├── .gitignore ├── Bibliography.bib ├── LICENSING ├── README.md ├── appendix ├── logic.tex └── main.tex ├── continuous ├── algebra.tex ├── complex.tex ├── derivatives.tex ├── derivatives │ ├── chainrule_1.eps │ ├── diffquot.eps │ ├── diffquot.svg │ ├── intcurves.eps │ ├── lin_ex1.eps │ ├── lineform.eps │ ├── lineform_slope.eps │ ├── lineform_slope.svg │ ├── tangent.eps │ ├── tangent.svg │ ├── x3.eps │ ├── x3_1.eps │ ├── x3_2.eps │ ├── x3_3.eps │ ├── x3_4.eps │ ├── x3vline.eps │ ├── xsquared.eps │ └── xsquared.svg ├── functions.tex ├── functions │ ├── 4xmx2.eps │ ├── function.eps │ ├── function.svg │ ├── halfxeg.eps │ ├── halfxeginv.eps │ ├── lawcosines.eps │ ├── pyth.eps │ ├── pyth.svg │ ├── sqrtx.eps │ ├── unitcirc.eps │ ├── vlt1.eps │ ├── vlt1.svg │ ├── vlt1.svg.png │ ├── vlt2.eps │ ├── vlt2.svg │ ├── x2.eps │ └── x2inv.eps ├── integration.tex ├── integration │ ├── 1sqrtx.eps │ ├── asectheta.eps │ ├── asectheta.svg │ ├── asintheta.eps │ ├── asintheta.svg │ ├── atantheta.eps │ ├── atantheta.svg │ ├── improper_01.eps │ ├── rei1.eps │ ├── rei1.svg │ ├── rei2.eps │ ├── rei2.svg │ ├── rei3.eps │ ├── rei3.svg │ ├── rei4.eps │ ├── rei4.svg │ ├── rei5.eps │ ├── rei5.svg │ ├── scatterplot-trend.eps │ ├── scatterplot.eps │ ├── secexample.eps │ └── secexample.svg ├── limits.tex ├── limits │ ├── 4xmx2.eps │ ├── circleeq.eps │ ├── indeterminate_form.eps │ ├── indeterminate_form.svg │ ├── infinited.eps │ ├── infinited.svg │ ├── jumps.eps │ ├── jumps.svg │ ├── llmt.eps │ ├── llmt.svg │ ├── sintheta.eps │ └── tangent.eps ├── main.tex ├── ode.tex ├── ode │ ├── 11df.eps │ ├── 13df.eps │ ├── 1df.eps │ ├── 3df.eps │ ├── 5df.eps │ ├── 7df.eps │ ├── 9df.eps │ ├── defn_e.eps │ ├── defn_e.svg │ ├── ept.eps │ ├── freefall.eps │ ├── freefall.svg │ ├── y0greater.eps │ └── y0less.eps ├── sequence │ ├── conv1.eps │ ├── conv2.eps │ ├── conv3.eps │ ├── lwrbnd.eps │ ├── nondecreasing.eps │ ├── nonincreasing.eps │ └── uprbnd.eps ├── sequences.tex ├── series.tex ├── series │ ├── 1storder.eps │ ├── 2ndorder.eps │ ├── etx.eps │ ├── etx2.eps │ ├── geopower.eps │ ├── henxs.eps │ ├── lnxtaylor.eps │ └── series-3n6n.eps ├── transcend │ ├── circleeq.svg │ └── natlog.eps ├── transcendental.tex ├── trig │ ├── basictrig.eps │ ├── basictrig.svg │ ├── lawcosines.eps │ ├── lawcosines.svg │ ├── pythcircle.eps │ └── pythcircle.svg ├── trigonometry.tex └── unitcircle.tex ├── discrete ├── algorithms.tex ├── counting.tex ├── inference.tex ├── main.tex ├── predicates.tex ├── proofs.tex ├── propositional.tex ├── recursion.tex ├── recursion │ ├── fibonacci.eps │ └── nfact.eps ├── sets.tex └── sets │ ├── equal.eps │ ├── equal.svg │ ├── intersection.eps │ ├── preunion.eps │ ├── subset.eps │ ├── subset.svg │ └── union.eps ├── fibonacci ├── fib.tex ├── fibgraph1.tex ├── fibpic.tex ├── grat.tex └── vitruvian.jpg ├── fitch.sty ├── frontmatter ├── cc-license.tex ├── main.tex ├── preface.tex └── toc.tex ├── graphs ├── 2pn6pn.eps ├── 3pn2pn6pn.eps ├── 3pn6pn.eps ├── 5p2sqn3.eps ├── 5p2sqn3p1.eps ├── 5p2sqxx3.eps ├── arctanx.eps ├── logabsx.eps ├── n51m2xs.eps ├── n5xp3x4.eps ├── nf2nfp1.eps ├── nf2nfp1.pdf ├── np22nm1.pdf ├── np22nm1.svg ├── oneovertwoton.eps ├── p1ch1xp2.eps ├── p1ch3x2m1xm1.eps ├── p1sin1x.pdf ├── pwlimex1.eps ├── sandwichtheorem.pdf ├── tanx.eps └── xsquared.eps ├── notes.tex ├── notes.tex.latexmain ├── photos ├── cauchy.jpg ├── cont2.png ├── cont_1.png ├── cover.png ├── desktop.png ├── googlenotes.png ├── legalpads.jpg ├── preview1.png ├── preview2.png ├── preview3.png ├── preview4.png ├── preview5.png ├── preview6.png ├── pythagoras.jpg ├── realset.eps ├── tabletnotes.jpg └── vim.png ├── physics ├── circuits.tex ├── electric.tex ├── main.tex └── newton.tex ├── preamble.tex ├── resources ├── 1qgraphoutline.svg ├── graphoutline.svg └── inkscape.tex └── titlepage.tex /.gitignore: -------------------------------------------------------------------------------- 1 | # Compiled source # 2 | ################### 3 | *.com 4 | *.class 5 | *.dll 6 | *.exe 7 | *.o 8 | *.so 9 | *.out 10 | 11 | # Packages # 12 | ############ 13 | # it's better to unpack these files and commit the raw source 14 | # git has its own built in compression methods 15 | *.7z 16 | *.dmg 17 | *.gz 18 | *.iso 19 | *.jar 20 | *.rar 21 | *.tar 22 | *.zip 23 | 24 | # Logs and databases # 25 | ###################### 26 | *.log 27 | *.sql 28 | *.sqlite 29 | 30 | # OS generated files # 31 | ###################### 32 | .DS_Store 33 | .DS_Store? 34 | ._* 35 | .Spotlight-V100 36 | .Trashes 37 | Icon? 38 | ehthumbs.db 39 | Thumbs.db 40 | 41 | # LaTeX # 42 | ######### 43 | *.bbl 44 | *.blg 45 | *.ilg 46 | *.log 47 | *.swp 48 | *.pdf 49 | *.toc 50 | *.idx 51 | *.swo 52 | *.ind 53 | *.pdf# 54 | *.aux 55 | *.eps-out 56 | *.eps_tex 57 | *.dvi 58 | 59 | *.fdb_latexmk 60 | *.fls 61 | *.lof 62 | *.lot 63 | *.xdv 64 | *.d 65 | *.brf 66 | 67 | # other 68 | *.old 69 | tags 70 | -------------------------------------------------------------------------------- /Bibliography.bib: -------------------------------------------------------------------------------- 1 | @misc{freenotes, 2 | title = "The Free Lecture Notes Page", 3 | url="http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/" 4 | } 5 | 6 | @misc{wiki:transcendental, 7 | author = "Wikipedia", 8 | title = "Transcendental function --- Wikipedia{,} The Free Encyclopedia", 9 | year = "2012", 10 | url = "http://en.wikipedia.org/w/index.php?title=Transcendental_function&oldid=478129834", 11 | note = "[Online; accessed 3-April-2012]" 12 | } 13 | 14 | @book{thomas, 15 | author = "George B. Thomas, Jr. and Maurice D. Weir and Joel Hass", 16 | title = "Thomas' Calculus", 17 | publisher = "Addison-Wesley", 18 | edition = "12", 19 | year = "2010", 20 | } 21 | 22 | @misc{wiktionary-calculus, 23 | title = "calculus - Wiktionary", 24 | url = "http://en.wiktionary.org/wiki/calculus", 25 | } 26 | 27 | @book{spivak, 28 | author = "Michael Spivak", 29 | title = "Calculus", 30 | publisher = "Publish or Perish, Inc.", 31 | edition = "3rd", 32 | year = "1994", 33 | } 34 | 35 | @book{pinter, 36 | author = "Charles C. Pinter", 37 | title = "A Book of Abstract Algebra", 38 | publisher = "McGraw-Hill", 39 | edition = "2nd", 40 | year = "1990", 41 | } 42 | 43 | @book{boycede, 44 | author = "William E. Boyce, Richard C. DiPrima", 45 | title = "Elementary Differential Equations and Boundary Value Problems", 46 | publisher = "Laurie Rosatone", 47 | edition = "9th", 48 | year = "2009", 49 | } 50 | 51 | @book{coddington, 52 | author = "Earl A. Coddington", 53 | title = "An Introduction To Ordinary Differential Equations", 54 | publisher = "Dover Publications", 55 | edition = "", 56 | year = "1989", 57 | } 58 | 59 | @book{mcsfull, 60 | author = "Eric Lehman, F Thompson Leighton, Albert R Meyer", 61 | title = "Mathematics for Computer Science", 62 | publisher = "MIT OpenCourseWare", 63 | year = "2012", 64 | } 65 | 66 | @book{serway, 67 | author = "Raymond A. Serway and John W. Jewett, Jr.", 68 | title= "Physics for Scientists and Engineers with modern physics", 69 | publisher = "Brooks/Cole", 70 | address = "10 Davis Drive Belmont, CA 94002-3098 USA", 71 | edition = "7", 72 | year = "2008", 73 | } 74 | 75 | @book{rosen, 76 | author = "Kenneth H. Rosen", 77 | title = "Discrete Mathematics and Its Applications", 78 | publisher = "McGraw-Hill", 79 | edition = "7", 80 | year = "2012", 81 | } 82 | 83 | @book{newton, 84 | author = "Sir Isaac Newton", 85 | title = "The Mathematical Principles of Natural Philosophy", 86 | publisher = "B. Motte", 87 | year = "1729", 88 | url = "http://books.google.com/books?id=Tm0FAAAAQAAJ", 89 | } 90 | 91 | @misc{britannica12, 92 | author = "Leonardo Pisano", 93 | title = "Encyclop\ae dia Britannica Online", 94 | year = "2012", 95 | url = 96 | "http://www.britannica.com/EBchecked/topic/336467/Leonardo-Pisano", 97 | note = "[Online; accessed 20-March-2012]" 98 | } 99 | 100 | @book{newton, 101 | author = "William Lidwell, Kritina Holden, Jill Butler", 102 | title = "Universal Principles of Design, Revised and Updated: 125 ways to 103 | Enhance Usability, Influence Perception, Increase Appeal, Make Better Design 104 | Decisions, and teach through Design", 105 | publisher = "Rockport Publishers", 106 | year = "2010", 107 | url = "http://books.google.com/books?id=Tm0FAAAAQAAJ", 108 | } 109 | 110 | @misc{mwbinet, 111 | author = "Eric W. Weisstein", 112 | title = "Binet's Fibonacci Number Formula", 113 | year = "2012", 114 | url = 115 | "http://mathworld.wolfram.com/BinetsFibonacciNumberFormula.html" 116 | } 117 | 118 | @misc{mwfib, 119 | author = "Pravin Chandra and Eric W. Weisstein", 120 | title = "Fibonacci Number", 121 | year = "2012", 122 | url = 123 | "http://mathworld.wolfram.com/FibonacciNumber.html" 124 | } 125 | 126 | @misc{mwgolden, 127 | author = "Eric W. Weisstein", 128 | title = "Golden Ratio", 129 | year = "2012", 130 | url = 131 | "http://mathworld.wolfram.com/GoldenRatio.html" 132 | } 133 | 134 | @misc{tikzunitcirc, 135 | author = "Supreme Aryal", 136 | title = "Unit circle", 137 | year = "2010", 138 | url = 139 | "http://www.texample.net/tikz/examples/unit-circle/", 140 | note = "[Online; accessed 14-April-2012]" 141 | } 142 | 143 | @book{gentle, 144 | author = "Maarten M. Fokkinga", 145 | title = "A Gentle Introduction to Category Theory", 146 | publisher = "University of Twente", 147 | year = "1994", 148 | } 149 | 150 | @book{aris-interp, 151 | author = "Aristotle", 152 | title = "On Interpretation", 153 | year = "350 B.C.E.", 154 | } 155 | 156 | @book{trudeau, 157 | author = "Richard J. Trudeau", 158 | title = "Introduction to Graph Theory", 159 | publisher = "Dover Publications", 160 | year = "1993", 161 | } 162 | -------------------------------------------------------------------------------- /LICENSING: -------------------------------------------------------------------------------- 1 | This text (and its source code, excluding images created by others or possible copyrighted text used by accident without attribution) is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. 2 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | texnotes 2 | -------- 3 | 4 | In order to compile this document, you will need a LaTeX distribution. I use 5 | texlive, so that's the best supported. On Arch Linux, this is provided by 6 | the package texlive-most. Other Linux distros will require different setups. 