $\ds \vb{S}^2 = \frac{3}{4}\hbar^2\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \quad \vb{S_+} = \hbar\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix} \quad \vb{S_-} = \hbar\begin{bmatrix}0 & 0 \\ 1 & 0 \end{bmatrix}$ 208 | 209 | Spin along $\vu{r}$:\\ 210 | \tab $\ds \vb{S}_r = \vb{S} \cdot \vu{r} = \frac{\hbar}{2}\begin{bmatrix}\cos\theta & e^{-i\phi}\sin\theta \\ e^{i\theta}\sin\theta & -\cos\theta\end{bmatrix}$\\ 211 | \tab $\ds \chi^r_+ = \begin{bmatrix}\cos(\theta/2) \\ e^{i\phi}\sin(\theta/2)\end{bmatrix} \qquad \chi^r_- = \begin{bmatrix} e^{-i\phi}\sin(\theta/2) \\ -\cos(\theta/2) \end{bmatrix}$ 212 | 213 | Lamor Precession:\\ 214 | \tab $\chi(t) = \begin{bmatrix}\cos(\alpha/2)e^{i\gamma B_0t/2} \\ \sin(\alpha/2)e^{-i\gamma B_0t/2}\end{bmatrix}$ 215 | 216 | \section*{The Hydrogen Atom} 217 | $n = j_{\text{max}} + l +1$ 218 | 219 | Wave Function: $\ds \psi_{nlm}(r, \theta, \phi) = R_{nl}(r)Y_l^m(\theta , \phi)$ 220 | 221 | Radial Equation:\\ 222 | \tab $\ds R_{nl}(r) = \frac{1}{r}\rho^{l+1}e^{-\rho}v(\rho) \qquad \ds \rho = \frac{r}{an}$\\ 223 | \tab $\ds v(\rho) = \sum_0^{j_\text{max}}c_j\rho^j \qquad c_{j+1} = \frac{2(j+l+1-n)}{(j+1)(j+2l+2)}c_j$ 224 | 225 | Energy:\\ 226 | \tab $\ds E_n = \frac{E_1}{n^2}$ \qquad $E_1 = -13.6$ eV 227 | 228 | \section*{Addition of Angular Momenta} 229 | Possible values of total spin $s$:\\ 230 | \tab $s = (s_1 + s_2),\, (s_1 + s_2 - 1), \,\ldots ,\, |s_1 - s_2|$\\ 231 | 232 | Operators:\\ 233 | \tab $S^2 = (S^{(1)})^2+ (S^{(2)})^2 + 2 \vb{S}^{(1)}\cdot \vb{S}^{(2)}$\\ 234 | \tab $2\vb{S}^{(1)}\cdot \vb{S}^{(2)} = S_+^{(1)}S_-^{(2)} + S_-^{(1)}S_+^{(2)} + S_z^{(1)}S_z^{(2)}$\\ 235 | \tab $\comm{S^2}{\vb{S}^{(1)}} = 2i\hbar (\vb{S}^{(1)} \cross \vb{S}^{(2)})$ 236 | 237 | Combined state with total spin $s$, and $z$-component $m$:\\ 238 | \tab $\ds \ket{s, m} = \sum_{m_1 + m_2 = m} C_{m_1m_2m}^{s_1s_2s}\ket{s_1, m_1}\ket{s_2, m_2}$\\ 239 | \tab $\ds \ket{s_1, m_1}\ket{s_2, m_2} = \sum_{s} C_{m_1m_2m}^{s_1s_2s} \ket{s, m}$ 240 | 241 | \section{Trig Identities} 242 | \tab $2\cos\theta\cos\phi = \cos(\theta - \phi) + \cos(\theta + \phi)$\\ 243 | \tab $2\sin\theta\sin\phi = \cos(\theta - \phi) - \cos(\theta + \phi)$\\ 244 | \tab $2\sin\theta\cos\phi = \sin(\theta + \phi) + \sin(\theta - \phi)$\\ 245 | \tab $2\cos\theta\sin\phi = \sin(\theta + \phi) - \sin(\theta - \phi)$\\ 246 | 247 | \columnbreak 248 | 249 | \section*{Integral Identities} 250 | \tab $\ds \int \sin ^{2}{ax}\,dx={\frac {x}{2}}-{\frac {1}{4a}}\sin 2ax+C$\\ 251 | \tab $\ds \int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}}-{\frac {3\cos ax}{4a}}+C$\\ 252 | \tab $\ds \int \cos ^{2}{ax}\,dx={\frac {x}{2}}+{\frac {1}{4a}}\sin 2ax+C$\\ 253 | \tab $\ds \int \cos^3{ax}\,dx = \frac{\sin ax}{a} - \frac{\sin^3{ax}}{3a} + C$\\ 254 | \tab $\ds \int \cos^{n}(ax)\sin(ax)\,dx = -\frac{\cos^{n+1}(ax)}{a(n+1)} + C$\\ 255 | \tab $\ds \int \sin^{n}(ax)\cos(ax)\,dx= \frac{\sin^{n+1}(ax)}{a(n+1)} + C$\\ 256 | \tab $\ds \int_0^\pi \cos^{2n+1}(ax)\sin(ax)\,dx = 0 \qquad n = 0, 1, 2 \ldots$\\ 257 | \tab $\ds \int_0^\pi \cos^{2n}(x)\sin(x)\,dx = \frac{2}{2n+1} \qquad n = 0, 1, 2 \ldots$\\ 258 | \tab $\ds \int_0^\pi \sin^{n}(ax)\cos(ax)\,dx = 0 \qquad n = 0, 1, 2 \ldots$\\ 259 | \tab $\ds \int_0^\infty x^n e^{-x/a}\,dx = n! a^{n+1}$\\ 260 | \tab $\ds \int_0^\infty x^{2n}e^{-x^2/a^2}\,dx=\sqrt{\pi}\frac{(2n)!}{n!}\left(\frac{a}{2}\right)^{2n+1}$\\ 261 | \tab $\ds \int_0^\infty x^{2n+1}e^{-x^2/a^2}\,dx = \frac{n!}{2}a^{2n+2}$ 262 | 263 | 264 | \renewcommand{\arraystretch}{1.7} 265 | \section*{Spherical Harmonics} 266 | \begin{tabular}{ l l l } 267 | \hline 268 | $l$ & $m$ & $Y_l^m(\theta, \phi)$\\ 269 | \hline 270 | 0 & 0 & $Y_0^0 = \sqrt{\frac{1}{4\pi}}$\\ 271 | 1 & 0 & $ Y_1^0 = \sqrt{\frac{3}{4\pi}}\cos\theta$\\ 272 | & $\pm 1$ & $Y_1^{\pm 1} = \mp \sqrt{\frac{3}{8\pi}}\sin\theta e^{\pm i\phi}$\\ 273 | 2 & 0 & $Y_2^0 = \sqrt{\frac{5}{16\pi}}(3\cos^2\theta - 1)$\\ 274 | & $\pm 1$ & $Y_2^{\pm 1}= \mp \sqrt{\frac{15}{8\pi}}\sin\theta\cos\theta e^{\pm i\phi}$\\ 275 | & $\pm 2$ & $ Y_2^{\pm 2}=\sqrt{\frac{15}{32\pi}}\sin^2\theta e^{\pm 2i\phi}$\\ 276 | 3 & 0 & $Y_3^0 = \sqrt{\frac{7}{16\pi}}(5\cos^3\theta - 3\cos\theta)$\\ 277 | & $\pm 1$ & $Y_3^{\pm 1} = \mp \sqrt{\frac{21}{64\pi}}\sin\theta(5\cos^2\theta - 1)e^{\pm i\phi}$\\ 278 | & $\pm 2$ & $Y_3^{\pm 2}\sqrt{\frac{105}{32\pi}}\sin^2\theta\cos\theta e^{\pm2 i\phi}$\\ 279 | & $\pm 3$ & $Y_3^{\pm 3}=\mp \sqrt{\frac{35}{64\pi}}\sin^3\theta e^{\pm 3i\phi}$ 280 | \end{tabular} 281 | 282 | \section*{Hydrogen Radial Wave Functions} 283 | \begin{tabular}{l} 284 | \hline 285 | $\ds R_{10} = 2a^{-3/2}e^{-r/a}$\\ 286 | \hline 287 | $\ds R_{20} = \frac{1}{\sqrt{2}}a^{-3/2}\left(1-\frac{1}{2}\frac{r}{a}\right)e^{-r/2a}$\\ 288 | $\ds R_{21} = \frac{1}{\sqrt{24}}a^{-3/2}\frac{r}{a}e^{-r/2a}$\\ 289 | \hline 290 | $\ds R_{30} = \frac{2}{\sqrt{27}}a^{-3/2}\left(1-\frac{2}{3}\frac{r}{a}+\frac{2}{27}\left(\frac{r}{a}\right)^2\right)e^{-r/3a}$\\ 291 | $\ds R_{31} = \frac{8}{27\sqrt{6}}a^{-3/2}\left(1-\frac{1}{6}\frac{r}{a}\right)\left(\frac{r}{a}\right)e^{-r/3a}$\\ 292 | $\ds R_{32} = \frac{4}{81\sqrt{30}}a^{-3/2}\left(\frac{r}{a}\right)^2 e^{-r/3a}$ 293 | \end{tabular} 294 | 295 | % Footer content 296 | \rule{0.3\linewidth}{0.25pt} 297 | \scriptsize\\ 298 | Updated \today\\ 299 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 300 | \end{multicols} 301 | \end{document} 302 | -------------------------------------------------------------------------------- /ELEC 221 - Signals and Systems/elec221fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape,letterpaper]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage[top=.25in,left=.25in,right=.25in,bottom=.25in]{geometry} 5 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 6 | \usepackage{hyperref} 7 | \usepackage{enumitem} 8 | \usepackage{upgreek} 9 | \usepackage{physics} 10 | \usepackage{mathtools} 11 | \usepackage{newtxtext,newtxmath} 12 | \usepackage{booktabs} 13 | \usepackage{xfrac} 14 | 15 | % Turn off header and footer 16 | \pagestyle{empty} 17 | 18 | 19 | % Redefine section commands to use less space 20 | \makeatletter 21 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 22 | {-1ex plus -.5ex minus -.2ex}% 23 | {0.5ex plus .2ex}%x 24 | {\normalfont\normalsize\bfseries}} 25 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 26 | {-1explus -.5ex minus -.2ex}% 27 | {0.5ex plus .2ex}% 28 | {\normalfont\small\bfseries}} 29 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 30 | {-1ex plus -.5ex minus -.2ex}% 31 | {1ex plus .2ex}% 32 | {\normalfont\footnotessize\bfseries}} 33 | \makeatother 34 | 35 | % Don't print section numbers 36 | \setcounter{secnumdepth}{0} 37 | 38 | \setlength{\parindent}{0pt} 39 | \setlength{\parskip}{1pt plus 0.5ex} 40 | 41 | \newcommand{\tab}{\hspace{0.02\textwidth}} 42 | \newcommand{\tabm}{\hspace*{0.07\textwidth}} 43 | \newcommand{\ds}{\displaystyle} 44 | 45 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 46 | \DeclareMathOperator{\sinc}{sinc} 47 | \DeclareMathOperator{\rect}{rect} 48 | 49 | % ----------------------------------------------------------------------- 50 | 51 | \begin{document} 52 | 53 | \raggedright 54 | \footnotesize 55 | \begin{multicols*}{3} 56 | 57 | 58 | % multicol parameters 59 | % These lengths are set only within the two main columns 60 | %\setlength{\columnseprule}{0.25pt} 61 | \setlength{\premulticols}{1pt} 62 | \setlength{\postmulticols}{1pt} 63 | \setlength{\multicolsep}{1pt} 64 | \setlength{\columnsep}{2pt} 65 | 66 | \begin{center} 67 | \Large{\underline{ELEC 221 Formula Sheet}} 68 | \end{center} 69 | 70 | \section{Continuous Time Signals} 71 | 72 | Even and Odd Components:\\ 73 | \tab $\ds x_e(t) = \frac{1}{2}[x(t) + x(-t)]$\\ 74 | \tab $\ds x_o(t) = \frac{1}{2}[x(t) - x(-t)]$ 75 | 76 | A signal is periodic with fundamental period $T_0$ if\\ 77 | \tab $x(t + kT_0) = x(t) \quad \forall t \in (-\infty, \infty)$ 78 | 79 | Energy:\\ 80 | \tab $\ds E = \int_{-\infty}^{\infty} \abs{x(t)}^2\,dt$ 81 | 82 | Power:\\ 83 | \tab $\ds P = \lim_{T \rightarrow \infty} \frac{1}{2T} \int_{-T}^{T}\abs{x(t)}^2\,dt$ 84 | 85 | Causality:\\ 86 | \begin{itemize} 87 | \itemsep0em 88 | \item Causal: $x(t) = 0$ for $t < 0$. 89 | \item Anti-Causal: $x(t) = 0$ for $t \geq 0$. 90 | \item A-Causal or Non-Causal: Both of the above. 91 | \end{itemize} 92 | 93 | \section{Continuous Time Systems} 94 | Dynamic systems have memory. Active systems can deliver energy to the outside world. 95 | \\ 96 | Linearity:\\ 97 | \tab $\mathcal{S}[\alpha x(t) + \beta y(t)] = \alpha\mathcal{S}[x(t)] + \beta\mathcal{S}[y(t)]$ 98 | 99 | Time Invariance:\\ 100 | \tab If $\mathcal{S}[x(t)] = y(t)$ then $\mathcal{S}[x(t \pm \tau)] = y(t \pm \tau)$. 101 | 102 | Zero-State Response:\\ 103 | \tab Due to the input as the initial conditions are zero. 104 | 105 | Zero-Input Response:\\ 106 | \tab Due to the initial conditions as the input is zero. 107 | 108 | Convolution:\\ 109 | \tab $\ds y(t) = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)\,d\tau$ 110 | 111 | Causality:\\ 112 | \tab A continuous time system $\mathcal{S}$ is causal if whenever $x(t) = 0$ and there are no initial conditions, $y(t) = 0$ and the output $y(t)$ does not depend on future inputs. 113 | \\~\\ 114 | Bounded-Input Bounded-Output Stability:\\ 115 | \tab If an input $x(t)$ bounded then the output of an BIBO system is also bounded.\\ 116 | \tab $\ds \int_{-\infty}^{\infty} \abs{h(t)}\,dt < \infty$ 117 | 118 | \section{Laplace Transform} 119 | $s = \sigma + j\omega$\\ 120 | 121 | Eigenfunction Property:\\ 122 | \tab $\mathcal{S}[e^{s_0t}] = H(s_0)e^{s_0t}$ 123 | 124 | One Sided Laplace Transform:\\ 125 | \tab $\ds F(s) = \mathcal{L}[f(t)u(t)] = \int_{0^-}^{\infty}f(t)e^{-st}\,dt$\\ 126 | \tab $\abs{H(s)}$ is greatest when $\omega$ is closest to the poles. 127 | 128 | \begin{tabular}{ll} 129 | \toprule 130 | Signal Support & ROC\\ 131 | \midrule 132 | Finite support & Entire s-plane.\\ 133 | Causal function & $\sigma > \max(\sigma_i)$, $-\infty < \omega < \infty$\\ 134 | Anti-causal & $\sigma < \min(\sigma_i)$, $-\infty < \omega < \infty$\\ 135 | Non-causal & $\mathcal{R} = \mathcal{R}_\text{causal} \cap \mathcal{R}_\text{anti-causal}$\\ 136 | \bottomrule 137 | \end{tabular} 138 | \vspace{0.5em} 139 | 140 | Initial Value Theorem:\\ 141 | \tab $\ds f(0+) \Leftrightarrow \lim_{s\rightarrow\infty} sF(s)$ 142 | 143 | Final Value Theorem:\\ 144 | \tab $\ds \lim_{t\rightarrow\infty} f(t) \Leftrightarrow \lim_{s\rightarrow 0} sF(s)$ 145 | 146 | Bounded-Input Bounded-Output Stability:\\ 147 | \tab If the region of convergence contains the $j\omega$-axis, then the system is BIBO stable. 148 | 149 | \section{Fourier Series} 150 | \tab Fourier analysis in the steady state. 151 | 152 | Eigenfunction Property:\\ 153 | \tab $\mathcal{S}[e^{j\omega_0t}] = H(j\omega_0)e^{j\omega_0t}$\\ 154 | \tab $x(t) = \sum_k X_k e^{j\omega_kt} \implies y(t) = \sum_k X_k H(j\omega_k)e^{j\omega_kt}$\\\vspace{1mm}\hspace{2.5cm}$= \sum_k X_k \abs{H(j\omega_k)}e^{j(\omega_kt + \angle H(\omega_k))}$ 155 | 156 | Fourier Series Coefficients (for any $t_0$):\\ 157 | \tab $\ds X_k = \frac{1}{T_0}\int_{t_0}^{t_0 + T_0}x(t)e^{-jk\omega_0t}\,dt$ 158 | 159 | Parseval's Power Relation (for any $t_0$):\\ 160 | \tab $\ds P = \frac{1}{T_0} \int_{t_0}^{t_0 + T_0}\abs{x(t)}^2\,dt = \sum_{k = -\infty}^{\infty}\abs{X_k}^2$ 161 | 162 | Symmetry of Line Spectra:\\ 163 | \tab $\abs{X_k} = \abs{X_{-k}}$\\ 164 | \tab $\angle X_k = -\angle X_{-k}$ 165 | 166 | Trigonometric Fourier Series:\\ 167 | \tab $x(t) = X_0 + 2\sum_{k = 1}^{\infty}\abs{X_k}\cos(k\omega t + \Theta_k)$\\ 168 | \tab $x(t) = c_0 + 2\sum_{k = 1}^{\infty}[c_k\cos(k\omega_0 t) + d_k\sin(k\omega_0 t)]$\\ 169 | \tab $c_k = \frac{1}{T_0} \int_{t_0}^{t_0 + T_0}x(t)\cos(k\omega_0 t)\,dt$ \quad $k$ = 0,1,2$\ldots$\\ 170 | \tab $d_k = \frac{1}{T_0} \int_{t_0}^{t_0 + T_0}x(t)\sin(k\omega_0 t)\,dt$\quad $k$ = 1,2,3$\ldots$\\ 171 | \tab $\Theta_k = -\arctan(d_k/c_k)$ 172 | 173 | \columnbreak 174 | Fourier Coefficients from Laplace Transform:\\ 175 | \tab If $x_1(t)$ is a single period of $x(t)$, then\\ 176 | \tab $X_k = \frac{1}{T_0}\mathcal{L}[x_1(t)] \big\rvert_{s = jk\omega_0}$ 177 | 178 | Response of LTI Systems to Periodic Signals:\\ 179 | \tab If the input to an LTI system has Fourier Series $x(t) = X_0 + 2\sum_{k=1}^{\infty}\abs{X_k}\cos(k\omega t + \angle X_k)$, then the steady state response is $\ds y(t) = X_0\abs{H(j0)} + 2\sum_{k=1}^{\infty}\abs{X_k}\abs{H(jk\omega_0)}\cos(k\omega_0 t + \angle X_k + \angle H(jk\omega_0))$ 180 | 181 | \section{Fourier Transform} 182 | \tab $X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt$\\ 183 | \tab $x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}\,d\omega$ 184 | 185 | Fourier Transform from Laplace Transform (if $X(s)$ contains the $j\omega$-axis):\\ 186 | \tab $\mathcal{F}[x(t)] = \mathcal{L}[x(t)]\big\rvert_{s = j\omega} = X(s)\big\rvert_{s = j\omega}$ 187 | 188 | Fourier Transform of Periodic Signals:\\ 189 | \tab $\ds \sum_k X_k e^{jk\omega_0 t} \xLeftrightarrow{\mathcal{F}} \sum_k 2\pi X_k \delta(\omega - k\omega_0)$ 190 | 191 | Parseval's Energy Relation:\\ 192 | \tab $\ds E = \int_{-\infty}^{\infty} \abs{x(t)}^2\,dt = \frac{1}{2\pi}\int_{-\infty}^{\infty}\abs{X(\omega)}^2\,d\omega$ 193 | 194 | Symmetry of Spectral Representations:\\ 195 | \tab $\abs{X(\omega)} = \abs{X(-\omega)}$\\ 196 | \tab $\Re[X(\omega)] = \Re[X(-\omega)]$\\ 197 | \tab $\angle X(\omega) = -\angle X(-\omega)$\\ 198 | \tab $\Im[X(\omega)] = -\Im[X(-\omega)]$ 199 | 200 | \section{Sampling Theory} 201 | \tab $x_s(t) = x(nT_s) = x(t)\rvert_{t = nT_s} = x(t)\sum_n \delta(t - nT_s)$\\ 202 | \tab $\ds X_s(\omega) = \frac{1}{T_s} \sum_{k=-\infty}^{\infty}X(\omega - k\omega_s)$ 203 | 204 | Nyquist-Shannon Sampling Rate:\\ 205 | \tab $\omega_s = \frac{2\pi}{T_s} \geq 2\omega_\text{max}$\\ 206 | \tab Aliasing occurs if $\omega_s < 2\omega_\text{max}$. 207 | 208 | Reconstruction $X(\omega)=X_s(\omega)H_\text{lp}(\omega)$: 209 | \vspace{-1em} 210 | \begin{equation*} 211 | \hspace{-8em} 212 | H_{\text{lp}}(\omega) = 213 | \begin{cases} 214 | T_s & \ds -\frac{\omega_s}{2} \leq \omega \leq \frac{\omega_s}{2}\\ 215 | 0 & \text{otherwise} 216 | \end{cases} 217 | \end{equation*} 218 | \tab (Inclusive bounds. If represented by a unit step function the bounds are inclusive for filters (Refer to lecture notes)) 219 | 220 | Signal Reconstruction from Sinc Interpolation:\\ 221 | \tab $\ds x_r(t) = \sum_{n=-\infty}^{\infty} x(nT_s) \frac{\sin(\pi(t-nT_s)/T_s)}{\pi(t-nT_s)/T_s}$ 222 | \end{multicols*} 223 | 224 | \newpage 225 | 226 | \begin{minipage}[t]{0.27\textwidth} 227 | \section{Discrete Time Signals} 228 | \tabm Define $x[n] = x(nT_0)$.\\ 229 | \tabm $x[n] = \sum_{k=-\infty}^{\infty}x[k]\delta[n-k]$ 230 | 231 | Periodicity:\\ 232 | \tabm $x[n + kN] = x[n]$ \qquad $\forall k\in \mathbb{Z}$ 233 | 234 | When sampling an analog sinusoid of fundamental period $T_0$, we obtain a periodic discrete sinusoid provided that $m$, $N$ not divisible by each-other:\\ 235 | \tabm $T_s/T_0 = m/N$ 236 | 237 | Aliasing occurs if:\\ 238 | \tabm $T_s > T_0/2$ 239 | 240 | Energy:\\ 241 | \tabm $\ds E = \sum_{n = -\infty}^{\infty}\abs{x[n]}^2$ 242 | 243 | Power:\\ 244 | \tabm $\ds P_x = \lim_{N \rightarrow \infty}\frac{1}{2N + 1}\sum_{n= -N}^{N}\abs{x[n]}^2$ 245 | 246 | Convolutional Sum:\\\ 247 | \tabm $\ds y[n] = \sum_{k = -\infty}^{\infty} x[k]h[n-k]$ 248 | 249 | Bounded-Input Bounded-Output Stability:\\ 250 | \tabm $\sum_k \abs{h[n]} < \infty$ 251 | 252 | Solution to Autoregressive Discrete System:\\ 253 | \tabm $y[n] = ay[n-1] + bx[n]$, $n \geq 0$\\ 254 | \tabm $y[n] = \sum_{k=0}^{n} ba^kx[n-k]$, $n\geq 0$ 255 | 256 | \section{Z-Transform} 257 | \tabm $\mathcal{Z}[x(nT_s)] = \mathcal{L}[x_s(t)]\rvert_{z=e^{sT_s}} = \sum_n x(nT_s)z^{-n}$ 258 | 259 | Convergence:\\ 260 | \tabm $|X(z)| = \sum_n \abs{x[n]}r^{-n} < \infty$ 261 | 262 | Initial Value Theorem:\\ 263 | \tabm $\ds x[0] = \lim_{z\rightarrow \infty}X(z)$ 264 | 265 | Final Value Theorem:\\ 266 | \tabm $\ds \lim_{n\rightarrow \infty}x[n] = \lim_{z\rightarrow 1}(z-1)X(z)$ 267 | 268 | BIBO Stability:\\ 269 | \tabm If the ROC contains radius $z=1$, then the system is BIBO stable. 