7 | 8 | If any packages are not included when you attempt to compile, they should be 9 | available at CTAN. I don't use anything too obscure in this document. 10 | 11 | To compile, run these commands in order: 12 | 13 | pdflatex notes.tex 14 | biblatex notes 15 | makeindex notes 16 | pdflatex notes.tex 17 | pdflatex notes.tex 18 | 19 | in the root folder. 20 | 21 | -------------------------------------------------------------------------------- /appendix/logic.tex: -------------------------------------------------------------------------------- 1 | \chapter{Logic Circuits} \index{logic circuits} 2 | \epigraph{ 3 | Look, everything we're putting into that box becomes ungrounded, and I don't 4 | mean grounded like to the earth, I mean, not tethered. I mean, we're blocking 5 | whatever keeps it moving forward and so they flip-flop. Inside the box it's like 6 | a street, both ends are cul-de-sacs. I mean, this isn't frame dragging or 7 | wormhole magic, this is basic mechanics and heat 101.} 8 | {\emph{Primer}, 2004} 9 | 10 | A logic circuit receives input signals \(p_1, p_2, \ldots, p_n\), each a bit, and produces output signals 11 | \(s_1, s_2, \ldots, s_n\), each a bit. 12 | 13 | \begin{figure}[h] 14 | \center{ 15 | \subfigure[and]{ 16 | \begin{circuitikz} \draw 17 | (0,2) node[and port] (myand1) {} 18 | (myand1.in 1) node[anchor=east] {} 19 | (myand1.in 2) node[anchor=east] {} 20 | (myand1.out) node[anchor=west] {}; 21 | \end{circuitikz} 22 | } 23 | \subfigure[or]{ 24 | \begin{circuitikz} \draw 25 | (0,2) node[or port] (myor1) {} 26 | (myor1.in 1) node[anchor=east] {} 27 | (myor1.in 2) node[anchor=east] {} 28 | (myor1.out) node[anchor=west] {}; 29 | \end{circuitikz} 30 | } 31 | \subfigure[nor]{ 32 | \begin{circuitikz} \draw 33 | (0,2) node[nor port] (mynor1) {} 34 | (mynor1.in 1) node[anchor=east] {} 35 | (mynor1.in 2) node[anchor=east] {} 36 | (mynor1.out) node[anchor=west] {}; 37 | \end{circuitikz} 38 | } 39 | \subfigure[xor]{ 40 | \begin{circuitikz} \draw 41 | (0,2) node[xor port] (myxor1) {} 42 | (myxor1.in 1) node[anchor=east] {} 43 | (myxor1.in 2) node[anchor=east] {} 44 | (myxor1.out) node[anchor=west] {}; 45 | \end{circuitikz} 46 | } 47 | \subfigure[not]{ 48 | \begin{circuitikz} \draw 49 | (0,2) node[not port] (mynot1) {} 50 | (mynot1.in) node[anchor=east] {} 51 | (mynot1.out) node[anchor=west] {}; 52 | \end{circuitikz} 53 | } 54 | } 55 | \caption{Basic logic gates.} 56 | \end{figure} 57 | 58 | \begin{comment} 59 | \begin{figure}[h] 60 | \begin{center} 61 | \begin{circuitikz} 62 | \draw 63 | (8, 2) node[and port] (and0) {} 64 | 65 | (3, 4) node[or port] (or0) {} 66 | (3, 0) node[or port] (or1) {} 67 | (5, 2) node[or port] (or2) {} 68 | 69 | (1, 4) node[not port] (not0) {} 70 | (2, 2) node[not port] (not1) {} 71 | (1, 0) node[not port] (not2) {} 72 | 73 | (not0.out) -- (or0.in 2) 74 | (not0.in) -- (0, 4) node[anchor=east] {\(p_2\)} 75 | (or0.in 1) -- (0, 5) node[anchor=east] {\(p_1\)} 76 | (not1.in) -- (0, 3) node[anchor=east] {\(p_3\)} 77 | (not2.in) -- (0, 0) node[anchor=east] {\(p_5\)} 78 | (or1.in) -- (0, 2) node[anchor=east] {\(p_4\)} 79 | (and0.out) -- (9,2) node[anchor=west] {\(s_1\)} 80 | (or1.out) -- (or2.in 2) 81 | (or0.out) -- (and0.in 1) 82 | (not1.out) -- (or2.in 1) 83 | (not2.out) -- (or1.in 2) 84 | (or2.out) -- (and0.in 2); 85 | \end{circuitikz} 86 | \end{center} 87 | \caption{A simple logic circuit} 88 | \end{figure} 89 | \end{comment} 90 | 91 | -------------------------------------------------------------------------------- /appendix/main.tex: -------------------------------------------------------------------------------- 1 | \chapter{Important Concepts} 2 | 3 | \section{Quadratic Formula} 4 | \index{quadratic formula} 5 | Quadratic formula\index{quadratic formula} 6 | \begin{equation} 7 | x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} 8 | \label{app:eq:quadratic} 9 | \end{equation} 10 | 11 | \section{Point-slope Formula} 12 | \label{sec:pointslope} 13 | \index{point-slope formula} 14 | The \textbf{point-slope formula} allows us, given a point $(x_1, y_1)$ and a slope $m$, to solve for $y$ as a function of $x$. 15 | \begin{equation} 16 | \label{eq:pointslope} 17 | y-y_1=m(x-x_1) 18 | \end{equation} 19 | 20 | \section{Conjugate}\label{app:def:conjugate} 21 | \index{conjugate} 22 | In algebra, the \textbf{conjugate}\index{conjugate} of a \emph{binomial} is another binomial formed by taking the opposite of the second term of the first binomial. For the initial binomial 23 | \[ a + b\] 24 | its conjugate would be 25 | \[a - b.\] 26 | 27 | Meanwhile, for the expression \[a^2+b^2\] we can factor this to produce \[(a-b)(a+b)\] where one expression is the conjugate of the other. 28 | 29 | \chapter{Proofs} 30 | 31 | \section{Power Rule for Derivatives} 32 | The \emph{power rule for derivatives}\index{power rule} states that 33 | \begin{equation} 34 | \label{eq:pwrrlprf} 35 | \ddx x^n=nx^{n-1} 36 | \end{equation} 37 | \begin{proof} 38 | To prove this, we use the limit definition of a derivative: 39 | \begin{equation} 40 | \label{eq:lmtdefprf} 41 | \frac{\ud f(x)}{\ud x}=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} 42 | \end{equation} 43 | And assume that $f(x)$ is of the form $f(x)=x^n$. 44 | 45 | Look at the term $(x+h)^n$. Take $n=1$. In this case: 46 | \begin{align*} 47 | (x+h)^1 &= x+h \\ 48 | \intertext{Try it for $n=2$.} 49 | (x+h)^2 &= (x+h)(x+h) \\ 50 | &= x^2+2hx+h^2 \\ 51 | \intertext{Now for $n=3$.} 52 | (x+h)^3 &= (x+h)(x+h)^2 \\ 53 | &= (x+h)(x^2+2hx+h^2) \\ 54 | &= x^3 + 2hx^2 + h^2x + hx^2 + 2h^2x + h^3 \\ 55 | &= x^3 +3hx^2 + 3h^2x +h^3 56 | \intertext{We are beginning to see a pattern in each of these sums: the first term is $x^n$, and each term after that has a common factor of $h$. 57 | Furthermore, it looks like there is always a term in the sum that has only one $h$ within. 58 | Let's find this for $n=4$ to be sure:} 59 | (x+h)^4 &= (x+h)(x+h)^3 \\ 60 | &= (x+h)(x^3+3hx^2+3h^2x+h^3) \\ 61 | &= x^4+3hx^3+3h^2x^2+h^3x+x^3h+3hx^2x^2+3h^3x+h^4 \\ 62 | \intertext{Simplify.} 63 | (x+h)^4&= x^4 +4hx^3+6h^2x^2+4h^3x+h^3x+x^3h+h^4 \\ 64 | \end{align*} 65 | It looks like the pattern follows for $n=4$. 66 | We can claim that this holds for any value $n$, supposing that $f(x)$ is of the form $x^n$. 67 | 68 | What does this mean for our limit equation \eqref{eq:lmtdefprf}? 69 | \begin{align*} 70 | \ddx f(x)&=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\ 71 | \intertext{Well, it means that the first term of our expanded polynomial should always cancel with the $\cdots -f(x)$ term if $n\neq 1$. 72 | It also means that we should always be able to divide our numerator by $h$ to calculate the limit, assuming, again, that $n\neq 1$. 73 | Our limit would then look like this:} 74 | \ddx f(x) &= \lim_{h \to 0} \frac{(x^n+n\cdot h x^{n-1}+\cdots +h^n)-x^n}{n} &n\neq 1\\ 75 | \intertext{$x^n$ terms cancel.} 76 | &=\lim_{h \to 0} \frac{n\cdot h x^{n-1}\cdots+h^n}{h}&n\neq 1\\ 77 | \\ 78 | \intertext{Divide by $h$} 79 | &=\lim_{h \to 0} nx^{n-1}\cdots+h^{n-1} &n\neq 1, \quad h\neq 0\\ 80 | \intertext{All terms following the first go to $0$ as $h \to 0$, and we are left with} 81 | \ddx f(x) &= nx^{n-1} 82 | \end{align*} 83 | Which is the same as \eqref{eq:pwrrlprf}. 84 | \end{proof} 85 | 86 | \section{Sandwich Theorem for Sequences} 87 | 88 | I didn't write any of this. Check the citations for my sources. 89 | 90 | \label{proof:sandwichsequence} 91 | \begin{theorem}[The Sandwich Theorem for Sequences]\index{The Sandwich 92 | Theorem for Sequences} 93 | \label{app:th:sandwichsequence} 94 | Let $\{a_n\}$, $\{b_n\}$, and $\{c_n\}$ be sequences of real numbers. If 95 | $a_n \leq b_n \leq c_n $ holds for all $n$ beyond some index $N$, and if 96 | $\lim_{n\to\infty} a_n = \lim_{n\to\infty} c_n = L$, then 97 | $\lim_{n\to\infty} b_n = L$ also. 98 | \end{theorem} 99 | \begin{theorem} 100 | \label{app:th:xninfty} 101 | If \(|x| < 1, \lim_{n \to \infty} x^n = 0\). 102 | \begin{proof} 103 | We need to show that to each $\varepsilon > 0$ there corresponds an 104 | integer $N$ so large that $\forall n > N \big( |x^n| < \varepsilon\big)$ 105 | Since $\varepsilon^{1/n}\to 1$, while $|x|<1$, $\exists N \big( 106 | \varepsilon^{1/n} > |x|$. In other words, 107 | \begin{equation}\label{app:eq:limxto1} 108 | \Big|x^N\Big| = \Big|x^N\Big| < \varepsilon. 109 | \end{equation} 110 | This is the integer we seek because, if $|x|< 1$, then 111 | \begin{equation}\label{app:eq:limxto2} 112 | \forall n > N \bigg( \Big|x^n\Big| < \Big|x^N\Big|\bigg) 113 | \end{equation} 114 | Combining \eqref{app:eq:limxto1} and \eqref{app:eq:limxto2} produces 115 | $|x^n|<\varepsilon$ for all $n > N$, concluding the 116 | proof.\cite[p.~AP-21]{thomas} 117 | \end{proof} 118 | \end{theorem} 119 | \label{proof:limxnnf} 120 | \begin{theorem} 121 | % Theorem and proof are a direct rip from Thomas' Calculus. 122 | For any number $x$, $\lim_{n\to\infty} \frac{x^n}{n!} = 0$. 123 | \cite[p.~AP-22]{thomas} 124 | \begin{proof} 125 | Since 126 | \[ - \frac{|x|^n}{n!} \leq \frac{x^n}{n!} \leq \frac{|x|^n}{n!},\] 127 | all we need to show is that $|x|^n/n! \to 0$. We can then apply theorem 128 | \ref{app:th:sandwichsequence} to conclude that $x^n/n! \to 0$. 129 | 130 | The first step in showing that $|x|^n/n! \to 0$ is to choose an integer $M 131 | > |x|$, so that $(|x|/M) < 1$. By Theorem \ref{app:th:xninfty}, we then have 132 | that $\left( |x|/M \right)^n \to 0$. We then restrict our attention to 133 | values of $n > M$. For these values of $n$, we can write 134 | \begin{align*} 135 | \frac{|x|^n}{n!}&= 136 | \frac{|x|^n}{1\cdot2\cdot\cdots\cdot M 137 | \cdot\underbrace{(M+1)\cdot(M+2)\cdot\cdots n}_{(n-M) \text{ 138 | factors}}} 139 | \\ 140 | &\leq \frac{|x|^n}{M!M^{n-M}}=\frac{|x|^n \cdot M^M}{M^n \cdot M!} = 141 | \frac{M^M}{M!} \left( \frac{|x|}{M} \right)^n . 142 | \end{align*} 143 | Thus, 144 | \[ 0 \leq \frac{|x|^n}{n!}\leq \frac{M^M}{M!}\left( \frac{|x|}{M} 145 | \right)^n .\] 146 | Now, the constant $M^M / M!$ does not change as $n$ increases. Thus 147 | Theorem \ref{app:th:sandwichsequence} tells us that $|x|^n/n! \to 0$ because 148 | $(|x|/M)^n \to 0$. 149 | \cite[p.