270 | 271 | \section{Misc. Identities} 272 | \tabm $\ds \sum_{k=0}^{n}x^k = \frac{1-x^{n+1}}{1-x} \quad\xrightarrow{n\rightarrow\infty \text{ and } x < 1}\quad \frac{1}{1-x}$\\ 273 | \tabm $\ds \cos\theta = \frac{e^{j\theta}+e^{-j\theta}}{2}$\\ 274 | \tabm $\ds \sin\theta = \frac{e^{j\theta}-e^{-j\theta}}{2j}$ 275 | 276 | % Footer content 277 | \rule{0.3\linewidth}{0.25pt} 278 | \scriptsize\\ 279 | Updated \today\\ 280 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 281 | \end{minipage}% 282 | \hspace{0.5em} 283 | \begin{minipage}[t]{0.35\textwidth} 284 | \subsection{Partial Fraction Decomposition} 285 | \begin{tabular}{lll} 286 | \toprule 287 | Fraction & Partial Fraction & Solution\\ 288 | \midrule\vspace{1mm} 289 | $\ds \frac{px + q}{(x-a)(x-b)}$ & $\ds \frac{A}{x-a} + \frac{B}{x-b}$ & $\ds A = \dfrac{pa + q}{a-b} \quad B =\dfrac{pb + q}{b-a}$\\\vspace{1mm} 290 | $\ds \frac{px + q}{(x-a)^2}$ & $\ds \frac{A}{x-a} + \frac{B}{(x-a)^2}$ & $\ds A = p \qquad\quad B = pa + q$\\\vspace{1mm} 291 | $\ds \frac{px^2 + qx + r}{(x-a)(x-b)(x-c)}$ & $\ds \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c}$ & $A=\dfrac{pa^2 + qa +r}{(a-b)(a-c)} \quad B =\dfrac{pb^2 + qb +r}{(b-a)(b-c)} \quad C =\dfrac{pc^2 + qc +r}{(c-a)(c-b)}$\\\vspace{1mm} 292 | $\ds \frac{px^2 + qx + r}{(x-a)^2(x-b)}$ & $\ds \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$ & \\\vspace{1mm} 293 | $\ds \frac{px^2 + qx + r}{(x-a)(x^2 + bx + c)}$ & $\ds \frac{A}{x-a} + \frac{Bx + C}{x^2 + bx + c}$ &\\ 294 | \bottomrule 295 | \end{tabular} 296 | \vspace{0.5em} 297 | \subsection{Two-Sided Z-Transforms} 298 | \begin{tabular}{lll} 299 | \toprule 300 | $f[n]$ & $F(z)$ & ROC\\ 301 | \midrule\vspace{1mm} 302 | $-u[-n-1]$ & $\ds \frac{1}{1-z^{-1}}$ & $\abs{z} < 1$\\\vspace{1mm} 303 | $-\alpha^nu[-n-1]$ & $\ds \frac{1}{1-\alpha z^{-1}}$ & $\abs{z} < \abs{\alpha}$\\\vspace{1mm} 304 | $-n\alpha^nu[-n-1]$ & $\ds \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^2}$ & $\abs{z} < \abs{\alpha}$\\\vspace{1mm} 305 | $\alpha^\abs{n}$, $\abs{\alpha} < 1$ & $\ds \frac{1}{1-\alpha z^{-1}} - \frac{1}{1-\alpha^{-1} z^{-1}}$ & $\abs{\alpha} < z < \abs{\frac{1}{\alpha}}$\\ 306 | \bottomrule 307 | \end{tabular} 308 | \vspace{0.5em} 309 | \subsection{Interconnection of LTI Systems} 310 | \begin{tabular}{lll} 311 | \toprule 312 | Connection & Time & Laplace/Z\\ 313 | \midrule\vspace{1mm} 314 | Series & $[h_1 * h_2](t)$ & $H_1(s)H_2(s)$\\ 315 | Parallel & $h_1(t) + h_2(t)$ & $H_1(s) + H_2(s)$\\ 316 | \bottomrule 317 | \end{tabular} 318 | 319 | 320 | \end{minipage} 321 | \begin{minipage}[t]{0.30\textwidth} 322 | \vspace{18em} 323 | \subsection{Finding Magnitudes} 324 | \tabm If $\ds H(s) = \frac{N(s)}{D(s)}$, then $\ds \abs{H(s)} = \frac{\abs{N(s)}}{\abs{D(s)}} = \frac{\sqrt{(N(s))^2}}{\sqrt{(D(s))^2}}$\\ 325 | and $\angle H(s) = \angle N(s) - \angle D(s)$ 326 | 327 | Complex Valued:\\ 328 | \tabm $\abs{H(s)}^2 = H(s)H(s)^*$ 329 | 330 | \subsection{Long Division For Z-Transform Inverse} 331 | Let $X(z) = B(z)/A(z)$ be a rational Z-Transform of a causal signal $x[n]$. We can then express $X(z)$ as 332 | $$X(z) = x[0] + x[1]z^{-1} + x[2]z^{-2} + \cdots$$ 333 | by dividing $B(z)$ by $A(z)$. To perform the long division, we let $X(z)$ be the series of $z^{-n}$ as above, and multiply both sides throiugh by $A(z)$ to obtain $A(z)X(z) = B(z)$. We then compare the coefficients in front of $z^{-n}$ of both sides to obtain values for the sequence $\{x[0], x[1], x[2], x[3], \ldots\}$. Then the inverse can be taken by inspection. 334 | 335 | \subsection{Fourier Transform of a Periodic Signal} 336 | Suppose $x(t)$ is periodic of period $T_0$. Let $x_p(t)$ be a single period of $x(t)$. Then $x(t) = \sum_k x_p(t - kT_0)$ and we can find $X(\omega)$ by using time shifting property and finding the Fourier Transform of $x_p(t)$ by using Laplace. 337 | \end{minipage} 338 | 339 | \end{document} 340 | -------------------------------------------------------------------------------- /PHYS 301 - Electricity and Magnetism/phys301fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{upgreek} 11 | \usepackage{physics} 12 | \usepackage{newtxtext,newtxmath} 13 | 14 | % This sets page margins to .5 inch if using letter paper, and to 1cm 15 | % if using A4 paper. (This probably isn't strictly necessary.) 16 | % If using another size paper, use default 1cm margins. 17 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 18 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 19 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 22 | } 23 | 24 | % Turn off header and footer 25 | \pagestyle{empty} 26 | 27 | 28 | % Redefine section commands to use less space 29 | \makeatletter 30 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 31 | {-1ex plus -.5ex minus -.2ex}% 32 | {0.5ex plus .2ex}%x 33 | {\normalfont\normalsize\bfseries}} 34 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 35 | {-1explus -.5ex minus -.2ex}% 36 | {0.5ex plus .2ex}% 37 | {\normalfont\small\bfseries}} 38 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 39 | {-1ex plus -.5ex minus -.2ex}% 40 | {1ex plus .2ex}% 41 | {\normalfont\footnotessize\bfseries}} 42 | \makeatother 43 | 44 | % Define BibTeX command 45 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 46 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 47 | 48 | % Don't print section numbers 49 | \setcounter{secnumdepth}{0} 50 | 51 | 52 | \setlength{\parindent}{0pt} 53 | \setlength{\parskip}{1pt plus 0.5ex} 54 | 55 | \newcommand{\tab}{\hspace{.02\textwidth}} 56 | \newcommand{\ds}{\displaystyle} 57 | 58 | \def\rcurs{{\mbox{$\resizebox{.09in}{.08in}{\includegraphics[trim= 1em 0 14em 0,clip]{ScriptR}}$}}} 59 | \def\brcurs{{\mbox{$\resizebox{.09in}{.08in}{\includegraphics[trim= 1em 0 14em 0,clip]{BoldR}}$}}} 60 | 61 | \renewcommand{\dv}[2]{\frac{d#1}{d#2}} 62 | 63 | % Redefine some commands for newtxmath boldness 64 | \renewcommand{\grad}{\nabla} 65 | \renewcommand{\curl}[1]{\nabla\times#1} 66 | \renewcommand{\div}[1]{\nabla\cdot#1} 67 | \renewcommand{\cross}{\times} 68 | 69 | % ----------------------------------------------------------------------- 70 | 71 | \begin{document} 72 | 73 | \raggedright 74 | \footnotesize 75 | \begin{multicols}{3} 76 | 77 | 78 | % multicol parameters 79 | % These lengths are set only within the two main columns 80 | %\setlength{\columnseprule}{0.25pt} 81 | \setlength{\premulticols}{1pt} 82 | \setlength{\postmulticols}{1pt} 83 | \setlength{\multicolsep}{1pt} 84 | \setlength{\columnsep}{2pt} 85 | 86 | \begin{center} 87 | \Large{\underline{PHYS 301 Formula Sheet}} 88 | \end{center} 89 | 90 | \section{Differential Maxwell's Equations} 91 | \hspace{3mm}\begin{tabular}{p{2cm}p{4cm}} 92 | $\ds \div{\vb{E}} = \frac{\rho}{\epsilon_0}$ & $\ds \curl{\vb{E}} = -\pdv{\vb{B}}{t}$\\ 93 | $\ds \div{\vb{B}} = 0$ & $\ds \curl{\vb{B}} = \mu_0\vb{J} + \mu_0\epsilon_0\pdv{\vb{E}}{t}$ 94 | \end{tabular} 95 | 96 | \section{Integral Maxwell's Equations} 97 | 98 | \begin{tabular}{p{2.75cm}p{5cm}} 99 | $\ds \oint \vb{E}\cdot d\vb{a} = \frac{Q_\text{enc}}{\epsilon_0}$ & $\ds \oint \vb{B}\cdot d\vb{a} = 0$\\ 100 | $\ds \oint \vb{E}\cdot d\vb{l} = -\dv{\Phi_B}{t}$ & $\ds \oint \vb{B}\cdot d\vb{l} = \mu_0 I_\text{enc} + \epsilon_0\mu_0 \dv{\Phi_E}{t}$ 101 | \end{tabular} 102 | \section{Electrostatics} 103 | \subsection{The Electric Field} 104 | Coulomb's Law:\\ 105 | \tab $\ds \vb{F} = \frac{1}{4\pi\epsilon_0} \frac{qQ}{\rcurs} \vu{\brcurs}$ 106 | 107 | The Electric Field:\\ 108 | \tab $\vb{F} = Q \vb{E}$ \qquad $\vb{E} = -\grad{V}$ 109 | 110 | Electric Field due to discrete point charges:\\ 111 | \tab $\ds \vb{E(\vb{r})} = \frac{1}{4\pi\epsilon_0} \sum_{i = 1}^{n} \frac{q_i}{\rcurs_i^2} \vu{\brcurs}_i$ 112 | 113 | Electric Field due to a continuous charge distribution:\\ 114 | \tab $\ds \vb{E(\vb{r})} = \frac{1}{4\pi\epsilon_0} \int \frac{\vu{\brcurs}}{\rcurs^2}\,dq$ 115 | 116 | \subsection{Electric Potential} 117 | \tab $\ds V(b) - V(a) = - \int_{a}^{b} \vb{E} \cdot d\vb{l}$ 118 | 119 | Poisson's Equation:\\ 120 | \tab $\ds \nabla^2 V = -\rho/\epsilon_0$ 121 | 122 | Potential due to a localized charge distribution:\\ 123 | \tab $\ds V(\vb{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vb{r'})}{\rcurs}\,dV$ 124 | 125 | \subsection{Work and Energy in Electrostatics} 126 | Energy stored in a point charge distribution:\\ 127 | \tab $\ds W = \frac{1}{2} \sum_{i=1}^{n} q_i V(\vb{r}_i)$ 128 | 129 | Energy of a continuous charge distribution:\\ 130 | \tab $\ds W = \frac{1}{2} \int \rho V\,d\uptau$ 131 | 132 | Total energy of a continuous charge distribution:\\ 133 | \tab $\ds W = \frac{\epsilon_0}{2} \int E^2\,d\uptau$ \quad (all space) 134 | 135 | \subsection{Conductors} 136 | \tab $\vb{E} = \vb{0}$ inside a conductor.\\ 137 | 138 | Electric Field immediately outside a conductor:\\ 139 | \tab $\ds \vb{E} = \frac{\sigma}{\epsilon_0} \vu{\vb{n}}$ 140 | 141 | Surface charge:\\ 142 | \tab $\ds \sigma = -\epsilon_0 \pdv{V}{n}$ 143 | 144 | Capacitors:\\ 145 | \tab $Q = CV$ 146 | \\ 147 | \tab $\ds W = \frac{1}{2}CV^2 = \frac{1}{2}\frac{Q^2}{C} = \frac{1}{2}QV$ 148 | 149 | \section{Potentials} 150 | Laplace's Equation:\\ 151 | \tab $\ds \nabla^2 V = \pdv[2]{V}{x} + \pdv[2]{V}{y} + \pdv[2]{V}{z}= 0$ 152 | 153 | \subsection{Separation of Variables} 154 | \begin{equation*} 155 | \hspace{-1cm}\int_0^a \sin(\frac{n\pi x}{a})\sin(\frac{m\pi x}{a})\,dx = 156 | \begin{cases} 157 | 0 & \text{if } n \neq m\\ 158 | a/2 & \text{if } n = m 159 | \end{cases} 160 | \end{equation*} 161 | 162 | Legendre Polynomials: 163 | \begin{itemize}[noitemsep,topsep=0pt] 164 | \item $P_0(x) = 1$ 165 | \item $P_1(x) = x$ 166 | \item $P_2(x) = (3x^2 - 1)/2$ 167 | \item $P_3(x) = (5x^3 - 3x)/2$ 168 | \end{itemize} 169 | 170 | Solution to Laplace in spherical ($\phi$ independent):\\ 171 | \tab $\ds V(r,\theta) = \sum_{l = 0}^{\infty}\left(A_l r^l + \frac{B_l}{r^{l + 1}}\right)P_l(\cos\theta)$ 172 | 173 | Solution to Laplace in cylindrical ($z$ independent):\\ 174 | \tab $\ds V(s,\phi) = A_0\ln(s) + B_0 +$\\\tab$\ds \sum_{n = 1}^{\infty}(A_n s^{-n} + B_ns^{-n})(C_n\cos(n\phi) + D_n\sin(n\phi))$ 175 | 176 | \subsection{Multipole Expansion} 177 | Potential at large distances ($\alpha$ is between $\vb{r}$ and $\vb{r'}$):\\ 178 | \tab $\ds V(\vb{r}) = \frac{1}{4\pi\epsilon_0}\sum_{n = 0}^{\infty}\frac{1}{r^{n + 1}}\int (r')^n P_n(\cos\alpha)\rho(\vb{r'})\,d\uptau$ 179 | 180 | Dipole Moment:\\ 181 | \tab $\ds \vb{p} = \sum_{i=1}^{n}q_i \vb{r'}_i = \int \vb{r'}\rho(\vb{r'})\,d\uptau'$ 182 | 183 | Electric Dipole Potential:\\ 184 | \tab $\ds V_{\text{dip}}(\vb{r}) = \frac{1}{4\pi\epsilon_0} \frac{\vb{p}\cdot\vu{r}}{r^2}$ 185 | 186 | \section{Electric Fields in Matter} 187 | 188 | Bound Charges:\\ 189 | \tab $\sigma_b = \vb{P}\cdot \vu{n} \qquad \rho_b = - \div{\vb{P}}$ 190 | 191 | The Electric Displacement:\\ 192 | \tab $\vb{D} = \epsilon_0\vb{E} + \vb{P}$ \quad\, $\div{\vb{D}} = \rho_f$ \quad\, $\ds \oint \vb{D} \cdot d\vb{a} = Q_{f,\text{enc}}$ 193 | 194 | \subsection{Linear Dielectrics} 195 | Polarization:\\ 196 | \tab $\vb{P} = \epsilon_0\chi_e \vb{E}$ 197 | 198 | Electric Displacement:\\ 199 | \tab $\vb{D} = \epsilon_0(1 + \chi_e)\vb{E} = \epsilon\vb{E}$ 200 | 201 | Energy in a Dielectric System:\\ 202 | \tab $\ds W = \frac{1}{2}\int \vb{D}\cdot\vb{E}\,d\uptau$ 203 | 204 | \subsection{Boundary Conditions in Electrostatics} 205 | \begin{itemize} 206 | \itemsep0em 207 | \item $D_{\text{above}}^{\perp} - D_{\text{below}}^{\perp} = \sigma_f$ 208 | \item $\vb{D}_{\text{above}}^{\parallel} - \vb{D}_{\text{below}}^{\parallel} = \vb{P}_{\text{above}}^{\parallel} - \vb{P}_{\text{below}}^{\parallel}$ 209 | \item $E_{\text{above}}^{\perp} - E_{\text{below}}^{\perp} = \sigma_f/\epsilon_0$ 210 | \item $\vb{E}_{\text{above}}^{\parallel} - \vb{E}_{\text{below}}^{\parallel} = \vb{0}$ 211 | \item $V_\text{above} = V_\text{below}$ 212 | \item $\epsilon_\text{above}E_\text{above}^\perp - \epsilon_\text{below}E_\text{below}^\perp = \sigma_f$ 213 | \item $\ds \epsilon_\text{above}\pdv{V_\text{above}}{n} - \epsilon_\text{below}\pdv{V_\text{below}}{n} = -\sigma_f$ 214 | \end{itemize} 215 | 216 | \section{Magnetostatics} 217 | Lorentz Force Law:\\ 218 | \tab $\vb{F} = q(\vb{E}+\vb{v}\cross\vb{B})$ 219 | 220 | Currents:\\ 221 | \tab $\vb{I} = \lambda \vb{v}$ \qquad $\vb{K} = \sigma\vb{v}$ \qquad $\vb{J} = \rho\vb{v}$ 222 | 223 | Continuity Equation:\\ 224 | \tab $\ds \grad \cdot \vb{J} = - \pdv{\rho}{t}$ 225 | 226 | Biot-Savart Law:\\ 227 | \tab $\ds \vb{B}(\vb{r}) = \frac{\mu_0}{4\pi}\int \frac{\vb{I}\cross\vu{\brcurs}}{\rcurs^2}\,dl' = \frac{\mu_0I}{4\pi}\int \frac{d\vb{l'}\cross\vu{\brcurs}}{\rcurs^2}$ 228 | 229 | \subsection{Magnetic Vector Potential} 230 | \tab $\vb{B} = \curl{\vb{A}}$ \quad where \quad $\div{\vb{A}} = 0$ 231 | 232 | Vector Potential Poisson's Equation:\\ 233 | \tab $\nabla^2\vb{A} = -\mu_0 \vb{J}$ 234 | 235 | Vector Potential when $\vb{J} \rightarrow \vb{0}$ at infinity:\\ 236 | \tab $\ds \vb{A}(\vb{r}) = \frac{\mu_0}{4\pi}\int \frac{\vb{J}(\vb{r'})}{\rcurs}\,d\uptau'$ 237 | 238 | Multipole Expansion of a current loop:\\ 239 | \tab $\ds \vb{A}(\vb{r}) = \frac{\mu_0}{4\pi} \sum_{n = 0}^{\infty} \frac{1}{r^{n+1}} \oint (r')^n P_n(\cos\alpha)\,d\vb{l'}$\\ 240 | \tab $\ds \vb{A}(\vb{r}) = \frac{\mu_0}{4\pi} \frac{\vb{m}\times\vu{r}}{r^2}$ 241 | 242 | Magnetic Dipole Moment for a vector area $\vb{a}$:\\ 243 | \tab $\ds \vb{m} = I\int d\vb{a} = I\vb{a}$ 244 | 245 | \section{Magnetic Fields in Matter} 246 | Bound Currents:\\ 247 | \tab $\vb{J}_B = \curl{\vb{M}}$ \qquad $\vb{K}_B = \vb{M}\cross\vb{\hat{n}}$ 248 | 249 | Auxiliary Field:\\ 250 | \tab $\ds \vb{H} = \frac{\vb{B}}{\mu_0} - \vb{M}$ \quad\, $\curl{\vb{H}} = \vb{J}_f$ \quad\, $\ds \oint \vb{H}\cdot d\vb{l} = I_\text{free}$ 251 | 252 | \subsection{Linear Media} 253 | Magnetization in linear media:\\ 254 | \tab $\vb{M} = \chi_m \vb{H}$ 255 | 256 | Auxiliary Field:\\ 257 | \tab $\vb{B} = \mu_0(\vb{H} + \vb{M}) = \mu_0(1 + \chi_m)\vb{H} = \mu \vb{H}$ 258 | 259 | Volume bound current:\\ 260 | \tab $\vb{J}_B = \chi_m \vb{J}_f$ 261 | 262 | 263 | \subsection{Boundary Conditions in Magnetostatics} 264 | \begin{itemize} 265 | \itemsep0em 266 | \item $B_\text{above}^\parallel - B_\text{below}^\parallel = \mu_0K$ 267 | \item $\vb{B}_\text{above} - \vb{B}_\text{below} = \mu_0(\vb{K}\cross\vu{n})$ 268 | \item $\vb{A}_\text{above} = \vb{A}_\text{below}$ 269 | \item $\ds \pdv{A_\text{above}}{n} - \pdv{A_\text{below}}{n} = -\mu_0\vb{K}$ 270 | \item $\vb{H}_\text{above}^\perp - \vb{H}_\text{below}^\perp = - (\vb{M}_\text{above}^\perp - \vb{M}_\text{below}^\perp) $ 271 | \item $\vb{H}_\text{above}^\parallel - \vb{H}_\text{below}^\parallel = \vb{K}_f \cross \vu{n}$ 272 | \end{itemize} 273 | 274 | \section{Vector Derivatives} 275 | \subsection{Cartesian} 276 | $d\vb{l} = dx\,\vu{x} + dy\,\vu{y} + dz\,\vu{z}$ 277 | \hspace{1cm} 278 | $d\uptau = dx\,dy\,dz$ 279 | 280 | Gradient:\\ 281 | \tab $\ds \grad{f} = \pdv{f}{x}\vu{x} + \pdv{f}{y}\vu{y} + \pdv{f}{z}\vu{z}$ 282 | 283 | Divergence:\\ 284 | \tab $\ds \div{\vb{v}} = \pdv{v_x}{x} + \pdv{v_y}{y} + \pdv{v_z}{z}$ 285 | 286 | Curl:\\ 287 | \vspace{-3mm} 288 | \tab $\ds \curl{\vb{v}} = \left(\pdv{v_z}{y} - \pdv{v_y}{z}\right)\vu{x} + \left(\pdv{v_x}{z} - \pdv{v_z}{x}\right)\vu{y} + \left(\pdv{v_y}{x} - \pdv{v_x}{y}\right)\vu{z}$ 289 | 290 | \subsection{Spherical} 291 | $d\uptau = r^2\sin\theta\,dr\,d\theta\,d\phi$ 292 | 293 | Gradient:\\ 294 | \tab $\ds \grad{f} = \pdv{f}{r}\vu{r} + \frac{1}{r}\pdv{f}{\theta}\vu*{\theta} + \frac{1}{r\sin\theta}\pdv{f}{\phi}\vu*{\phi}$ 295 | 296 | Divergence:\\ 297 | \tab \hspace{-0.2mm}$\ds \div{\vb{v}} = \frac{1}{r^2}\pdv{r}(r^2v_r) + \frac{1}{r\sin\theta}\pdv{\theta}(\sin\theta v_\theta) + \frac{1}{r\sin\theta}\pdv{v_\phi}{\phi}$ 298 | 299 | Curl:\\ 300 | \tab $\ds \curl{\vb{v}} = \frac{1}{r\sin\theta}\left[\pdv{\theta}(\sin\theta\, v_\phi)- \pdv{v_\theta}{\phi}\right]\vu{r}\,+$\\ 301 | \tab \tab $\ds \frac{1}{r}\left[\frac{1}{\sin\theta}\pdv{v_r}{\phi}-\pdv{r}(r v_\phi)\right]\vu*{\theta} + \frac{1}{r}\left[\pdv{r}(r v_\theta)-\pdv{v_r}{\theta}\right]\vu*{\phi}$ 302 | 303 | \subsection{Cylindrical} 304 | $d\uptau = s\,ds\,d\phi\,dz$ 305 | 306 | Gradient:\\ 307 | \tab $\ds \grad{f} = \pdv{f}{s}\vu{s}+\frac{1}{s}\pdv{f}{\phi}\vu*{\phi}+\pdv{f}{z}\vu{z}$ 308 | 309 | Divergence:\\ 310 | \tab $\ds \div{\vb{v}} = \frac{1}{s}\pdv{s}(sv_s)+\frac{1}{s}\pdv{v_\phi}{\phi} + \pdv{v_z}{z}$ 311 | 312 | Curl:\\ 313 | \tab $\ds \curl{\vb{v}} = \left[\frac{1}{s}\pdv{v_z}{\phi}-\pdv{v_\phi}{z}\right]\vu{s}+\left[\pdv{v_s}{z}-\pdv{v_z}{s}\right]\vu*{\phi}+\frac{1}{s}\left[\pdv{s}(sv_\phi)-\pdv{v_s}{\phi}\right]\vu{z}$ 314 | 315 | \section{Fundamental Theorems} 316 | Fundamental Theorem of Line Integrals:\\ 317 | \tab $\ds \int_{\vb{a}}^{\vb{b}}(\grad{f})\cdot d\vb{l} = f(\vb{b})-f(\vb{a})$ 318 | 319 | Divergence Theorem:\\ 320 | \tab $\ds \int (\div \vb{A})\, d\uptau = \oint \vb{A}\cdot d\vb{a}$ 321 | 322 | Stoke's Theorem:\\ 323 | \tab $\ds \int (\curl{\vb{A}})\cdot d\vb{a} = \oint \vb{A}\cdot d\vb{l}$ 324 | 325 | \section{Vector Identities} 326 | \tab $\ds \div{\left(\frac{\vu{\brcurs}}{\rcurs^2}\right)} = 4\pi \delta^3(\brcurs)$ 327 | \\ 328 | \tab $\ds \grad{\left(\frac{1}{\rcurs}\right)} = -\frac{\vu{\brcurs}}{\rcurs}$ 329 | \\ 330 | \tab $\ds \delta(kx) = \frac{1}{\abs{k}}\delta(x)$ 331 | 332 | \section{Spherical Coordinates} 333 | \tab $x = r\sin\theta\cos\phi$ 334 | \\ 335 | \tab $y = r\sin\theta\sin\phi$ 