~AP-22]{thomas} 150 | \end{proof} 151 | \end{theorem} 152 | 153 | \section{L'Hospital's Rule} 154 | \label{proof:lhospital} 155 | \begin{theorem}[L'Hospital's Rule] 156 | Suppose that $f(a)=g(a)=0$, that $f$ and $g$ are differentiable on an open interval $I$ containing $a$, and that $g'(x) \neq 0$ on $I$ if $x \neq a$. Then 157 | \[ \lim_{x \to a} \frac {f(x)}{g(x)} \=H \lim_{x \to a} \frac{f'(x)}{g'(x)} \] 158 | assuming that the limit on the right side of this equation exists. 159 | \end{theorem} 160 | \begin{proof} 161 | We first establish the limit equation for the case $x \to a^+$. The method needs almost no change to apply to $x \to a^{-}$, and the combination of these two cases establishes the result. 162 | 163 | Suppose that $x$ lies to the right of $a$. Then $g'(x) \neq 0$, and we can apply Cauchy's Mean Value Theorem to the closed interval from $a$ to $x$. This step produces a number $c$ between $a$ and $x$ such that 164 | $$ \frac{f'(c)}{g'(c)}=\frac{f(x)-f(a)}{g(x)-g(a)} $$ 165 | But $f(a)=g(a)$, so 166 | $$ \frac{f'(c)}{g'(c)}=\frac{f(x)}{g(x)} $$ 167 | As $x$ approaches $a$, $c$ approaches $a$ because it always lies between $a$ and $x$. Therefore, 168 | $$ \lim_{x \to a^+} \frac{f(x)}{g(x)}=lim_{c \to a} \frac{f'(c)}{g'(c)} = lim_{x \to a^+} \frac{f'(x)}{g'(x)} $$ 169 | which establishes l'Hospital's Rule for the case where $x$ approaches $a$ from 170 | above. The case where $x$ approaches $a$ from below is proved by applying 171 | Cauchy's Mean Value Theorem (found in Section \ref{sec:lhospital}) to the closed interval $[x,a], x < 172 | a$.\cite{thomas} 173 | \end{proof} 174 | \input{appendix/logic.tex} 175 | 176 | \chapter{The Fibonacci Sequence and the Golden Ratio} 177 | \input{fibonacci/fib.tex} 178 | -------------------------------------------------------------------------------- /continuous/algebra.tex: -------------------------------------------------------------------------------- 1 | \chapter{Algebra} 2 | 3 | In order to understand functions, we need to understand basic algebra. It will 4 | give us a powerful set of tools that we can use to solve problems down the road, 5 | like partial fraction decomposition (\secref{sec-integration-pfd}). 6 | 7 | \section{Laws} 8 | If $a$, $b$, and $c$ are any numbers,\footnote{% 9 | I am sure these all hold for real numbers, and presumably for complex as well, 10 | though other number systems may have different laws. 11 | I have not explored these possibilities. 12 | }then the following laws hold:\footnote{% 13 | Most of this is from the opening chapter of~\cite{spivak}, but bits and pieces 14 | are collected from elsewhere and cited as such. 15 | } 16 | \subsection{Associative law for additon} 17 | \label{sec:alg:assoc:add} 18 | \index{addition!associative law} 19 | The associative law extends our ability to discuss the operation $(+)$ on any 20 | two elements to three elements, without changing the order of these elements: 21 | \begin{equation} 22 | a + ( b + c ) = (a + b) + c. 23 | \end{equation} 24 | It follows from this (though the proof is somewhat complicated, see~\cite[p.~4]{spivak}), 25 | that we may write sums without regard for parentheses. 26 | This means that we may write, for instance 27 | \begin{equation*} 28 | a_1 + a_2 + a_3 + a_4 + \cdots + a_n, 29 | \end{equation*} 30 | without any ambiguity as to what order the operation must be performed. 31 | 32 | \subsection{Existence of an additive identity} 33 | \index{addition!identity element} 34 | The identity element for addition is 0. 35 | This means that the sum of any element and 0 is always the original element. 36 | We write this: 37 | \begin{equation} 38 | a + 0 = 0 + a = a. 39 | \end{equation} 40 | 41 | \subsection{Existence of additive inverses} 42 | \index{addition!inverse} 43 | \begin{equation} 44 | a + (-a) = (-a) + a = 0. 45 | \end{equation} 46 | In this case, we mean that every element in the set $\mathbb{R}$ of real numbers 47 | has an inverse with respect to the operation $(+)$.\cite[p.~14]{pinter} 48 | 49 | \subsection{Commutative law for addition} 50 | \index{addition!commutative law} 51 | \label{sec:alg:comm:add} 52 | This states that the value of $a + b$ or $b + a$ is independent of the order 53 | in which $a$ and $b$ are taken.\cite[p.~14]{pinter} 54 | \begin{equation} 55 | a + b = b + a. 56 | \end{equation} 57 | 58 | \subsection{Associative law for multiplication} 59 | \index{multiplication!associative law} 60 | The associative law for multiplication is analagous with the one for 61 | addition from \secref{sec:alg:assoc:add}. 62 | \begin{equation} 63 | a \cdot (b \cdot c) = (a \cdot b) \cdot c. 64 | \end{equation} 65 | 66 | \subsection{Existence of a multiplicative identity} 67 | \index{multiplication!identity element} 68 | \label{sec:mult:id} 69 | Multiplication of real numbers has an identity element, $1$, 70 | such that multiplying any number by this element gives us the original number: 71 | \begin{equation} 72 | a \cdot 1 = 1 \cdot a = a, \qquad \text{for } 1 \neq 0. 73 | \label{eq:mult:id} 74 | \end{equation} 75 | The notation here is a little strange. 76 | We know that $1$ is the identity element for multiplication, but it also 77 | refers to the number $1$, so why do we state that $1 \neq 0$? 78 | Of course one is not equal to zero! 79 | 80 | The reason for this is that we are talking about the \emph{element} 1, this 81 | being the identity element for multiplication, and not simply the \emph{number} 82 | 1. We may just as well have written: 83 | \begin{equation*} 84 | a \cdot e = e \cdot a = a, \qquad \text{for } e \neq 0, 85 | \end{equation*} 86 | but writing 1 instead of $e$ as in \eref{eq:mult:id} here makes sense, 87 | since 1 is, in fact, both the number and the element in question. 88 | 89 | \subsection{Existence of multiplicative inverses} 90 | For every element $a$ in $\mathbb{R}$, there is an element $a^{-1}$ in 91 | $\mathbb{R}$ such that $a \cdot a^{-1}$ gives us the identity element 92 | from \secref{sec:mult:id}. 93 | \begin{equation} 94 | a \cdot a^{-1} = a^{-1} \cdot a = 1, \qquad \text{for } a \neq 0. 95 | \end{equation} 96 | 97 | \subsection{Commutative law for multiplication} 98 | As with the commutative law for addition (\secref{sec:alg:comm:add}), 99 | this states that the value of $a * b$ or $b * a$ is independent of the order 100 | in which $a$ and $b$ are taken.\cite[p.~14]{pinter} 101 | \begin{equation} 102 | a \cdot b = b \cdot a. 103 | \end{equation} 104 | 105 | \subsection{Distributative law} 106 | \index{distributive law} 107 | The distributive law is a relationship between multiplication and addition. 108 | It allows us to manipulate the order of application when we are combining these 109 | two operations. 110 | \begin{equation} 111 | a \cdot (b + c) = a \cdot b + a \cdot c. 112 | \end{equation} 113 | 114 | \section{Inequality} 115 | \index{inequalities} 116 | When we say $a$ is 117 | \emph{less than}\index{inequalities!less than} 118 | $b$, we write $a < b$, 119 | and take it to mean the same thing as saying $b$ is 120 | \emph{greater than}\index{inequalities!greater than} 121 | $a$ ($b > a$).\cite[p.~9]{spivak} 122 | Thus the numbers $a$ satisfying $a > 0$ are called \emph{positive}, while 123 | those numbers $a$ satisfying $a < 0$ are called \emph{negative}. 124 | 125 | \section{More laws} 126 | \begin{theorem}[Trichotomy law] 127 | \index{trichotomy law} 128 | For every number $a$, one and only one of the following holds: 129 | \begin{enumerate} 130 | \item $a = 0$, 131 | \item $a$ is in the collection $P$, 132 | \item $-a$ is in the collection $P$. 133 | \end{enumerate} 134 | \cite[p.~9]{spivak} 135 | \end{theorem} 136 | 137 | \begin{theorem}[Closure under addition] 138 | \index{addition!closure} 139 | If $a$ and $b$ are in $P$, then $a + b$ is in $P$. 140 | \cite[p.~9]{spivak} 141 | \end{theorem} 142 | 143 | \begin{theorem}[Closure under multiplication] 144 | \index{multiplication!closure} 145 | If $a$ and $b$ are in $P$, then $a \cdot b$ is in $P$. 146 | \cite[p.~9]{spivak} 147 | \end{theorem} 148 | 149 | \begin{defn} 150 | \index{absolute value} 151 | For any number $a$, we define the \emph{absolute value}\index{absolute value} 152 | $|a|$ of $a$ to be:\cite[p.~11]{spivak} 153 | \begin{equation} 154 | |a| = \begin{dcases} 155 | a, &a \geq 0\\ 156 | -a, & a \leq 0. 157 | \end{dcases} 158 | \end{equation} 159 | \end{defn} 160 | 161 | %%% Local Variables: 162 | %%% mode: latex 163 | %%% TeX-master: "../notes" 164 | %%% End: 165 | -------------------------------------------------------------------------------- /continuous/complex.tex: -------------------------------------------------------------------------------- 1 | \chapter{Complex Numbers}\label{ch:complex} 2 | Earl A. Coddington, professor of mathematics at UCLA, offers an extremely helpful crash-course in complex numbers in his book \emph{An Introduction To Ordinary Differential Equations}, Chapter 0 \cite{coddington}. 3 | Most of the initial knowledge in this chapter comes from my notes on that chapter, but I will attempt to provide pictures and examples where I found the source text lacking. 4 | \begin{defn} 5 | A \keyword{complex number}{complex number} is an ordered pair of real numbers $(x, y)$. 6 | If $z$ is a complex number, we write 7 | \begin{equation} 8 | z = (x,y). 9 | \end{equation} 10 | \end{defn} 11 | \begin{defn} 12 | The \keyword{sum}{complex sum} $z_1+z_2$ is the complex number given by 13 | \begin{equation} 14 | z_1 + z_2 = (x_1 + x_2, y_1 + y_2). 15 | \label{eq:complexsum} 16 | \end{equation} 17 | \end{defn} 18 | \begin{defn} 19 | If $z=(x,y)$, the \keyword{negative}{negative} of $z$, denoted $-z$, is defined to be the number 20 | \begin{equation} 21 | -z = (-x, -y). 22 | \end{equation} 23 | \end{defn} 24 | \begin{defn} 25 | The \keyword{zero}{zero} complex number, written simply 0, is defined as 26 | \begin{equation} 27 | 0 = (0, 0). 28 | \end{equation} 29 | \end{defn} 30 | Since \eref{eq:complexsum} defines complex sums in terms of just real number addition operations, and we know that these real number operations are commutative, it follows that 31 | \begin{equation} 32 | z_1+z_2 = z_2 + z_1. 33 | \end{equation} 34 | Likewise does the associative property of addition for real numbers hold for complex numbers: 35 | \begin{equation} 36 | (z_1 + z_2) + z_2 = z_1 + (z_2 + z_3). 37 | \end{equation} 38 | And the number $0$ provides our additive identity: 39 | \begin{equation} 40 | z + 0 = z. 41 | \end{equation} 42 | Finally, we have an additive inverse for complex numbers 43 | \begin{equation} 44 | z+(-z)=0. 45 | \end{equation} 46 | For additional information on these properties as they apply to the set of real numbers, I will direct the reader to Michael Spivak's \emph{Calculus, Third Edition}, perhaps the single greatest introduction to ``real mathematics'' ever written. 47 | These properties, and their importance with regard to real numbers, is detailed extensively in the first chapter. 48 | \begin{defn} 49 | The \keyword{difference}{difference}, $z_1-z_2$, is defined by 50 | \begin{equation} 51 | z_1-z_2 = z_1 + (-z_2). 52 | \end{equation} 53 | \end{defn} 54 | \begin{defn} 55 | The \keyword{product}{product} $z_1z_2$ is defined by 56 | \begin{equation} 57 | z_1z_2 = (x_1x_2 - y_1y_2, x_1 y_2 + x_2 y_1). 58 | \label{eq:complex_product} 59 | \end{equation} 60 | \end{defn} 61 | \begin{remark} 62 | \eref{eq:complex_product} can be found by performing basic multiplication on the following form of the numbers: 63 | \begin{align*} 64 | z_1 &= x_1 + \iu y_1 \\ 65 | z_2 &= x_2 + \iu y_2 \\ 66 | z_1z_2 &= (x_1+\iu y_1)(x_2 + \iu y_2) 67 | \end{align*} 68 | In order to use this, however, we must define the following units: 69 | \end{remark} 70 | \begin{defn} 71 | The \keyword{unit}{unit} complex number is the number $(1,0)$. 72 | This may be multiplied by any complex number $z=(x,y)$ and the product will always be $z$. 73 | \end{defn} 74 | \begin{defn} 75 | The \keyword{imaginary unit}{imaginary unit} is defined to be the number \[\iu = (0,1).\] 76 | \end{defn} 77 | From those definitions, we see that if $z=(x,y)$ we can write it in terms of its real and imaginary parts as follows: 78 | \begin{align} 79 | z&=x(1,0)+y(0,1), 80 | \intertext{which is equivalent to stating} 81 | z&=x+\iu y. 82 | \end{align} 83 | 84 | -------------------------------------------------------------------------------- /continuous/functions/pyth.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 18 | 20 | 53 | 56 | 57 | 59 | 60 | 62 | image/svg+xml 63 | 65 | 66 | 67 | 68 | 69 | 73 | 75 | 82 | 90 | 97 | 98 | c b a 142 | 143 | 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-------------------------------------------------------------------------------- /continuous/main.tex: -------------------------------------------------------------------------------- 1 | \part{Mathematical Analysis} 2 | \thispagestyle{empty} 3 | Here we are going to study functions, series, sequences, and applications of these concepts. 4 | The numbers we will find this section are almost always going to be \emph{real numbers}, 5 | which basically means they behave the way we expect numbers to behave and we are able to plot them on number lines, graph them, and all that fun stuff. 6 | 7 | Most of this part is going to focus on what schools typically call ``calculus,'' 8 | and it covers the mathematics I have learned in my first three semesters as an undergraduate at Christopher Newport University. 9 | 10 | \setcounter{section}{0} 11 | \input{continuous/algebra} 12 | \input{continuous/functions} 13 | \input{continuous/trigonometry} 14 | \input{continuous/limits} 15 | \input{continuous/derivatives} 16 | \input{continuous/transcendental} 17 | \input{continuous/integration} 18 | \input{continuous/sequences} 19 | \input{continuous/series} 20 | \input{continuous/complex} 21 | \input{continuous/ode} 22 | -------------------------------------------------------------------------------- /continuous/ode.tex: -------------------------------------------------------------------------------- 1 | \chapter{Ordinary Differential Equations} 2 | \begin{defn} 3 | A \textbf{differential equation} is an equation involving derivatives. 4 | \end{defn} 5 | \begin{defn} 6 | A \textbf{direction field} tells us the slope of a function at any given place. 7 | \end{defn} 8 | \begin{ex} 9 | In physics, we often define acceleration to be a vector relative to another 10 | vector, velocity. 11 | Here, we will just consider them as scalars for the sake of argument. 12 | Acceleration is a change in velocity, so 13 | \begin{equation} 14 | a = \leib{v}{t}, 15 | \label{eq:acceleration} 16 | \end{equation} 17 | where $v$ represents velocity and $t$ represents time. 18 | Now we integrate both sides of \eref{eq:acceleration} with respect to $t$, 19 | \begin{equation} 20 | \int a \ud t = \int \leib{v}{t} \ud t. 21 | \label{eq:intaccel} 22 | \end{equation} 23 | Now, assuming\footnote{This must be explained later, but as a warning: no, 24 | the $\ud t$ in the derivative operation and the $\ud t$ in the integration 25 | operation do not simply cancel.} 26 | \begin{equation} 27 | \int \leib{v}{t} \ud t = \int \ud v, 28 | \label{eq:handwave} 29 | \end{equation} 30 | then using \eref{eq:handwave}, we see that 31 | \begin{equation} 32 | \int a \ud t = \int \ud v. 33 | \label{eq:naughtint} 34 | \end{equation} 35 | From here we simplify, finding that 36 | \begin{equation} 37 | at + c_1 = v + c_2. 38 | \label{eq:almostvelocity} 39 | \end{equation} 40 | The constants in \eref{eq:almostvelocity} are simply constants and may be combined into another constant, $C$. 41 | Also, the equation may be rearranged to put it in more familiar form, yielding 42 | \begin{equation} 43 | v = at + C, 44 | \label{eq:velocity} 45 | \end{equation} 46 | which we recognize as the classical mechanics equation for velocity. 47 | Integrating once more, and replacing $v$ with the definition of velocity as change in position, we find 48 | \begin{align} 49 | \int \leib{x}{t} \ud t &= \int (at + C) \ud t, \nonumber \\ 50 | \int \leib{x}{t} \ud t &= \int at \ud t + \int C \ud t, \nonumber \\ 51 | x + c_3 &= a \frac{t^2}{2} + Ct + c_4. \nonumber \\ 52 | \intertext{Now we may simply combine the constants once more, defining $C_1$ to constitute the difference of $c_4$, and $c_3$,} 53 | x &= a \frac{t^2}{2} + Ct + C_1. \label{eq:phys_const} 54 | \end{align} 55 | \eref{eq:phys_const} may be rewritten in its more common form: 56 | \begin{equation} 57 | x(t) = \frac{1}{2} a t^2 + v_0 t + x_0. 58 | \label{eq:position} 59 | \end{equation} 60 | \end{ex} 61 | 62 | \section{Second-order differential equations with linear combination solutions} 63 | 64 | \section{Linear, homogeneous} 65 | 66 | \[ ay'' + by'' +cy = 0 \] 67 | 68 | Each coefficient is a constant. We come up with a 69 | characteristic equation\index{characteristic equation} 70 | 71 | \[ ar^2 +br + c = 0 \] 72 | 73 | Which is quadratic. 74 | 75 | \begin{enumerate} 76 | \item We get two distinct solutions, meaning $r1 \neq r2$. this implies 77 | $r1, r2 \in \mathbb{R}$ and our characteristic equation is of the form 78 | $y = c_1 e^{r_1t}+c_2e^{r_2t}$. 79 | \item $r1 = r2 $, and $r1, r2 \in \mathbb{C}$, 80 | meaning we still have a general solution of the form 81 | $y = c_1 e^{r_1t} + c_2 e^{r_2t}$, where $r_1,\quad r_2$ are of the form 82 | $\alpha \pm \beta i$. Note that our general solution here is in what 83 | is called ``linear combination form,'' and we will change this later. 84 | \item $r_1=r_2=r$, where $r \in \mathbb{R}$. 85 | This implies a general solution of the form $y=c_1e^{rt}+c_2te^{rt}$. 86 | \end{enumerate} 87 | 88 | \begin{ex} 89 | \begin{equation} 90 | y'' + 4y' + 4y = 0 91 | \label{eq:sec_ord_ode} 92 | \end{equation} 93 | \begin{enumerate} 94 | \item[(a)] Find one solution, $y_1 (t)$. 95 | \item[(b)] Show that $y_2(t) = ty_1(t)$ is also a solution. 96 | \item[(c)] Give the general solution. 97 | \end{enumerate} 98 | \begin{sol} 99 | \begin{enumerate} 100 | \item[(a)] Characteristic equation: 101 | \begin{align} 102 | r^2 + 4r + 4 &= 0 \\ 103 | (r + 2) ^2 &= 0 104 | \end{align} 105 | This implies that $r = - 2$, and therefore $y_1(t)= e^{-2t}$ 106 | is a solution to \eref{eq:sec_ord_ode}. 107 | \item[(b)] 108 | \begin{align} 109 | y_2 (t) &= t e^{-2t} \\ 110 | y_2 ' (t) &= e^{-2t} - 2 t e^{-2t} \\ 111 | y_2 '' (t) &= -2 e^{-2t} - 2 e^{-2t} + 4t e^{-2t} \\ 112 | \end{align} 113 | To test this, we show that 114 | \[ -2e^{-2t} - 2e^{-2t} + 4te^{-2t} + 4e^{-2t} -8te^{-2t} + 4te^{-2t} = 0, \] 115 | which it does, so our solution is correct. 116 | \item[(c)] 117 | The general solution will be a linear combination of $y_1$ 118 | and $y_2$: 119 | \[ y = c_1 e^{-2t} + c_2te^{-2t}. \] 120 | \end{enumerate} 121 | \end{sol} 122 | \end{ex} 123 | 124 | \section{Principle of Superposition} 125 | If $y_1, y_2$ are solutions to $L(y) = y'' +g(t) y' + r(t) y = 0$, then 126 | $y=c_1 y_1 + c_2 y_2$ is a solution to $L(y) = 0$. 127 | 128 | The proof for this is found by plugging $y''$ and $y'$ into $L(y)$, and showing 129 | that the result equals zero. 130 | 131 | \begin{theorem} 132 | If $y_1, y_2$ are solutions to $L(y) = 0$, then $y=c_1y_1 + c_2 y_2$ is the 133 | general solution, iff: 134 | 135 | \begin{equation} 136 | \wronk{y_1, y_2} = 137 | \begin{vmatrix} 138 | y_1(t) & y_2(t) \\ 139 | y_1'(t) & y_2'(t) 140 | \end{vmatrix} 141 | \neq 0 142 | \end{equation} 143 | \end{theorem} 144 | 145 | \begin{ex} 146 | If $y_1=t$ and $\wronk{y_1, y_2} = t^2 e^t$, find $y_2(t)$. 147 | \begin{sol} 148 | \begin{align*} 149 | y_1(t) y_2 ' (t) - y_2 (t) y_1'(t) &= t^2 e^t \\ 150 | y_1(t)\leib{y_2 (t)}{t} - y_2(t) \leib{y_1(t)}{t} &= t^2 e^t \\ 151 | t \leib{y_2(t)}{t} - y_2(t) &= t^2 e^t \\ 152 | \leib {y_2}{t} - \frac{y_2(t)}{t} &= t e^t 153 | \end{align*} 154 | Let $\mu (t) = e^{\ln t} = - 1/t$ and we find that 155 | \[ \frac{y_2(t)}{t} = e^t + C. \] 156 | \end{sol} 157 | \end{ex} 158 | \section{Second-order linear homogeneous differential equations with constant coefficients} 159 | A \textbf{second-order linear homogeneous} differential equation with 160 | \textbf{constant coefficients} is of the form 161 | \begin{equation} 162 | a y'' + by' + cy = 0. 163 | \end{equation} 164 | \begin{ex} 165 | Solve the initial value problem 166 | \begin{equation} 167 | y'' + 4y' +5y = 0, 168 | \label{eq:2013-nov-ivp} 169 | \end{equation} 170 | where $y(0) = 1$ and $y'(0) = 0$. 