336 | \\ 337 | \tab $z = r\cos\theta$ 338 | \\ 339 | \vspace{3mm} 340 | \tab $\vu{x} = \sin\theta\cos\phi\,\vu{r} + \cos\theta\cos\phi\,\vu*{\theta} - \sin\phi\,\vu*{\phi}$ 341 | \\ 342 | \tab $\vu{y} = \sin\theta\sin\phi\,\vu{r} + \cos\theta\sin\phi\,\vu*{\theta} + \cos\phi\,\vu*{\phi}$ 343 | \\ 344 | \tab $\vu{z} = \cos\theta \, \vu{r} - \sin\theta \, \vu*{\theta}$ 345 | \\ 346 | \vspace{3mm} 347 | \tab $r=\sqrt{x^2 + y^2 + z^2}$ 348 | \\ 349 | \tab $\theta = \tan^{-1}\left(\sqrt{x^2+y^2}/z \right)$ 350 | \\ 351 | \tab $\phi = \tan^{-1}(y/x)$ 352 | \\ 353 | \vspace{3mm} 354 | \tab $\vu{r}=\sin\theta\cos\phi\,\vu{x} + \sin\theta\sin\phi\,\vu{y}+\cos\theta\,\vu{z}$ 355 | \\ 356 | \tab $\vu*{\theta}=\cos\theta\cos\phi\,\vu{x}+\cos\theta\sin\phi\,\vu{y}-\sin\theta\,\vu{z}$ 357 | \\ 358 | \tab $\vu*{\phi} = -\sin\phi\,\vu{x}+\cos\phi\,\vu{y}$ 359 | 360 | \section{Cylindrical Coordinates} 361 | \tab $x = s\cos\phi$ 362 | \\ 363 | \tab $y = s \sin\phi$ 364 | \\ 365 | \tab $z=z$ 366 | \\ 367 | \vspace{3mm} 368 | \tab $\vu{x}=\cos\phi\,\vu{s}-\sin\phi\vu*{\phi}$ 369 | \\ 370 | \tab $\vu{y} = \sin\phi\,\vu{s}+\cos\phi\,\vu*{\phi}$ 371 | \\ 372 | \tab $\vu{z} = \vu{z}$ 373 | \\ 374 | \vspace{3mm} 375 | \tab $s = \sqrt{x^2+y^2}$ 376 | \\ 377 | \tab $\phi = \tan^{-1}(y/x)$ 378 | \\ 379 | \tab $z = z$ 380 | \\ 381 | \vspace{3mm} 382 | \tab $\vu{s} = \cos\phi\,\vu{x}+\sin\phi\,\vu{y}$ 383 | \\ 384 | \tab $\vu*{\phi}=-\sin\phi\,\vu{x}+\cos\phi\,\vu{y}$ 385 | \\ 386 | \tab $\vu{z} = \vu{z}$ 387 | 388 | % Footer content 389 | \rule{0.3\linewidth}{0.25pt} 390 | \scriptsize\\ 391 | Updated \today\\ 392 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 393 | \end{multicols} 394 | \end{document} 395 | -------------------------------------------------------------------------------- /MATH 400 - Applied Partial Differential Equations/math400fs.tex: -------------------------------------------------------------------------------- 1 | % !Tex program = pdflatex 2 | 3 | \documentclass[12pt,landscape]{article} 4 | \usepackage{multicol} 5 | \usepackage{calc} 6 | \usepackage{ifthen} 7 | \usepackage{mathtools} 8 | \usepackage[landscape]{geometry} 9 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 10 | \usepackage{color,graphicx,overpic} 11 | \usepackage{hyperref} 12 | \usepackage{enumitem} 13 | \usepackage{upgreek} 14 | \usepackage[italicdiff]{physics} 15 | \usepackage{newtxtext,newtxmath} 16 | \usepackage{mdframed} 17 | \usepackage{amsbsy} 18 | 19 | % This sets page margins to .5 inch if using letter paper, and to 1cm 20 | % if using A4 paper. (This probably isn't strictly necessary.) 21 | % If using another size paper, use default 1cm margins. 22 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 23 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 24 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 25 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 26 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 27 | } 28 | 29 | % Turn off header and footer 30 | \pagestyle{empty} 31 | 32 | 33 | % Redefine section commands to use less space 34 | \makeatletter 35 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 36 | {-1ex plus -.5ex minus -.2ex}% 37 | {0.5ex plus .2ex}%x 38 | {\normalfont\normalsize\bfseries}} 39 | \renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}% 40 | {-1explus -.5ex minus -.2ex}% 41 | {0.5ex plus .2ex}% 42 | {\normalfont\small\bfseries}} 43 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 44 | {-1ex plus -.5ex minus -.2ex}% 45 | {1ex plus .2ex}% 46 | {\normalfont\footnotessize\bfseries}} 47 | \makeatother 48 | 49 | % Define BibTeX command 50 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 51 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 52 | 53 | % Don't print section numbers 54 | \setcounter{secnumdepth}{0} 55 | 56 | 57 | \setlength{\parindent}{0pt} 58 | \setlength{\parskip}{1pt plus 0.5ex} 59 | 60 | \newcommand{\tab}{\hspace*{1em}} 61 | \newcommand{\ds}{\displaystyle} 62 | 63 | % Redefine some commands for newtxmath boldness 64 | \renewcommand{\grad}{\nabla} 65 | \renewcommand{\curl}[1]{\nabla\times#1} 66 | \renewcommand{\div}[1]{\nabla\cdot#1} 67 | \renewcommand{\cross}{\times} 68 | \newcommand{\defn}[1]{\textbf{Def} (\emph{#1})} 69 | \newcommand{\thm}[1]{\textbf{Thm} (\emph{#1})} 70 | 71 | \newcommand{\Var}[1]{\mathrm{Var}(#1)} 72 | \newcommand{\Cov}[1]{\mathrm{Cov}(#1)} 73 | 74 | \mdfsetup{skipabove=2pt,skipbelow=2pt, innertopmargin=-6pt, innerbottommargin=2pt, innerleftmargin=2pt, innerrightmargin=2pt} 75 | \theoremstyle{definition} 76 | \newmdtheoremenv{theorem}{Theorem} 77 | 78 | % ----------------------------------------------------------------------- 79 | 80 | \begin{document} 81 | 82 | \raggedright 83 | \footnotesize 84 | \begin{multicols*}{3} 85 | 86 | \raggedcolumns 87 | 88 | % multicol parameters 89 | % These lengths are set only within the two main columns 90 | %\setlength{\columnseprule}{0.25pt} 91 | \setlength{\premulticols}{1pt} 92 | \setlength{\postmulticols}{1pt} 93 | \setlength{\multicolsep}{1pt} 94 | \setlength{\columnsep}{2pt} 95 | 96 | \begin{center} 97 | \Large{\underline{MATH 400 Formula Sheet}} 98 | \end{center} 99 | 100 | \section{Linear First Order Equations} 101 | First Order Linear PDE:\\ 102 | \tab $a(x,y)u_x + b(x,y)u_y + c(x,y)u = f(x,y)$ 103 | 104 | \subsection{Simple Transport Equation} 105 | $$u_t + cu_x = 0 \,, -\infty < x < \infty, t > 0$$ 106 | $$u(x,0) = \phi(x) \,, -\infty < x < \infty$$ 107 | Solution:\\ 108 | \tab $u(x,t) = \phi(x - ct)$ 109 | 110 | \section{Linear Second Order Equations} 111 | Quasilinear PDE:\\ 112 | \tab $a_{11}(x, y, u, u_x, u_y)u_{xx} + 2a_{12}(x, y, u, u_x, u_y)u_{xy} + a_{22}(x, y, u, u_x, u_y)u_{yy} + a_{00}(x, y, u, u_x, u_y) = 0$ 113 | 114 | Semilinear PDE:\\ 115 | \tab $a_{11}(x, y)u_{xx} + 2a_{12}(x, y)u_{xy} + a_{22}(x, y)u_{yy} + a_{00}(x, y, u, u_x, u_y) = 0$ 116 | 117 | Linear PDE:\\ 118 | \tab $a_{11}(x, y)u_{xx} + 2a_{12}(x, y)u_{xy} + a_{22}(x, y)u_{yy} + a_{1}(x,y)u_x + a_{2}(x,y)u_y + a_{0}(x,y)u = f(x,y)$ 119 | 120 | Discriminant for semilinear PDEs:\\ 121 | \tab $\mathcal{D}(x,y) = [a_{12}(x,y)]^2 - a_{11}(x,y)a_{22}(x,y)$ 122 | 123 | Classification of semilinear PDEs: 124 | \begin{equation*} 125 | \begin{cases} 126 | \mathcal{D}(x,y) > 0 & \text{Hyperbolic}\\ 127 | \mathcal{D}(x,y) < 0 & \text{Elliptic}\\ 128 | \mathcal{D}(x,y) = 0 & \text{Parabolic} 129 | \end{cases} 130 | \end{equation*} 131 | 132 | Change of variables: 133 | \begin{equation*} 134 | \begin{cases} 135 | U_{\xi\eta} + b_{00}(\xi, \eta, U, U_\xi, U_\eta) = 0 & \text{Hyperbolic}\\ 136 | U_{\xi\xi} + U_{\eta\eta} + b_{00}(\xi, \eta, U, U_\xi, U_\eta) = 0 & \text{Elliptic}\\ 137 | U_{\xi\xi} + b_{00}(\xi, \eta, U, U_\xi, U_\eta)= 0 & \text{Parabolic} 138 | \end{cases} 139 | \end{equation*} 140 | 141 | \subsection{Wave Equation} 142 | General Solution:\\ 143 | \tab $u(x,t) = f(x + ct) + g(x-ct)$ 144 | 145 | Initial Value Problem on the real line:\\ 146 | $$u_{tt} - c^2u_{xx} = f(x,t) \,, -\infty < x < \infty, t > 0$$ 147 | $$u(x,0) = \phi(x) \,, u_t(x,0) = \psi(x) -\infty < x < \infty$$ 148 | 149 | d'Alembert's Formula:\\ 150 | \tab $\ds u(x,t) = \frac{1}{2}[\phi(x-ct) + \phi(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct}\psi(s)\mathop{ds} + \frac{1}{2c} \int_{0}^{t}\int_{x-c(t-s)}^{x+c(t-s)}f(y,s)\mathop{dy}\mathop{ds}$ 151 | 152 | \subsection{Non-Homogenous Boundary Conditions} 153 | \tab Shift the data ($u = v + w$): 154 | 155 | Dirichlet, $u(0,t) = a(t)$, $u(L,t) = b(t)$:\\ 156 | \tab $\ds w(x,t) = a(t) + \frac{x}{L}[b(t) - a(t)]$ 157 | 158 | Neumann, $u_x(0,t) = a(t)$, $u_x(L,t) = b(t)$\\ 159 | \tab $\ds w(x,t) = xa(t) + \frac{x^2}{2L}[b(t) - a(t)]$ 160 | 161 | Mixed 1, $u(0,t) = a(t)$, $u_x(L,t) = b(t)$\\ 162 | \tab $w(x,t) = a(t) + xb(t)$ 163 | 164 | Mixed 2, $u_x(0,t) = a(t)$, $u(L,t) = b(t)$\\ 165 | \tab $w(x,t) = (x-L)a(t) + b(t)$ 166 | 167 | \subsection{Sturm-Liouville Theory} 168 | Consider the homogeneous linear second-order PDE:\\ 169 | \tab $r(x)u_t - (p(x)u_x)_x + q(x)u = 0$ 170 | 171 | Separation of Variables:\\ 172 | \tab $\ds -\frac{T'(t)}{T(t)} = -\frac{(p(x)X'(x))'}{r(x)X(x)} + \frac{q(x)}{r(x)} = \lambda$ 173 | 174 | ODEs: 175 | $$T' + \lambda T = 0$$ 176 | $$-(p(x)X')' + q(x)X = \lambda r(x)X$$ 177 | 178 | Associated Eigenvalue Problem:\\ 179 | \tab $-(p(x)X')' + q(x)X = \lambda r(x) X$ 180 | 181 | General Boundary Conditions:\\ 182 | \tab $\mathcal{B}_1X = \alpha_1 X(a) + \beta_1 X(b) + \gamma_1 X'(a) + \delta_1 X'(b) = 0$\\ 183 | \tab $\mathcal{B}_2X = \alpha_2 X(a) + \beta_2 X(b) + \gamma_2 X'(a) + \delta_2 X'(b) = 0$ 184 | 185 | Boundary conditions are separated when:\\ 186 | \tab $\beta_1 = \delta_1 = 0 \hspace{1cm} \alpha_2 = \gamma_2 = 0$ 187 | 188 | Lagrange's Identity. BCs are symmetric if for all functions $f$, $g$ that satisfy the BCs:\\ 189 | \tab $\tab \left[-p\left(f'g - fg'\right)\right]_a^b = 0$ 190 | 191 | Sturm-Liouville Problem ($p(x) > 0, r(x) > 0$): 192 | \begin{equation*} 193 | \begin{cases} 194 | \mathcal{L}X(x) = \lambda r(x)X, \, a < x < b &\\ 195 | \mathcal{B}_1 X = 0, \,\mathcal{B}_2 X = 0&\\ 196 | \end{cases} 197 | \end{equation*} 198 | 199 | Theorem 5.3.1:\\ 200 | \tab If you have symmetric BCs, then any to eigenfunctions of a SL problem that correspond to distinct eigenvalues are orthogonal. If any function is expanded in a series of these eigenfunctions, the coefficients are determined. 