171 | \begin{sol} 172 | From \eref{eq:2013-nov-ivp} we have the characteristic equation 173 | \begin{equation} 174 | r^2 +4r +5 = 0, 175 | \label{eq:2013-nov-ivp-char} 176 | \end{equation} 177 | which implies that 178 | \begin{align*} 179 | r_{1,2} &= \frac{ 180 | -4 \pm \sqrt{4^2 - 4 \times 5} 181 | }{ 182 | 2 183 | } \\ 184 | r_{1,2} &= \frac { 185 | -4 \pm \sqrt{4-5} 186 | }{ 187 | 2 188 | }\\ 189 | r_{1,2} &= -2 \pm \sqrt{-1} \\ 190 | r_{1,2} &= -2 \pm \iu 191 | \end{align*} 192 | Thus 193 | \[ y_{1,2} = \ec ^ {(-2 \pm \iu ) t} \] 194 | are solutions to \eref{eq:2013-nov-ivp}. 195 | \end{sol} 196 | \end{ex} 197 | \section{Euler's identity} 198 | \begin{equation} 199 | \ec ^{it} = \cos t + \iu \sin t 200 | \end{equation} 201 | \begin{equation} 202 | \ec^{\pi \iu} = -1 203 | \end{equation} 204 | \begin{equation} 205 | \ec^{\pi \iu} + 1 = 0 206 | \label{eq:euler-identity} 207 | \end{equation} 208 | \begin{remark} 209 | The number $\ec$ is defined in the equation 210 | \begin{equation} 211 | \int^\ec_1 \frac{1}{t} \ud t 212 | \end{equation} 213 | \begin{figure}[H] 214 | \begin{center} 215 | \includegraphics[width=0.4\textwidth]{continuous/ode/defn_e} 216 | \caption{$\ec$ is the number that makes the shaded area equal to $1$.} 217 | \end{center} 218 | \end{figure} 219 | \end{remark} 220 | % \section{Integrating Factors} 221 | % We will observe differential equations of the form 222 | % \begin{equation} 223 | % \leib{y}{t} = g(t) y + r(t) 224 | % \end{equation} 225 | % \begin{enumerate} 226 | % \item $\leib{y}{t} = t^2 y + \cos{t}$ 227 | % \item $t y +3=\leib{y}{t}-2t$ 228 | % \end{enumerate} 229 | -------------------------------------------------------------------------------- /continuous/ode/defn_e.eps: -------------------------------------------------------------------------------- 1 | %!PS-Adobe-3.0 EPSF-3.0 2 | %%Creator: cairo 1.12.16 (http://cairographics.org) 3 | %%CreationDate: Thu Nov 7 11:35:59 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163.832 148.902 c 165.254 148.902 166.402 150.055 166.402 151.473 c h 174 | S Q 175 | 0.862745 0 0 rg 176 | 201.039 41.901 m 201.039 40.479 199.887 39.327 198.469 39.327 c 197.047 177 | 39.327 195.898 40.479 195.898 41.901 c 195.898 43.319 197.047 44.471 198.469 178 | 44.471 c 199.887 44.471 201.039 43.319 201.039 41.901 c h 179 | f 180 | 1 0.254902 0.254902 rg 181 | q 1 0 0 -1 0 209.010254 cm 182 | 201.039 167.109 m 201.039 168.531 199.887 169.684 198.469 169.684 c 197.047 183 | 169.684 195.898 168.531 195.898 167.109 c 195.898 165.691 197.047 164.539 184 | 198.469 164.539 c 199.887 164.539 201.039 165.691 201.039 167.109 c h 185 | S Q 186 | 0.862745 0 0 rg 187 | 237.695 34.612 m 237.695 33.194 236.543 32.042 235.125 32.042 c 233.703 188 | 32.042 232.551 33.194 232.551 34.612 c 232.551 36.034 233.703 37.182 235.125 189 | 37.182 c 236.543 37.182 237.695 36.034 237.695 34.612 c h 190 | f 191 | 1 0.254902 0.254902 rg 192 | q 1 0 0 -1 0 209.010254 cm 193 | 237.695 174.398 m 237.695 175.816 236.543 176.969 235.125 176.969 c 233.703 194 | 176.969 232.551 175.816 232.551 174.398 c 232.551 172.977 233.703 171.828 195 | 235.125 171.828 c 236.543 171.828 237.695 172.977 237.695 174.398 c h 196 | S Q 197 | 0 g 198 | BT 199 | 22.4 0 0 22.4 277.224072 20.767625 Tm 200 | /f-0-0 1 Tf 201 | (L)Tj 202 | ET 203 | 0.862745 0 0 rg 204 | 270.312 30.573 m 270.312 29.151 269.16 27.999 267.738 27.999 c 266.32 27.999 205 | 265.168 29.151 265.168 30.573 c 265.168 31.991 266.32 33.143 267.738 33.143 206 | c 269.16 33.143 270.312 31.991 270.312 30.573 c h 207 | f 208 | 1 0.254902 0.254902 rg 209 | q 1 0 0 -1 0 209.010254 cm 210 | 270.312 178.438 m 270.312 179.859 269.16 181.012 267.738 181.012 c 266.32 211 | 181.012 265.168 179.859 265.168 178.438 c 265.168 177.02 266.32 175.867 212 | 267.738 175.867 c 269.16 175.867 270.312 177.02 270.312 178.438 c h 213 | S Q 214 | Q Q 215 | showpage 216 | %%Trailer 217 | end restore 218 | %%EOF 219 | -------------------------------------------------------------------------------- /continuous/transcend/circleeq.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 18 | 20 | 42 | 49 | 50 | 52 | 53 | 55 | image/svg+xml 56 | 58 | 59 | 60 | 61 | 62 | 66 | 70 | 75 | 80 | 81 | 91 | r 143 | x y 170 | 171 | -------------------------------------------------------------------------------- /continuous/transcendental.tex: -------------------------------------------------------------------------------- 1 | \chapter{Transcendental Functions} 2 | 3 | A \textbf{transcendental function} is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials. 4 | In other words, 5 | 6 | \begin{quote} 7 | [...] a transcendental number is a (possibly complex) number that is not algebraic---that is, it is not a root of a non-zero polynomial equation with rational coefficients.\footnote{% 8 | \texttt{http://en.wikipedia.org/w/index.php?title=Transcendental\_number\&oldid=609933437}} 9 | \end{quote} 10 | 11 | A transcendental function is a function that ``transcends'' algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction. 12 | 13 | Examples of transcendental functions include the \emph{exponential function}, the \emph{logarithm}, and the \emph{trigonometric functions}. 14 | %\cite{wiki:transcendental} 15 | 16 | Formally, 17 | 18 | \begin{defn} 19 | An analytic function \(f(z)\) of the real or complex variables \(z_1, \ldots, z_n\) is \textbf{transcendental} if the \(n+1\) functions \(z_1, \ldots, z_n\) are algebraically independent. 20 | \cite{wiki:transcendental} 21 | \end{defn} 22 | 23 | \section{Natural Logarithms} 24 | \begin{figure}[h] 25 | \begin{center} 26 | \includegraphics[width=0.4\textwidth]{continuous/transcend/natlog} 27 | \end{center} 28 | \caption{A plot of $f(x) =\ln x$.} 29 | \label{fig:natlog} 30 | \end{figure} 31 | 32 | \begin{defn} 33 | The \textbf{natural logarithm}\index{natural logarithm} is the function given by 34 | \begin{equation} 35 | \ln x = \int ^{x} _{1} \frac{1}{t} \ud t \text{,} \qquad x \in \mathbb{N} 36 | \end{equation} 37 | \end{defn} 38 | \begin{defn} 39 | The \textbf{number $e$} is that number in the domain of the natural logarithm satisfying 40 | \[ \ln{e}=1 \] 41 | It is roughly equal to 42 | \[2.7182818284590452353602874713526624977572470936999595\ldots\] 43 | \end{defn} 44 | \subsection{Algebraic Properties of the Natural Logarithm} 45 | 46 | For any numbers $b>0$ and $x>0$, the natural logarithm satisfies the following rules: 47 | \begin{table}[H] 48 | \begin{tabular}{p{3in}>\(p{3in}<\)} 49 | Product Rule & \displaystyle{\ln{bx}=\ln b + \ln x} \\\\ 50 | Quotient Rule & \displaystyle{\ln{\frac{b}{x}}=\ln b - \ln x} \\ \\ 51 | Reciprocal Rule & \displaystyle{\ln{\frac{1}{x}}=-\ln x} \\\\ 52 | Power Rule & \displaystyle{\ln{x^r}=r \ln x \qquad \forall r \in \mathbb{R}} 53 | \end{tabular} 54 | \end{table} 55 | % \begin{equation} 56 | % \ln{bx}=\ln b + \ln x 57 | % \end{equation} 58 | % \begin{equation} 59 | % \ln{\frac{b}{x}}=\ln b - \ln x 60 | % \end{equation} 61 | % \begin{equation} 62 | % \ln{\frac{1}{x}}=-\ln x 63 | % \end{equation} 64 | % \begin{equation} 65 | % \forall r \in \mathbb{R} \quad \ln{x^r}=r \ln x 66 | % \end{equation} 67 | 68 | %The following table was sourced from \url{www.math.ualberta.ca/~apotapov/MATH115/ln-logs.pdf}: 69 | \section{Logarithmic Identities} 70 | \begin{align*} 71 | a^xa^y &=a^{x+y} & \log_a{(uv)}&=\log_a u+\log_a v \\ 72 | (a^x)^y &= a^{xy} & \log_a{(u^y)} &= y\log_a u \\ 73 | a^{-x} &= \frac{1}{a^x} & \log_a{\left(\frac{1}{u}\right)} &= -\log_a u \\ 74 | \frac{a^x}{a^y} &= a^{x-y} & \log_a {\frac{u}{v}}&=\log_a u-\log_a v 75 | \end{align*} 76 | 77 | The number $e$ and its relationship to logarithms becomes especially important in integration, 78 | where we manipulate its properties in calculus to solve equations and integrate functions we would not 79 | otherwise be able to handle. 80 | 81 | The inverse equations for $e^x$ and $\ln x$ are 82 | \begin{equation} 83 | \forall (x>0)\big[e^{\ln x}=x\big] 84 | \label{eq:exinv1} 85 | \end{equation} 86 | \begin{equation} 87 | \forall x\big[\ln{(e^x)} =x\big] 88 | \label{eq:exinv2} 89 | \end{equation} 90 | 91 | The derivative of $e^x$ is very special, and it is 92 | \begin{equation} 93 | \ddx e^x = e^x \ud x. 94 | \label{eq:ddxex} 95 | \end{equation} 96 | 97 | 98 | \section{Hyperbolic Functions} 99 | Both \(\cos x\) and \(\sin x\) come from the formula for a circle. 100 | \begin{equation} 101 | x^2 + y^2=r^2 102 | \label{eq:circle} 103 | \end{equation} 104 | 105 | But we can define other useful functions using the equation for a hyperbola. 106 | \begin{equation} 107 | x^2-y^2=1 108 | \label{eq:hyperbola} 109 | \end{equation} 110 | Namely, \(\cosh x\) and \(\sinh x\). 111 | 112 | In \ref{eq:hyperbola}, let \[ y \to \frac{e^x-e^{-x}}{2}\] to get \(\sinh x\). 113 | Let \[ x \to \frac{e^x+e^{-x}}{2}\] to find \(\cosh x\). 114 | 115 | We can prove that these still satisfy equation \ref{eq:hyperbola}: 116 | 117 | \begin{proof} 118 | \begin{align*} 119 | 1&=x^2-y^2 \\ 120 | 1&=\left( \frac{e^x+e^{-x}}{2} \right) - \left( \frac{e^x - e^{-x}}{2} 121 | \right)^2 \\ 122 | 1&=\frac{e^{2x}+2e^xe^{-x}+e^{-2x}}{4}-\frac{e^{2x}-2e^xe^{-x}+e^{-2x}}{4} 123 | \qedhere 124 | \end{align*} 125 | \end{proof} 126 | -------------------------------------------------------------------------------- /continuous/trig/basictrig.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 18 | 36 | 38 | 40 | 41 | 43 | image/svg+xml 44 | 46 | 47 | 48 | 49 | 50 | 54 | 62 | 69 | 76 | 82 | Θ c b a 156 | 157 | -------------------------------------------------------------------------------- /continuous/trig/pythcircle.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 18 | 20 | 42 | 49 | 50 | 52 | 53 | 55 | image/svg+xml 56 | 58 | 59 | 60 | 61 | 62 | 66 | 76 | 80 | 85 | 90 | 91 | r 143 | x y 170 | 171 | -------------------------------------------------------------------------------- /continuous/unitcircle.tex: -------------------------------------------------------------------------------- 1 | \begin{figure}[h] 2 | \begin{tikzpicture}[scale=5.3,cap=round,>=latex] 3 | % draw the coordinates 4 | \draw[->] (-1.5cm,0cm) -- (1.5cm,0cm) node[right,fill=white] {$x$}; 5 | \draw[->] (0cm,-1.5cm) -- (0cm,1.5cm) node[above,fill=white] {$y$}; 6 | 7 | % draw the unit circle 8 | \draw[thick] (0cm,0cm) circle(1cm); 9 | 10 | \foreach \x in {0,30,...,360} { 11 | % lines from center to point 12 | \draw[gray] (0cm,0cm) -- (\x:1cm); 13 | % dots at each point 14 | \filldraw[black] (\x:1cm) circle(0.4pt); 15 | % draw each angle in degrees 16 | \draw (\x:0.6cm) node[fill=white] {$\x^\circ$}; 17 | } 18 | 19 | % draw each angle in radians 20 | \foreach \x/\xtext in { 21 | 30/\frac{\pi}{6}, 22 | 45/\frac{\pi}{4}, 23 | 60/\frac{\pi}{3}, 24 | 90/\frac{\pi}{2}, 25 | 120/\frac{2\pi}{3}, 26 | 135/\frac{3\pi}{4}, 27 | 150/\frac{5\pi}{6}, 28 | 180/\pi, 29 | 210/\frac{7\pi}{6}, 30 | 225/\frac{5\pi}{4}, 31 | 240/\frac{4\pi}{3}, 32 | 270/\frac{3\pi}{2}, 33 | 300/\frac{5\pi}{3}, 34 | 315/\frac{7\pi}{4}, 35 | 330/\frac{11\pi}{6}, 36 | 360/2\pi} 37 | \draw (\x:0.85cm) node[fill=white] {$\xtext$}; 38 | 39 | \foreach \x/\xtext/\y in { 40 | % the coordinates for the first quadrant 41 | 30/\frac{\sqrt{3}}{2}/\frac{1}{2}, 42 | 45/\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2}, 43 | 60/\frac{1}{2}/\frac{\sqrt{3}}{2}, 44 | % the coordinates for the second quadrant 45 | 150/-\frac{\sqrt{3}}{2}/\frac{1}{2}, 46 | 135/-\frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2}, 47 | 120/-\frac{1}{2}/\frac{\sqrt{3}}{2}, 48 | % the coordinate on s for the third quadrant 49 | 210/-\frac{\sqrt{3}}{2}/-\frac{1}{2}, 50 | 225/-\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2}, 51 | 240/-\frac{1}{2}/-\frac{\sqrt{3}}{2}, 52 | % the coordinates for the fourth quadrant 53 | 330/\frac{\sqrt{3}}{2}/-\frac{1}{2}, 54 | 315/\frac{\sqrt{2}}{2}/-\frac{\sqrt{2}}{2}, 55 | 300/\frac{1}{2}/-\frac{\sqrt{3}}{2}} 56 | \draw (\x:1.25cm) node[fill=white] {$\left(\xtext,\y\right)$}; 57 | 58 | % draw the horizontal and vertical coordinates 59 | % the placement is better this way 60 | \draw (-1.25cm,0cm) node[above=1pt] {$(-1,0)$} 61 | (1.25cm,0cm) node[above=1pt] {$(1,0)$} 62 | (0cm,-1.25cm) node[fill=white] {$(0,-1)$} 63 | (0cm,1.25cm) node[fill=white] {$(0,1)$}; 64 | \end{tikzpicture} 65 | \caption{The unit circle, thanks to \cite{tikzunitcirc}.\label{fig:tikzunitcirc}} 66 | \end{figure} 67 | -------------------------------------------------------------------------------- /discrete/algorithms.tex: -------------------------------------------------------------------------------- 1 | \chapter{Algorithms} 2 | 3 | An \textbf{algorithm} is a finite sequence of precise instructions for 4 | performing a computation on or for solving a problem. 5 | 6 | \begin{algorithm} 7 | \caption{Finding the Maximum Element in a Finite Sequence.\cite[p.~193]{rosen}} 8 | \begin{algorithmic}[1] 9 | \Procedure {max}{$a_1$, $a_2$, $\ldots$, $a_n$} 10 | \State $max := a_1$ 11 | \For {$i:=2$, $i< n$} 12 | \If{$max < a_i$} 13 | $max := a_1$ 14 | \EndIf 15 | \EndFor 16 | \EndProcedure 17 | \end{algorithmic} 18 | \end{algorithm} 19 | 20 | The following two algorithms return $location$, the subscript of the term that 21 | equals $x$, or $0$ if $x$ is not found. 22 | \begin{algorithm} 23 | \caption{The Linear Search Algorithm.\cite[p.~194]{rosen}} 24 | \begin{algorithmic}[1] 25 | \Procedure{linear search}{$x:$ integer, $a_1, a_2, \ldots, a_n:$ distinct 26 | integers} 27 | \State $i:=1$ 28 | \While{$i \leq n$ and $x \neq a_i$} 29 | \State $i:= i+1$ 30 | \EndWhile 31 | \If{$i \leq n$} 32 | \State location $:= i$ 33 | \Else 34 | \State location $:= 0$ 35 | \EndIf 36 | \State \Return location 37 | \EndProcedure 38 | \end{algorithmic} 39 | \end{algorithm} 40 | \begin{algorithm} 41 | \begin{algorithmic} 42 | \Procedure {binary search}{$x:$ integer,$a_1,a_2,\ldots,a_n:$ increasing 43 | integers} 44 | \State $i:= 1$ 45 | \State $j:=n$ 46 | \While {$ia_m$} 49 | \State $i:=m+1$ 50 | \Else 51 | \State $j:=m$ 52 | \EndIf 53 | \EndWhile 54 | \If{$x=a_i$} 55 | \State $location:=i$ 56 | \Else 57 | \State $location:=0$ 58 | \EndIf 59 | \State \Return location 60 | \EndProcedure 61 | \end{algorithmic} 62 | \end{algorithm} 63 | 64 | -------------------------------------------------------------------------------- /discrete/counting.tex: -------------------------------------------------------------------------------- 1 | \chapter{Counting} 2 | 3 | The basis of much of counting is the idea of the \emph{binomial coefficient}. 4 | \begin{defn} 5 | The \textbf{binomial coefficient} of $n$ and $k$, read ``$n$ choose $k$,'' is written 6 | \[ n \choose k \] 7 | and refers to the number of subsets with $k$ elements that we could find for a set of $n$ elements. 8 | \end{defn} 9 | 10 | We will often see these with the binomial formula, 11 | \begin{equation} 12 | (a+b)^n = \sum^\infty_{n=0} a^{n-k}b^k. 13 | \label{eq:binform} 14 | \end{equation} 15 | 16 | %[This chapter remains unfinished.] 17 | % 18 | % We can use counting techniques to determine the complexity of an algorithm. 19 | % Counting is very important not only for computer science but also for any job. 20 | % For example, counting problems are common in job interviews to see how a potential employee reacts. 21 | % 22 | % Counting, ultimately, is a very simple theoretical process governed by some basic rules. 23 | % 24 | % \section{Rules of Counting} 25 | % 26 | % \subsection{The Sum Rule}\index{sum rule for counting} 27 | % If a first task can be done in \(n_1\) ways and a second task can be done in \(n_2\) ways, 28 | % and if these tasks cannot be done at the same time, then there are \(n_1+n_2\) ways to do either task. 29 | % 30 | % In set notation: if \(A\) and \(B\) are \emph{disjoint}, then \(|A \cup B|=|A|+|B|\). 31 | % 32 | % \subsection{The Product Rule}\index{product rule for counting} 33 | % 34 | % Suppose that a procedure can be broken down into two tasks. 35 | % If there are \(n_1\) ways to do the first task and 36 | % \(n_2\) ways to do the second task 37 | % \emph{after} the first task has been done, 38 | % n there are \(n_1n_2\) ways to do the procedure. 39 | % 40 | % In set notation: 41 | % \[|A \times B| = |A||B| \] 42 | % 43 | % \subsection{Examples} 44 | % 45 | % \begin{ex} 46 | % Let \(D = \{x, y, z\}\). Let \(R=\{1,2,3,4,5\}\). 47 | % \begin{itemize} 48 | % \item[a) ] How many functions are there from \(D\to R\)? 49 | % \item[b) ] How many \emph{one-to-one} functions are there? 50 | % \item[c) ] How many onto functions are there? 51 | % \end{itemize} 52 | % \begin{sol} 53 | % We only have two rules for counting right now. 54 | % For the product rule, we must assume we are going to define mapping for \(x\), then for \(y\), then for \(z\). 55 | % With the sum rule, then we can define mapping for \(x\), or \(y\), or \(z\). 56 | % 57 | % Then we go back to ``how do we define the function?'' Do we have to find a mapping for every element in the domain? 58 | % 59 | % Yes. By definition of functions, we must. 60 | % 61 | % 62 | % \begin{itemize} 63 | % \item[a) ] There are \(5 \times 5 \times 5\) possible mappings from \(D \to R\). 64 | % \item[b) ] \(5\times4\times3\). 65 | % \item[c) ] It is impossible for the function to be onto. There are not enough elements in the domain to have values in the range to map to them.:w 66 | % \end{itemize} 67 | % \end{sol} 68 | % \end{ex} 69 | % \begin{ex} 70 | % A typical PIN is a sequence of any four numbers chosen form the 26 letters and the ten digits. 71 | % \begin{itemize} 72 | % \item[a) ] How many different PINs are possible if repetition is allowed? 73 | % \item[b) ] What if repetition is not allowed? 74 | % \end{itemize} 75 | % \end{ex} 76 | % \begin{ex} 77 | % The ASCII character set is represented by 7 binary bits. How many characters are there in the set? 78 | % % \begin{sol} 79 | % % We have to make seven choices in sequence to come up with an ASCII character. 80 | % % For each choice, we have two choices. 81 | % % \end{sol} 82 | % \end{ex} 83 | % \begin{ex} 84 | % Count the number of binary bit strings of length 4 or less. 85 | % \end{ex} 86 | % \begin{ex} 87 | % A student can choose a computer project from one of three lists. The three lists contain 10, 20, or 30 possible projects. There is no overlap among that list. How many projects are there to choose from? 88 | % \begin{sol} 89 | % \[10+20+30 \text{ projects}\] 90 | % \end{sol} 91 | % \end{ex} 92 | % 93 | % \section{The Pidgeonhole Problem}\index{pidgeonhole problem} 94 | % \begin{quote} 95 | % A flock of 13 pideons roosts in a set of 12 pidgeonholes. One of the 96 | % pidgeonholes must have more than one pidgeon. 97 | % \end{quote} 98 | % If $k$ is a positive integer and $k+1$ objects are placed into $k$ boxex, then 99 | % at least one box must have more than one object. 100 | % \begin{proof} 101 | % We can prove this by contradiction. Suppose all of the pidgeons fit in to $k$ 102 | % boxes exclusively. Therefore, there must be $k$ pidgeons, which is not equal 103 | % to $k+1$. 104 | % \end{proof} 105 | % \begin{corollary} 106 | % A function $f$ from a set with $k+1$ elements to a set with $k$ elements is 107 | % not \emph{one-to-one}. 108 | % \begin{proof} 109 | % Say we have eight boxes. We want to divide the objects evenly among the 110 | % boxes, so we place $2$ in each box. The number of boxes over the number of 111 | % elements is equal to $2$ objects per box. 112 | % 113 | % For nine boxes, we must take the ceiling function of $9/4$ and find 3. 114 | % \end{proof} 115 | % \end{corollary} 116 | % \begin{theorem} 117 | % \label{th:pidgeonhole} 118 | % If $N$ objects are placed into $k$ boxes, then there is at least one box 119 | % containing at least $N/K$ objects. 120 | % \end{theorem} 121 | % \begin{ex} 122 | % Among 100 people there are at least [100/12]=9 who were born in the same 123 | % month. 124 | % \end{ex} 125 | % \begin{ex} 126 | % How many cards must be selected from a standard deck of 52 cards to guarantee 127 | % that at least three cards of the same suit are selected. After generalizing 128 | % the pidgeonhole problem, we find that at least one box contains at least 129 | % $[N/4]$ cards. At least three cards of one suit are selected 130 | % \end{ex} 131 | % \section{Combination Rule} 132 | % 133 | % For the addition rule, we know the values and don't know the positions. 134 | % Where we know the position and don't know the values, we use the combination 135 | % rule to solve the problems. 136 | % 137 | % For the rule of products, things are more general. We can select the same 138 | % elements, and we are determining value rather than location. 