201 | 202 | Theorem 5.3.2:\\ 203 | \tab Under Theorem 5.3.1, all the eigenvalues are real numbers and the eigenfunctions can be chosen to be real-valued. 204 | 205 | Theorem 5.3.3:\\ 206 | \tab Under 5.3.1, if $q(x) \geq 0$ for all $a \leq x \leq b$ and if for all real-valued functions $f$ satisfying the BCs we satisfy 207 | $$[p(x)f(x)f'(x)]_a^b \leq 0$$ 208 | then there can be no negative eigenvalues. 209 | 210 | Theorem 5.4.1:\\ 211 | \tab For any regular of periodic SL problem, there are an infinite amount of eigenvalues. 212 | 213 | \section{Laplace's Equation} 214 | 215 | Polar Coordinates:\\ 216 | \tab $\ds u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2} u_{\theta \theta} = 0$ 217 | 218 | Spherical Coordinates:\\ 219 | \tab $\ds u_{rr} + \frac{2}{r}u_r + \frac{1}{r^2 \sin\theta}\left[\sin\theta \,u_\theta\right]_\theta + \frac{1}{r^2\sin^2\theta}u_{\phi\phi} = 0$ 220 | 221 | Maximum Principle:\\ 222 | \tab Let $D$ be a connected bounded open set of $\mathbb{R}^n$, $n = 2, 3$. If $u(\vb{x})$ is harmonic in $D$ and continuous on $D \cup \partial D$ then $u(\vb{x})$ attains its maximum and minimum values on $\partial D$. 223 | 224 | \section{PDEs in 3 Dimensions} 225 | Elliptic PDE:\\ 226 | \tab $-\div[p(\vb{x})\grad{u}] + q(\vb{x}) u = f(\vb{x})$ 227 | 228 | Parabolic PDE:\\ 229 | \tab $r(\vb{x})u_t -\div[p(\vb{x})\grad{u}] + q(\vb{x}) u = f(\vb{x}, t)$ 230 | 231 | Hyperbolic PDE:\\ 232 | \tab $r(\vb{x})u_{tt} -\div[p(\vb{x})\grad{u}] + q(\vb{x}) u = f(\vb{x}, t)$ 233 | 234 | Eigenvalue Problem:\\ 235 | \tab $-\div[p(\vb{x})\grad{X}] + q(\vb{x}) X = \lambda r(\vb{x})X$ 236 | 237 | Green's First Identity:\\ 238 | \tab Let $D$ be an open connected domain in $\mathbb{R}^3$ and let $\partial D$ be its piecewise smooth boundary Let $f$ and $g$ be smooth functions on $D \cup \partial D$. Then\\ 239 | \tab $\ds \oint_{\partial D} f \pdv{g}{n}\mathop{ds} = \iint_{D}\grad{F}\cdot\grad{g}\mathop{d\vb{x}} + \iint_{D} f \grad^2{g}\mathop{d\vb{x}}$ 240 | 241 | Bessel's Equation:\\ 242 | \tab $\rho^2 y'' + \rho y + (\rho^2 -n^2)y = 0$\\ 243 | \tab $y(\rho) = c_1 J_n(\rho) + c_2 Y_n(\rho)$ 244 | 245 | Associated Legendre Equation:\\ 246 | \tab $\ds \dv{x}\left[(1-x^2)\dv{x}P_l^m(x)\right] + \left[l(l+1) - \frac{m^2}{1-x^2}\right]P_l^m(x)=0$ 247 | 248 | Spherical Harmonics:\\ 249 | \tab $\ds \grad^2_{S^2} Y_l^m = \frac{1}{\sin\theta}[\sin\theta\,Y]_\theta + \frac{1}{\sin^2\theta}Y_{\phi\phi} = l(l+1)Y_l^m$ 250 | \tab $Y_l^m(\theta,\phi) = P_l^\abs{m}(\cos\theta)e^{im\phi}$ 251 | \tab $l = 0, 1, 2 \ldots \quad m = -l, -l+1,\ldots, l-1, l$ 252 | 253 | \section{Integral Transform Methods} 254 | Fourier Transform:\\ 255 | \tab $\ds \hat{f}(k) = \int_{-\infty}^{\infty} f(x)e^{-ikx}\mathop{dx}$ 256 | 257 | Inverse Fourier Transform:\\ 258 | \tab $\ds \hat{f}(k) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(x)e^{ikx}\mathop{dx}$ 259 | 260 | Error Function:\\ 261 | \tab $\ds \erf(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-p^2} \mathop{dp}$ 262 | \tab $\erf(0) = 0 \quad \lim_{x\rightarrow\infty}\erf(x) = 1$ 263 | 264 | Laplace Transform:\\ 265 | \tab Let $f(t)$ be a piecewise continuous function on $[0,\infty)$ and suppose $\exists K, \gamma$ such that $\abs{f(t)} \leq K e^{\gamma t} \, \forall t > 0$. Then 266 | \tab $\ds F(s) = \int_{0}^{\infty}f(t)e^{-st}\mathop{dt}$ 267 | 268 | Inverse Laplace Transform:\\ 269 | \tab $\ds f(t) = \frac{1}{2\pi i}\int_{\alpha - i\infty}^{\alpha + i \infty}F(s)e^{st}\mathop{ds} \quad \alpha > \gamma$ 270 | 271 | 272 | 273 | 274 | \newpage 275 | 276 | \section{Solved 1D Eigenvalue Problems} 277 | \subsection{Dirichlet} 278 | \begin{equation*} 279 | \begin{rcases} 280 | -X'' = \lambda X\\ 281 | X(0) = 0, X(L) = 0 282 | \end{rcases} \Rightarrow \begin{cases} 283 | \lambda_n = \left(\frac{n\pi}{L}\right)^2, \, n = 1, 2, 3 \ldots\\ 284 | X_n(x) = \sin(\frac{n\pi}{L}x) 285 | \end{cases} 286 | \end{equation*} 287 | 288 | \subsection{Neumann} 289 | \begin{equation*} 290 | \begin{rcases} 291 | -X'' = \lambda X\\ 292 | X'(0) = 0, X'(L) = 0 293 | \end{rcases} \Rightarrow \begin{cases} 294 | \lambda_n = \left(\frac{n\pi}{L}\right)^2, \, n = 0, 1, 2 \ldots\\ 295 | X_n(x) = \cos(\frac{n\pi}{L}x) 296 | \end{cases} 297 | \end{equation*} 298 | 299 | \subsection{Periodic} 300 | \begin{equation*} 301 | \begin{rcases} 302 | -X'' = \lambda X\\ 303 | X(-\pi) = X(\pi)\\ 304 | X'(-\pi) = X'(\pi) 305 | \end{rcases} \Rightarrow \begin{cases} 306 | \lambda_n = n^2, \, n = 0, 1, 2 \ldots\\ 307 | X_n(x) \in \{1, \cos(n\theta), \sin(n\theta)\} 308 | \end{cases} 309 | \end{equation*} 310 | 311 | \subsection{Mixed 1} 312 | \begin{equation*} 313 | \begin{rcases} 314 | -X'' = \lambda X\\ 315 | X(0) = 0, X'(L) = 0 316 | \end{rcases} \Rightarrow \begin{cases} 317 | \lambda_n = \left(\frac{(2n+1)\pi}{2L}\right)^2, \, n = 0, 1, 2 \ldots\\ 318 | X_n(x) = \sin(\frac{(2n+1)\pi}{2L}x) 319 | \end{cases} 320 | \end{equation*} 321 | 322 | \subsection{Mixed 2} 323 | \begin{equation*} 324 | \begin{rcases} 325 | -X'' = \lambda X\\ 326 | X'(0) = 0, X(L) = 0 327 | \end{rcases} \Rightarrow \begin{cases} 328 | \lambda_n = \left(\frac{(2n+1)\pi}{2L}\right)^2, \, n = 0, 1, 2 \ldots\\ 329 | X_n(x) = \cos(\frac{(2n+1)\pi}{2L}x) 330 | \end{cases} 331 | \end{equation*} 332 | 333 | 334 | \section{Solved PDE Problems} 335 | \subsection{Problem 1} 336 | $$\Delta u = 0 \quad x^2 + y^2 < a^2$$ 337 | $$u = h(x,y) \quad \text{on} \quad x^2 + y^2 = a^2$$ 338 | Solution (Oct 21):\\ 339 | \tab $\ds u(r,\theta) = \frac{a^2 - r^2}{2\pi} \int_{-\pi}^{\pi} \frac{h(\phi)}{a^2 - 2ar\cos(\phi - \theta) + r^2}\mathop{d\phi}$ 340 | 341 | \subsection{Problem 2} 342 | $$\Delta u = 0 \quad 0 < r < a, \quad 0 < \theta < \beta$$ 343 | $$u(r, 0) = 0, \quad u(r, \beta) = 0, \quad 0 < r < a$$ 344 | $$u_r(a,\theta) = h(\theta) \quad 0 < \theta < \beta$$ 345 | Solution (Oct 23):\\ 346 | \tab $\ds u(r,\theta) = \sum_{n=1}^{\infty} C_n r^{n\pi/\beta}\sin\left(\frac{n\pi}{\beta}\theta\right)$\\ 347 | \tab $\ds C_n = \frac{2}{n\pi}a^{-n\pi/\beta + 1}\int_{0}^{\beta}h(\phi)\sin\left(\frac{n\pi}{\beta}\theta\right)$ 348 | 349 | \subsection{Problem 3} 350 | $$\Delta u = 0 \quad a < r < b, \quad 0 < \theta < 2\pi$$ 351 | General Solution (Oct 23):\\ 352 | \tab $\ds u(r, \theta) = \frac{1}{2} \left(C_0 + D_0\ln(r)\right) + \sum_{n=1}^{\infty}\left(C_n^{(1)}r^n + D_n^{(1)} r^{-n}\right)\cos(n\theta) + \left(C_n^{(2)}r^n + D_n^{(2)}r^{-n}\right)\sin(n\theta)$ 353 | 354 | \subsection{Problem 4} 355 | $$S_t - DS_{xx}=0 \quad -\infty < x < \infty, \quad t > 0$$ 356 | $$S(x, 0) = \delta(x - y), \quad y \geq 0$$ 357 | Solution (Nov 16):\\ 358 | \tab $\ds S(x,t) = \frac{1}{\sqrt{4\pi Dt}}e^{-(x - y)^2/(4Dt)}$ 359 | 360 | \subsection{Problem 5} 361 | $$u_t - Du_{xx}=0 \quad -\infty < x < \infty, \quad t > 0$$ 362 | $$u(x, 0) = \varphi(x), \quad -\infty < x < \infty$$ 363 | Solution (Nov 18):\\ 364 | \tab $\ds u(x,t) =\int_{-\infty}^{\infty} \varphi(y)S(x - y, t)\mathop{dy}$ 365 | 366 | \subsection{Problem 6} 367 | $$u_t - Du_{xx}=0 \quad 0 < x < \infty, \quad t > 0$$ 368 | $$u(0, t) = 0, \quad t > 0$$ 369 | $$u(x, 0) = \varphi(x), \quad -\infty < x < \infty$$ 370 | Solution (Nov 18):\\ 371 | \tab $\ds u(x,t) =\int_{0}^{\infty} \phi(y)[S(x-y, t) - S(x + y, t)]\mathop{dy}$ 372 | 373 | \subsection{Problem 7} 374 | $$\Delta u =0 \quad -\infty < x < \infty, \quad y > 0$$ 375 | $$u(x,0) = h(x), \quad -\infty < x < \infty$$ 376 | Solution (Nov 20):\\ 377 | \tab $\ds u(x,y) = \int_{-\infty}^{\infty}h(z) \frac{y}{\pi [(x - z)^2 - y^2]}\mathop{dz}$ 378 | 379 | \subsection{Problem 8} 380 | $$u_t - Du_{xx}=0 \quad 0 < x < \infty, \quad t > 0$$ 381 | $$u(0, t) = h(t), \quad t > 0$$ 382 | $$u(x, 0) = 0, \quad -\infty < x < \infty$$ 383 | Solution (Nov 23):\\ 384 | \tab $\ds u(x,t) = \begin{cases} 385 | 0 & t < b\\ 386 | 1 - \erf\left(\frac{x}{\sqrt{4D(t-b)}}\right) & t > b 387 | \end{cases}$ 388 | 389 | 390 | 391 | % Footer content 392 | \rule{0.3\linewidth}{0.25pt} 393 | \scriptsize\\ 394 | Updated \today\\ 395 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 396 | \end{multicols*}% 397 | 398 | \end{document} 399 | -------------------------------------------------------------------------------- /MATH 217 - Multivariable and Vector Calculus/math217fs.