139 | % 140 | % \begin{ex} 141 | % How many bit strings of length $100$ have at least $2$ ones? 142 | % \begin{sol} 143 | % The solution is given by the combination rule: 144 | % \[ C(100 2) \] 145 | % For exactly 3 ones, we do 146 | % \[ C(100, 3) \] 147 | % One hundred $1$s: 148 | % \[ C(100,100) \] 149 | % Or 150 | % \[ C(100, 2) + C(100, 3),+ \cdots + C(100,100) \] 151 | % We can do this using the combination rule as follows: 152 | % \[ 2^{100} - C(100,0) - C(100,1) \] 153 | % which is the total number of bit strings with at least 2 ones. 154 | % \end{sol} 155 | % \end{ex} 156 | % 157 | % \section{Counting the Complement} 158 | % 159 | % This is an applicaiton of the set decomposition principle, which states that the 160 | % total number of objects is equal to the number of objects that have a certain 161 | % property plus the number of objects that do not have the property. 162 | % \begin{ex} 163 | % Passwords of lenght 8 are made of lowercase letters and decimal digits. How 164 | % many of such passwords contain at least one decimal digit? 165 | % 166 | % In the past, we solved this as follows: 167 | % \[ (26+10)^8 = \text{ number of passwords with more than one digit} + 26^8 \] 168 | % 169 | % The combination rule will tell us:location of digit -> value of digit -> 170 | % value of letters 171 | % 172 | % 173 | % Number of passwords with one digit: 174 | % \[ C(8,1) \times 10 \times 26^7 \] 175 | % The number of passwords with two digits: 176 | % \[ C(8,2)\times10^2\times26^6 \] 177 | % And so on. The sum of these numbers provides our answer. 178 | % \end{ex} 179 | % 180 | % \section{The Binomial Theorem} 181 | % \begin{ex} 182 | % Find the expansion of 183 | % \[(x+y)^2,\, (x+y)^3\] 184 | % \begin{sol} 185 | % \[(x+y)^2 = x^2 + 2\times y+ y^2\] 186 | % That is to say, 187 | % \[ C(2, 0)+ C(2, 1)+ C(2,2) \] 188 | % For 189 | % \[(x+y)^3\] 190 | % we get 191 | % \begin{align*} 192 | % (x+y)^3&=C(3,0)+C(3,1)+C(3,2)+C(3,3) \\ 193 | % &= x^3 + 3x^2 y + 3 x y^2 + y^3 194 | % &= (x+y)(x+y)(x+y) 195 | % \end{align*} 196 | % \end{sol} 197 | % \end{ex} 198 | % This gives us the \textbf{binomial theorem}. 199 | % \[ (x+y)^n = \sum^n_{j=0} C(n, j)x^{n-j}y^j \] 200 | % \begin{ex} 201 | % What is the coefficient of $x^{25}y^{75}$ in the expansion of $(2x-5y)^{100}$? 202 | % \begin{sol} 203 | % Let $2x=a$ and $5y =b$. 204 | % \[ (a+b)^n= \sum^n_{j=0} C(n,j) a^{n-j} b^j \] 205 | % Now we solve for the variables. We know $a$, $b$, and $n$, so we must solve for 206 | % $j=75$. 207 | % Now, we put together our sum. 208 | % \[ C(100,75)(2x)^{100-75}(-5y)^{75} \] 209 | % \[ = C(100,75) 2^{25} \cdot x^{25} \cdot (-5)^{75} \cdot y^{75} \] 210 | % Which makes our answer 211 | % \[ C(100, 75) \cdot 2^{25}\cdot(-5)^{25}\] 212 | % \end{sol} 213 | % \end{ex} 214 | % \begin{homework} 215 | % Section $6.3$ (p.413): $17,20,33,34,37$. 216 | % Section $6.4$ (p.421): $3, 5, 9$. 217 | % 218 | % On Monday, we will get the even numbered answers for sections $6.1-6.4$, 219 | % around six or seven questions. We will also receive the review question 220 | % answers. This homework will be due on Tuesday, along with a quiz on counting. 221 | % Review the self-assessment on the counting sction. 222 | % \end{homework} 223 | -------------------------------------------------------------------------------- /discrete/inference.tex: -------------------------------------------------------------------------------- 1 | \chapter{Rules of Inference} 2 | \epigraph{I don't want to believe. I want to know.} 3 | {Carl Sagan} 4 | \label{ch:rules-of-inference} 5 | Proofs are used to establish the truth of mathematical statements. In order to 6 | make a proof, we must use the \textbf{rules of inference}\index{rules of 7 | inference} to establish the truth of more complicated logical arguments. An 8 | \textbf{argument} is a sequence of propositions that ends with a conclusion. A 9 | \textbf{valid} argument is one in which the last proposition follows from those 10 | propositions before it. 11 | 12 | When we are writing mathematical proofs, it's not common to actually cite the rules of inference in our text. 13 | However, they should form the logical connectors between the claims we make in our proofs and should be present in the implicit form. 14 | Becoming familiar with these rules, and how to use them, will allow us to both write more coherent proofs and to avoid logic errors in our writing. 15 | 16 | \section{Rules of Inference for Propositions} 17 | 18 | We will present the rules of inference using a variant of \emph{Fitch diagrams}. 19 | Each step in a Fitch diagram includes a number for the step, a proposition or conclusion, 20 | and a justification for the step. 21 | 22 | \subsection{\emph{Modus ponens}}\label{modus_ponens}\index{\emph{modus ponens}} 23 | \begin{equation*} 24 | \begin{fitch} 25 | \fb p & assumption \\ 26 | \fa p \to q & assumption \\ 27 | \fa q & $\big(p \wedge (p \to q)\big) \to q$ 28 | \end{fitch} 29 | \end{equation*} 30 | 31 | Another way you might see this written is 32 | 33 | \begin{array}{rl} 34 | 1. & p \rightarrow q \\ 35 | 2. & p \\ 36 | \hline 37 | \therefore & q 38 | \end{array} 39 | 40 | where the three dots ($\therefore$) is read as ``therefore''. 41 | We will usually try to stick with the fitch diagrams. 42 | 43 | \emph{Modus ponens} is Latin for ``mode that affirms,'' and comes from the 44 | tautology $\big(p \wedge (p \to q)\big) \to q$. It is the simplest valid 45 | \textbf{argument}\index{argument}, a sequence of statements that ends with a conclusion. 46 | 47 | \subsection{\emph{Modus tollens}}\index{\emph{modus tollens}} 48 | \begin{equation*} 49 | \begin{fitch} 50 | \fb \neg q & assumption \\ 51 | \fa p \to q & assumption \\ 52 | \fa \neg p & $\big(\neg q \wedge (p \to q)\big) \to \neg p$ 53 | \end{fitch} 54 | \end{equation*} 55 | 56 | \subsection{Hypothetical syllogism}\index{hypothetical syllogism} 57 | \begin{equation*} 58 | \begin{fitch} 59 | \fb p \to q& assumption \\ 60 | \fa q \to r& assumption \\ 61 | \fa p \to r& $\big((p \to q) \wedge (q \to r)\big) \to (p \to r)$ 62 | \end{fitch} 63 | \end{equation*} 64 | A hypothetical syllogism is sometimes thought of as \emph{double modus ponens}. 65 | 66 | \subsection{Disjunctive syllogism}\index{disjunctive syllogism} 67 | \begin{equation*} 68 | \begin{fitch} 69 | \fb p \vee q & assumption \\ 70 | \fa \neg p & assumption \\ 71 | \fa q & $(p \lor q ) \land \neg p \to q$ 72 | \end{fitch} 73 | \end{equation*} 74 | 75 | \subsection{Addition}\index{addition} 76 | \begin{equation*} 77 | \begin{fitch} 78 | \fb p & assumption \\ 79 | \fa p \lor q & $ p \to (p \lor q)$ 80 | \end{fitch} 81 | \end{equation*} 82 | 83 | \subsection{Simplification}\index{simplification} 84 | \begin{equation*} 85 | \begin{fitch} 86 | \fb p \land q & assumption \\ 87 | \fa p & $(p \land q) \to p$ 88 | \end{fitch} 89 | \end{equation*} 90 | 91 | \subsection{Conjunction}\index{conjunction} 92 | \begin{equation*} 93 | \begin{fitch} 94 | \fb p & assumption \\ 95 | \fa q & assumption \\ 96 | \fa p \land q & $\big( (p) \land (q)\big) \to (p \land q)$ 97 | \end{fitch} 98 | \end{equation*} 99 | 100 | \subsection{Resolution}\index{resolution} 101 | \begin{equation*} 102 | \begin{fitch} 103 | \fb p \lor q & assumption \\ 104 | \fa \neg p \lor r & assumption \\ 105 | \fa q \lor r & $ \big( (p \lor q) \land (\neg p \lor r ) \big) \to (q \lor r)$ 106 | \end{fitch} 107 | \end{equation*} 108 | 109 | \section{Rules of Inference for Quantified Statements} 110 | 111 | \subsection{Universal Generalization} 112 | 113 | \textbf{Universal generalization} states that given $P(c)$ for all elements $c$ 114 | in the domain, $\forall x P(x)$ is true. 115 | \begin{equation} 116 | \begin{fitch} 117 | \fb P(c) \text{ for some arbitrary $c$} & assumption \\ 118 | \fa \forall x P(x) & universal generalization 119 | \end{fitch} 120 | \label{eq:univ_gen} 121 | \end{equation} 122 | 123 | \subsection{Universal Instantiation}\label{univ_inst} 124 | 125 | \textbf{Universal instantiation} states that given $\forall x P(x)$, $P(c)$ is 126 | true for a particular element $c$ in the domain. 127 | \begin{equation} 128 | \begin{fitch} 129 | \fb \forall x \big(P(x) \to Q(x)\big) & proposition \\ 130 | \fa P(a) & universal instantiation 131 | \end{fitch} 132 | \label{eq:univ_inst} 133 | \end{equation} 134 | 135 | \subsection{Existential Generalization} 136 | 137 | \textbf{Existential generalization}\index{existential generalization} concludes 138 | that, given a particular element $c$ for which $P(c)$ is known to be true, $\exists x P(x)$. 139 | 140 | \subsection{Existential Instantiation} 141 | 142 | \textbf{Existential instantiation}\index{existential instantiation} states that if 143 | $\exists x P(x)$ is true, $P(c)$ for some element $c$. 144 | 145 | \subsection{Universal \emph{Modus Ponens}} 146 | 147 | \textbf{Universal \emph{modus ponens}} combines universal instantiation 148 | (Section \ref{univ_inst}) and \emph{modus ponens} (Section \ref{modus_ponens}) to 149 | tell us that if $\forall x (P(x) \to Q(x) )$ is true, and if $P(a)$ is true for a 150 | particular element $a$ in the domain of the universal quantifier, then $Q(a)$ must 151 | also be true. 152 | \begin{equation} 153 | \begin{fitch} 154 | \fb \forall x (P(x) \to Q(x)) \\ 155 | \fa P(a), \text{ where $a$ is a particular element in the domain} \\ 156 | \fa Q(a) 157 | \end{fitch} 158 | \label{eq:univ_mod_pon} 159 | \end{equation} 160 | 161 | \subsection{Universal \emph{Modus Tollens}} 162 | 163 | \textbf{Universal \emph{modus tollens}} states that 164 | \begin{equation} 165 | \begin{fitch} 166 | \fb \forall x (P(x) \to Q(x)) \\ 167 | \fa \neg Q(a), \text{ where $a$ is a particular element in the domain} \\ 168 | \fa \neg P(a) 169 | \end{fitch} 170 | \label{eq:univ_mod_tol} 171 | \end{equation} 172 | 173 | -------------------------------------------------------------------------------- /discrete/main.tex: -------------------------------------------------------------------------------- 1 | \part{Discrete Mathematics} 2 | 3 | This is the part of this text which won't focus on numbers. 4 | Were it not for specific examples, no numbers would exist in this part altogether. 5 | This part sets the stage for the logical theory behind mathematics. We will start with \emph{propositional logic}, a very simple form of logic that establishes the groundwork for logic statements. 