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,landscape]{article} 2 | \usepackage{multicol} 3 | \usepackage{calc} 4 | \usepackage{ifthen} 5 | \usepackage[landscape]{geometry} 6 | \usepackage{amsmath,amsthm,amsfonts,amssymb} 7 | \usepackage{color,graphicx,overpic} 8 | \usepackage{hyperref} 9 | \usepackage{enumitem} 10 | \usepackage{physics} 11 | \usepackage{newtxtext,newtxmath} 12 | 13 | % This sets page margins to .5 inch if using letter paper, and to 1cm 14 | % if using A4 paper. (This probably isn't strictly necessary.) 15 | % If using another size paper, use default 1cm margins. 16 | \ifthenelse{\lengthtest { \paperwidth = 11in}} 17 | { \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} } 18 | {\ifthenelse{ \lengthtest{ \paperwidth = 297mm}} 19 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 20 | {\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} } 21 | } 22 | 23 | % Turn off header and footer 24 | \pagestyle{empty} 25 | 26 | % Redefine section commands to use less space 27 | \makeatletter 28 | \renewcommand{\section}{\@startsection{section}{1}{0mm}% 29 | {-1ex plus -.5ex minus -.2ex}% 30 | {0.5ex plus .2ex}%x 31 | {\normalfont\large\bfseries}} 32 | \renewcommand{\subsubsection}{\@startsection{subsection}{2}{0mm}% 33 | {-1explus -.5ex minus -.2ex}% 34 | {0.5ex plus .2ex}% 35 | {\normalfont\normalsize\bfseries}} 36 | \renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}% 37 | {-1ex plus -.5ex minus -.2ex}% 38 | {1ex plus .2ex}% 39 | {\normalfont\small\bfseries}} 40 | \makeatother 41 | 42 | % Define BibTeX command 43 | \def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em 44 | T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}} 45 | 46 | % Don't print section numbers 47 | \setcounter{secnumdepth}{0} 48 | 49 | \setlength{\parindent}{0pt} 50 | \setlength{\parskip}{1pt plus 0.5ex} 51 | 52 | % Custom Commands 53 | \newcommand{\ds}{\displaystyle} 54 | \newcommand{\tab}{\hspace{.02\textwidth}} 55 | % ----------------------------------------------------------------------- 56 | 57 | \begin{document} 58 | \raggedright 59 | \footnotesize 60 | \begin{multicols}{3} 61 | 62 | 63 | % multicol parameters 64 | % These lengths are set only within the two main columns 65 | %\setlength{\columnseprule}{0.25pt} 66 | \setlength{\premulticols}{1pt} 67 | \setlength{\postmulticols}{1pt} 68 | \setlength{\multicolsep}{1pt} 69 | \setlength{\columnsep}{2pt} 70 | 71 | \begin{center} 72 | \Large{\underline{MATH 217 Formula Sheet}} 73 | \end{center} 74 | 75 | \section{Vectors \& Geometry of Space} 76 | Distance between two points:\\ 77 | \tab $d = \sqrt{(x_1-x_0)^2 + (y_1-y_0)^2 + (z_1 - z_0)^2}$ 78 | 79 | Equation of a sphere:\\ 80 | \tab $R^2 = (x_1-x_0)^2 + (y_1-y_0)^2 + (z_1 - z_0)^2$ 81 | 82 | Unit vector:\\ 83 | \tab $\vb{u} = \frac{\vb{a}}{|\vb{a}|}$ 84 | 85 | Dot Product:\\ 86 | \tab $\vb{a}\cdot\vb{b}=\|\vb{a}\|\ \|\vb{b}\|\cos(\theta)$ 87 | 88 | Scalar projection of $\vb{a}$ on to $\vb{b}$:\\ 89 | \tab $\text{comp}_{\vb{b}}\vb{a}=|\vb{a} |\cos \theta =|\vb{a} |{\frac {\vb{a} \cdot \vb{b} }{|\vb{a} |\,|\vb{b} |}}={\frac {\vb{a} \cdot \vb{b} }{|\vb{b} |}}$ 90 | 91 | Vector projection of $\vb{a}$ on to $\vb{b}$:\\ 92 | \tab $\text{proj}_{\vb{b}}\vb{a} ={\frac {\vb{a} \cdot \vb{b} }{|\vb{b} |}}{\frac {\vb{b} }{|\vb{b} |}}$ 93 | 94 | Orthogonal projection of $\vb{a}$ on to $\vb{b}$:\\ 95 | \tab $\vb{v} = \vb{a} - \text{proj}_{\vb{b}}\vb{a}$ 96 | 97 | Cross product:\\ 98 | \tab $\vb{a} \times \vb{b} =\left\|\vb{a} \right\|\left\|\vb{b} \right\|\sin(\theta )$ 99 | 100 | \subsubsection{Lines \& Planes} 101 | 102 | Vector equation of a line:\\ 103 | \tab $\vb{r}(t) = \vb{r_0} + \vb{v}(t)$ 104 | 105 | Vector equation of a plane:\\ 106 | \tab $(\vb{r} - \vb{r}_0) \cdot \vb{n} = 0$ 107 | 108 | Scalar equation of a plane, where $a,b,c$ are components of the normal vector:\\ 109 | \tab $a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$ 110 | 111 | Distance from the point $P=(x_1,y_1,z_1)$ to the plane $Ax+By+Cz+D=0$:\\ 112 | \tab $d = \frac{|Ax_1+By_1+Cz_1 +D|}{\sqrt{A^2+B^2+C^2}}$ 113 | 114 | \subsubsection{Quadric Surfaces} 115 | 116 | Equation of an ellipsoid:\\ 117 | \tab ${x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1$ 118 | 119 | Equation of an elliptic paraboloid:\\ 120 | \tab ${x^2 \over a^2} + {y^2 \over b^2} =z$ 121 | 122 | Equation of a hyperbolic paraboloid:\\ 123 | \tab ${x^2 \over a^2} - {y^2 \over b^2} = z$ 124 | 125 | Equation of a elliptic hyperboloid of one sheet:\\ 126 | \tab ${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1$ 127 | 128 | Equation of an elliptic hyperboloid of two sheets:\\ 129 | \tab ${x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1$ 130 | 131 | \section{Vector Functions} 132 | Arc length of a vector function:\\ 133 | \tab $\int_a^b|\vb{r}'(t)|\,dt$ 134 | 135 | Length of the curve $y=f(x)$:\\ 136 | \tab $\int_a^b \sqrt{1+(f'(x))^2}\,dx$ 137 | 138 | If $\vb{r}'(t)$ is differentiable at $t=a$ and $ \vb{r}'(a) \neq \vb{r}(0)$, the tangent line to the curve given by $\vb{r}'(t)$ is the line through $\vb{r}'(a)$ in the direction of $\vb{r}'(a)$ 139 | 140 | \section{Partial Derivatives} 141 | $f(x,y)$ is continuous at $(a,b)$ if\\ 142 | \tab $\lim_{(x,y)\to (a,b)}f(x,y)=f(a,b)$\\ 143 | 144 | Suppose that $z=f(x,y)$, $f$ is differentiable, $x=g(t)$, and $y=h(t)$. Assuming that the relevant derivatives exist,\\ 145 | \tab $\pdv{z}{t} = \pdv{z}{x}\pdv{x}{t}+\pdv{z}{y}\pdv{y}{t}$ 146 | 147 | If $f(x,y)$ is defined on a domain $D$ that contains the point $(a,b)$. If $\pdv{z}{y}{x}$ and $\pdv{z}{x}{y}$ are continuous on $D$, then\\ 148 | \tab $\pdv{z}{y}{x} = \pdv{z}{x}{y}$ 149 | 150 | Equation of a tangent plane:\\ 151 | \tab $z=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)+f(x_0,y_0)$ 152 | 153 | The differential of $f(x,y)$ is:\\ 154 | \tab $df = \pdv{f}{x}dx + \pdv{f}{y}dy$ 155 | 156 | Suppose that $f(x,y)$ is a function and $x=g(s,t)$ and $y=h(s,t)$:\\ 157 | \tab $\pdv{f}{s}=f_xg_s+f_yh_s\qquad \pdv{f}{t}=f_xg_t+f_yh_t.$ 158 | 159 | The slope of a surface given by $z=f(x,y)$ in the direction of a vector $\vb{u}$ is called the directional derivative of $f$, written $D_uf$\\ 160 | \tab $D_{\vb{u}}f=\nabla f\cdot \vb{u}=|\nabla f||\vb{u}|\cos\theta= 161 | |\nabla f|\cos\theta$ 162 | 163 | The maximum value of the directional derivative $D_{\vb{u}} f(\vb{v})$ is $|\nabla f(\vb{v})|$and it occurs when $\vb{u}$ has the same direction as the gradient vector $\nabla f(\vb{v})$.\\ 164 | \vspace{0.2cm} 165 | The gradient vector $\nabla f(a,b,c)$ is orthogonal to the level surface $S$ through $(a,b,c)$\\ 166 | \vspace{0.2cm} 167 | Discriminant of $f(x,y)$:\\ 168 | \tab $D(x,y) = \det\mqty[f_{xx}& f_{xy}\\f_{yx} & f_{yy} ]$\\ 169 | \vspace{0.2cm} 170 | If $f_{xx} > 0$ or $f_{yy} > 0$ and $D(a,b) > 0$, then $f(a,b)$ is a local minimum.\\ 171 | If $f_{xx} < 0 $ or $f_{yy} < 0$ and $D(a,b) > 0$, then $f(a,b)$ is a local maximum.\\ 172 | If $D(a,b) < 0$, then $f(a,b)$ is a saddle point.\\ 173 | If $D(a,b) = 0$, no information is given.\\ 174 | 175 | \subsubsection{Optimization} 176 | The extreme values of $f(x,y)$ can only occur at:\\ 177 | \begin{itemize} 178 | \vspace{-0.5em} 179 | \setlength\itemsep{-0.3em} 180 | \item Interior critical points, where both partials exist. 181 | \item Boundary points of the domain of the function. 182 | \end{itemize} 183 | \vspace{-0.25cm} 184 | To maximize or minimize $f(x,y)$ subject to the constraint $g(x,y) = C$, we solve: 185 | \begin{itemize} 186 | \vspace{-0.5em} 187 | \setlength\itemsep{-0.3em} 188 | \item $\nabla f(x,y) = \lambda \nabla g(x,y)$ 189 | \item $g(x,y) = C$ 190 | \end{itemize} 191 | 192 | \section{Multiple Integrals} 193 | 194 | Fubini's Theorem: If $f(x,y)$ is continuous on the domain $D$:\\ 195 | \tab $\iint_D f(x,y)\,dA = \int_{c}^{d} \hspace{-1mm} \int_{a}^{b}f(x,y)\,dx\,dy = \int_{a}^{b} \hspace{-1mm} \int_{c}^{d}f(x,y)\,dy\,dx$ 196 | 197 | Area of the domain $D$:\\ 198 | \tab $ A = \iint_D\,dA$ 199 | 200 | Average value of a $f(x,y)$ over domain $D$:\\ 201 | \tab $f_{\text{avg}} = \frac{1}{A}\iint_D f(x,y)\,dA$ 202 | 203 | Mass of a lamina $D$ with density $\rho (x,y)$:\\ 204 | \tab $m = \iint_D \rho (x,y)\,dA$ 205 | 206 | Moment of a mass about the $x$-axis:\\ 207 | \tab $M_x = \iint_D x\rho (x,y) \,dA$ 208 | 209 | Center of mass of a lamina:\\ 210 | \tab $\bar{x} = M_x/m \hspace{1cm} \bar{y} = M_y/m \hspace{1cm} \bar{z} = M_z/m$ 211 | 212 | Surface area of $f(x,y)$:\\ 213 | \tab $\int_{x_0}^{x_1}\int_{y_0}^{y_1} \sqrt{f_x^2+f_y^2+1}\,dy\,dx.$ 214 | 215 | \subsubsection{Cylindrical Coordinates} 216 | \tab $x=r\cos\theta \hspace{1cm} y=r\sin\theta \hspace{1cm} z = z$\\ 217 | \tab $dA = r\,dr\,d\theta \hspace{1cm} dV = r\,dr\,dz\,d\theta$\\ 218 | \tab $x^2 + y^2 = r^2$ 219 | \subsubsection{Spherical Coordinates} 220 | \tab $x= \rho\sin\phi\cos\theta$\\ 221 | \tab $y= \rho\sin\phi\sin\theta$\\ 222 | \tab $z= \rho\cos\phi$\\ 223 | \tab $x^2+y^2+z^2 = \rho^2$\\ 224 | \tab $x^2+y^2 = \rho^2\sin^2\phi$\\ 225 | \tab $dV = \rho^2\sin\phi\,d\rho\,d\phi\,d\theta$\\ 226 | 227 | \subsubsection{Change of Variables} 228 | Suppose that $f(x,y)$ is continuous on $R$ and that $R$ and $S$ are type I or type II plane regions. Suppose also that $T$ is one-to-one, except perhaps on the boundary of $S$:\\ 229 | $\iint_R F(x,y) \, dV = 230 | \iint_S F(x(u,v),y(u,v)) 231 | \left|{\partial(x,y)\over\partial(u,v)}\right| \,du\,dv$ 232 | $\small\text{3D case:}\iiint_R F(x,y,z) \, dV =$\\ 233 | \hspace{-0.5cm}\tab $\iiint_S F(x(u,v,w),y(u,v,w),z(u,v,w)) 234 | \left|{\partial(x,y,z)\over\partial(u,v,w)}\right| \,du\,dv\,dw$\\ 235 | Jacobian:\\ 236 | \tab $\ds \left|{\partial(x,y)\over\partial(u,v)}\right| = \det\mqty[ x_u& x_v\\ y_u& y_v ]$\\ 237 | \tab $\ds \left|{\partial(x,y,z)\over\partial(u,v,w)}\right| = \det\mqty[ 238 | x_u& x_v&x_w\\ 239 | y_u& y_v&y_w \\ 240 | z_u&z_v&z_w]$ 241 | 242 | \section{Vector Calculus} 243 | A vector field $\vb{F}$ is conservative if there is a scalar function $f$ such that: $\vb{F} = \nabla f$\\ 244 | \tab $\text{curl }\vb{F} = \curl{\vb{F}}$\\ 245 | \tab $\text{div }\vb{F} = \div{\vb{F}}$\\ 246 | \tab $\nabla\times(\grad{f}) = \vb{0}$ 247 | 248 | If $\div{\vb{F}} \neq 0$, then $\vb{F}$ cannot be a curl of another vector field. 249 | 250 | \subsubsection{Line Integrals} 251 | \tab $\int_C f(x,y,z)\,ds = \int_{a}^{b} f(\vb{r}(t))\,|\vb{r}'(t)|\,dt$\\ 252 | where $\vb{r}(t)$ is the parametrization of $C$. 253 | 254 | \tab $\int_C \vb{F}\cdot d\vb{r} = \int_{a}^{b} \vb{F(\vb{r}(t))}\cdot \vb{r}'(t)\,dt = \int_{a}^{b} \vb{F}\cdot\vb{T}\,ds$\\ 255 | where $\vb{T}$ is the unit tangential vector $\frac{\vb{r}'(t)}{|\vb{r}'(t)|}$ 256 | 257 | Fundamental Theorem of Line Integrals:\\ 258 | \tab $\int_C \grad{f} \cdot d\vb{r} = f(\vb{r}(b)) - f(\vb{r}(a))$ 259 | 260 | \subsubsection{Independence of Path \& Conservativeness} 261 | \begin{itemize}[leftmargin=0.5cm] 262 | \vspace{-0.5em} 263 | \setlength\itemsep{-0.3em} 264 | \item Let $\vb{F}$ be a continuous vector field on the domain $D$. We have independence of path in $D$ if:\\ 265 | \tab $\int_{C_1} \vb{F}\cdot d\vb{r} = \int_{C_2} \vb{F}\cdot d\vb{r}$\\ 266 | \item If $\vb{F}$ is conservative, then $\int_{C} \vb{F}\cdot d\vb{r}$ is independent of path. 267 | 268 | \item If Clairaut's Theorem fails, then $\vb{F}$ is not conservative. 269 | 270 | \item If $D$ is an open (boundary points are not on the domain) and connected region and $\vb{F}$ is a continuous vector field of $D$, then if $\int_{C_1} \vb{F}\cdot d\vb{r}$ is independent of path in $D$, then $\vb{F}$ is conservative. 271 | 272 | \item Green's Theorem evaluates to 0 for any conservative vector field. 273 | 274 | \item $\vb{F}$ is conservative if and only if $\nabla \times \vb{F} = 0$ 275 | \end{itemize} 276 | 277 | \subsubsection{Green's Theorem} 278 | Let $C$ be a simple piecewise smooth curve that bounds a region $D$ in the plane. If $P(x,y)$ and $Q(x,y)$ have continuous partials in an open region containing $D$, then\\ 279 | \tab $\int_C P\,dx +Q\,dy = \iint_D \pdv{Q}{x} 280 | -\pdv{P}{y} \,dA$ 281 | 282 | If $\vb{F}$ is a vector field with third component 0 defined on a domain $D$ enclosed by boundary $C$ then\\ 283 | \tab $\oint_{C} \vb{F}\cdot d\vb{r} = \iint_D (\curl{\vb{F}})\cdot\vb{k}\,dA.$ 284 | 285 | Similarly, if $C$ is defined by $\vb{r}(t) = \left$\\ 286 | \tab $\oint_{C} \vb{F}\cdot \vb{n}\,ds = \iint_D \div{\vb{F}}\,dA$ 287 | 288 | \subsubsection{Parametric Surfaces} 289 | Plane through $\vb{r}_0$ parallel to the non-parallel vectors $\vb{v}_1$, $\vb{v}_2$:\\ 290 | \tab $\vb{r}(s,t) = \vb{r}_0 + s\vb{v}_1 + t\vb{v}_2$ 291 | 292 | Graph of a function $z = f(x,y)$:\\ 293 | \tab $\vb{r}(x,y) = \left$ 294 | 295 | Cylinder about the x-axis:\\ 296 | \tab $\vb{r}(x,\theta) = \left$ 297 | 298 | A cone given by $z = a\sqrt{x^2 + y^2}$:\\ 299 | \tab $\vb{r}(r,\theta) = \left$ 300 | 301 | An ellipsoid given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$\\ 302 | \tab $\vb{r}(\phi,\theta) = \left$ 303 | 304 | A smooth parametric surface given by the equation $\vb{r}(u,v) = \left$ in a domain $D$ and $S$ is covered once throughout the parameter domain $D$, then the area of $S$ is:\\ 305 | \tab $A(S) = \iint_D |\vb{r}_u \times \vb{r}_v|\,dA$ 306 | 307 | Surface area of a graph of a function:\\ 308 | \tab $A(S) = \iint_D \sqrt{1 + (f_x)^2 + (f_y)^2}\,dA$ 309 | 310 | 311 | \subsubsection{Surface Integrals} 312 | 313 | $\iint_S f(x,y,z)\,d\vb{S} = \iint_D f(\vb{r}(u,v))\, |\vb{r}_u \times \vb{r}_v|\,dA$\\ 314 | \vspace{0.2cm} 315 | Graph of a function $z = g(x,y)$:\\ 316 | $\iint_S f(x,y,z)\,d\vb{S} = \iint_D f(x,y,g(x,y)) \sqrt{1 + (g_x)^2 + (g_y)^2}\,dA$\\ 317 | \vspace{0.2cm} 318 | A surface $S$ given by a graph $z = g(x,y)$:\\ 319 | \tab $\vb{F} \cdot (\vb{r}_x \times \vb{r}_y) = \left< P,Q,R \right> \cdot \left<-\pdv{g}{x},-\pdv{g}{y},1\right> \text{ so}$\\ 320 | \tab $\iint_S \vb{F} \cdot \,d\vb{S} = \iint_D (-P\pdv{g}{x} - Q\pdv{g}{y} + R)\, dA$\\ 321 | Vector field:\\ 322 | $\iint_S f(x,y,z)\,d\vb{S} = \iint_S \vb{F}\cdot\vb{n}\,dS = \iint_D \vb{F} \cdot (\vb{r}_u \times \vb{r}_v)\, dA$ 323 | 324 | \subsubsection{Stokes' Theorem} 325 | Let $S$ be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve $C$ with positive orientation. Let $\vb{F}$ be a vector field whose components have continuous partial derivatives on an open region in $\mathbb{R}^3$ that contains $S$. Then\\ 326 | $\oint_{C} \vb{F}\cdot d\vb{r} 327 | =\iint_D(\nabla\times \vb{F})\,d\vb{S} = \iint_D(\nabla\times \vb{F})\cdot \vb{n}\,dA$ 328 | 329 | \subsubsection{Divergence Theorem} 330 | Let $E$ be a solid region in $\mathbb{R}^3$ with piecewise smooth boundary surface $S$ (given the outward orientation). Let $\vb{F}$ be a vector field with continuous partial derivatives on a region containing $E$. Then\\ 331 | \tab $\iint_D \vb{F}\cdot\,d\vb{S} = \iint_D \vb{F}\cdot\vb{n}\,dS=\iiint_E \nabla\cdot\vb{F}\,dV.$\\ 332 | The flux of $\vb{F}$ across the boundary surface of $E$ is equal to the triple integral of the divergence of $\vb{F}$ over $E$.\\ 333 | For cases where it is too difficult to parameterize a surface, the surface is not closed, and we cannot use Stokes' Theorem, we can close the surface and apply the Divergence Theorem. (Later subtract the contribution from the closed surface) 334 | 335 | \subsubsection{Trigonometric Identities} 336 | \begin{itemize}[leftmargin=0.5cm] 337 | \vspace{-0.5em} 338 | \setlength\itemsep{-0.3em} 339 | \item $\ds \sin^2 x = \frac{1 - \cos(2x)}{2}$ 340 | \item $\ds \cos^2 x = \frac{1 + \cos(2x)}{2}$ 341 | \item $\sin(2x) = 2\sin x\cos x$ 342 | \item $\cos(2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x$ 343 | \end{itemize} 344 | 345 | % Footer content 346 | \rule{0.3\linewidth}{0.25pt} 347 | \scriptsize\\ 348 | Updated \today\\ 349 | \href{https://github.com/DonneyF/formula-sheets}{https://github.com/DonneyF/formula-sheets} 350 | 351 | \end{multicols} 352 | \end{document} --------------------------------------------------------------------------------