6 | From there, we will describe \emph{predicate logic}, essentially a more powerful variant of propositional logic. 7 | We will then analyze the idea of mathematical \emph{proofs}, eventually working our way toward \emph{set theory}, the groundwork of what most people know as mathematics. 8 | 9 | \clearpage 10 | \dictum[Morpheus, \emph{The Matrix}]{This is your last chance. After this, there is no turning back. You take the blue pill---the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill---you stay in Wonderland and I show you how deep the rabbit-hole goes.} 11 | 12 | \setcounter{section}{0} 13 | \input{discrete/propositional} 14 | %\input{discrete/predicates} 15 | %\input{discrete/inference} 16 | %\input{discrete/proofs} 17 | %\input{discrete/sets} 18 | %\input{discrete/recursion} 19 | %\input{discrete/counting} 20 | -------------------------------------------------------------------------------- /discrete/predicates.tex: -------------------------------------------------------------------------------- 1 | \chapter{Predicates and Quantifiers} 2 | \label{ch:predicates} 3 | 4 | Predicates are based on the idea that we can replace parts of our propositions with variables in order to separate our discussion of logic from the irrelevant details of the problem. 5 | 6 | We saw in the previous chapter how the specifics of propositions seemed largely irrelevant; most examples were useless gibberish like ``my house is red'' with all of the focus on the theoretical relationships between these statements. 7 | 8 | Predicate logic is a more powerful system of logic which allows us to state ``there exists'' and ``for all'' using logic. 9 | 10 | \section{Predicate Logic} 11 | 12 | Propositional logic is too simple for us to make many types of conclusions. 13 | Instead, we use \definitionintro{predicate logic}, which allows us to make general 14 | statements about objects and their properties. 15 | Predicate logic is a generalization of propositional logic in which variables may 16 | be assigned to individual parts of statements, and then we can peroform 17 | analysis on the statements in general---instead of in just one specific instance. 18 | 19 | \begin{defn} 20 | A \definition{term}{term} is a variable. 21 | \end{defn} 22 | 23 | \begin{defn} 24 | A \definition{propositional function}{propositional function}, 25 | $P(x)$, is a 26 | type of \textbf{predicate}\index{predicate} in 27 | predicate logic. 28 | \end{defn} 29 | 30 | The important thing about propositional functions is that their truth value depends on the value of a variable, $x$. 31 | A propositional function becomes a proposition when a value is assigned to $x$, and then it has a truth value and we can evaluate it. 32 | \begin{ex} 33 | \[ P(n)=\text{``$n$ is prime''} \] 34 | \begin{remark} 35 | $P(n)$, a propositional function with a truth value, is different from the numerical function $p(n)$. 36 | When we talk about functions in the context of propositional logic, we must be careful not to confuse them with their possible numerical counterparts. 37 | \end{remark} 38 | \end{ex} 39 | 40 | 41 | \subsection{Domain of Discourse}\index{domain of discourse} 42 | Just as values for a variable must be stated in order for a propositional function to have a truth value, a \textbf{domain of discourse}\index{domain of discourse} must be specified in addition to the universal quantification. 43 | This is often referred to as just the \emph{domain} of the function. 44 | 45 | 46 | For example, for propositional functions talking about numbers, we often assume $D \to\mathbb{R}$.\footnote{We use $D$ as shorthand referring to the domain of discourse of a function. 47 | $\mathbb{R}$ means ``all real numbers.'' That is, all rational and irrational numbers. It does not include, for example, complex numbers which include $i$, the ``imaginary unit.''} 48 | 49 | 50 | \section{Quantification}\index{quantification} 51 | %If we wish to state that a given propositional function is true for all possible values in a domain, we use the \emph{universal quantification} of that function. 52 | %% Yeah, but we use this for so much more than that. Commenting this out for now. 53 | 54 | The \textbf{universal quantification}\index{universal quantification} of $P(x)$ 55 | \begin{equation} 56 | \forall x P(x) 57 | \end{equation} 58 | is the statement 59 | ``$P(x)$ for all values of $x$ in the domain.'' 60 | 61 | To show that the universal quantification of $P(x)$ is false for a domain, simply find a single value of $x$ for which $P(x)$ is false. 62 | 63 | \section{Existential Quantification}\index{existential quantification} 64 | 65 | If we wish to state that an element exists in a domain, we use the \emph{existential quantification} of a propositional function. 66 | 67 | The \textbf{existential quantification}\index{existential quantification} of $P(x)$ is the proposition 68 | ``There exists an element $x$ in the domain such that $P(x)$.'' 69 | We use the notation \[\exists x P(x)\] for the existential quantification of $P(x)$. 70 | 71 | \begin{note} 72 | In order to show that the existential quantification of $P(x)$ is false, we must 73 | show that $P(x)$ is false for every possible value of $x$ in the domain. 74 | \end{note} 75 | 76 | 77 | \subsection{Uniqueness Quantifier} 78 | A specific case of existential quantification is defined by the 79 | \textbf{uniqueness quantifier}\index{uniqueness quantifier}, $\exists!$ or $\exists_1$. The notation 80 | \[ \exists! x P(x) \] 81 | is the statement ``There exists a unique $x$ such that $P(x)$ is true.'' The 82 | downside to the uniqueness quantifier is that the rules of inference for 83 | existential quantification cannot be used on it. Since propositional logic can 84 | be used to express uniqueness already, we should try to avoid use of uniqueness 85 | quantification. 86 | 87 | To demonstrate uniqueness using propositional logic, we make a statement such as the following: 88 | \[ \exists x \Big( P(x) \land \forall y \big( P(y) \implies (x=y)\big)\Big) \] 89 | 90 | \section{Logical Equivalence of Quantified Propositions}\index{logical equivalence} 91 | 92 | In order for two statements involving predicates and quantifiers to be logically equivalent, 93 | they must have the same truth value regardless of the values of their propositional variables 94 | and the domain of discourse used. 95 | 96 | DeMorgan's Laws are an important logical equivalence even when quantified propositions are discussed. 97 | As stated in our definition of logical equivalence, they hold regardless of the values of their variables. 98 | 99 | %The following is a quote from Rosen's \emph{Discrete Mathematics and its Applications} on the issue: 100 | % 101 | %\begin{quote} 102 | % Statements involving predicates and quantifiers are \textbf{logically 103 | % equivalent} if and only if they have the same truth value no matter what 104 | % predicates are substituted into the statements and which the domain of 105 | % discourse is used for the variables in these propositional functions. We use the 106 | % notation $S \equiv T$ to indicate that two statements $S$ and $T$ involving 107 | % predicates and quantifiers are logically equivalent. 108 | % 109 | % \hfill\cite[p.~45]{rosen} 110 | %\end{quote} 111 | 112 | \subsection{DeMorgan's Laws for Quantifiers}\index{DeMorgan's laws for quantifiers} 113 | 114 | DeMorgan's Laws for quantifiers allow us to radically simplify logical expressions involving quantifiers. 115 | 116 | \begin{equation} 117 | \neg \exists x P(x) \equiv \forall x \neg P(x) 118 | \label{eq:dmq1} 119 | \end{equation} 120 | \begin{equation} 121 | \neg \forall x P(x) \equiv \exists x \neg P(x) 122 | \label{eq:dmq2} 123 | \end{equation} 124 | 125 | \begin{ex} 126 | For example, let's take \emph{Euler's conjecture},\footnote{Pronounced ``oiler.''} first proposed in 1769.\footnote{Eventually disproved in 1987. Solution at the end of the example.} 127 | 128 | Let us first define the propositional function $P(a,b,c,d)$.\footnote{ $ : : = $ is used to mean ``equals by definition,'' and is sometimes used in order to contrast with regular ``equals.''} 129 | \[P(a,b,c,d) : : = 130 | a^4 + b^4 + c^4 = d^4\] 131 | Now, Euler proposed that there are no positive integers $a, b, c,$ and $d$ such that $P(a,b,c,d)$ is true. We state this by writing 132 | \[ E(a,b,c,d) : : = 133 | \forall a \in \mathbb{Z^+} 134 | \forall b \in \mathbb{Z^+} 135 | \forall c \in \mathbb{Z^+} 136 | \forall d \in \mathbb{Z^+} 137 | \big(\neg P(a,b,c,d) \big).\] 138 | Let's break this apart. 139 | The ``$a \in \mathbb{Z^+}$ is used to describe our \emph{domain of discourse}. 140 | $\mathbb{Z^+}$ refers to the set of all positive integers. 141 | In general use, we can simplify this statement by writing 142 | \[ E(a,b,c,d) : : = 143 | \forall a,b,c,d, \in \mathbb{Z^+} \big( \neg P(x)\big)\] 144 | but for our purposes, we want to work with the original proposition, because we wish to use DeMorgan's Laws on it. 145 | 146 | Using DeMorgan's first law for quantifiers, equation \eqref{eq:dmq1}, we can change the last part of this proposition: 147 | \[\forall d \in \mathbb{Z^+} \big(\neg P(a,b,c,d)\big)\equiv \neg \exists d \in \mathbb{Z^+} \big(P(a,b,c,d)\big)\] 148 | Now, we continue up the chain, reversing each of the negated statements as if everything to the right of the negation sign were one single proposition. 149 | Here's our new statement: 150 | \begin{align*} 151 | E(a,b,c,d) : : &= 152 | \forall a \in \mathbb{Z^+} 153 | \forall b \in \mathbb{Z^+} 154 | \forall c \in \mathbb{Z^+} 155 | \neg\exists d \in \mathbb{Z^+} 156 | \big( P(a,b,c,d) \big)\\ 157 | \intertext{Now, continuing DeMorgan's Laws,} 158 | E(a,b,c,d) : : &= 159 | \forall a \in \mathbb{Z^+} 160 | \forall c \in \mathbb{Z^+} 161 | \neg\exists c \in \mathbb{Z^+} 162 | \exists d \in \mathbb{Z^+} 163 | \big( P(a,b,c,d) \big)\\ 164 | E(a,b,c,d) : : &= 165 | \forall a \in \mathbb{Z^+} 166 | \neg\exists c \in \mathbb{Z^+} 167 | \exists c \in \mathbb{Z^+} 168 | \exists d \in \mathbb{Z^+} 169 | \big( P(a,b,c,d) \big)\\ 170 | \intertext{Arriving finally at} 171 | E(a,b,c,d) : : &= 172 | \neg\exists a \in \mathbb{Z^+} 173 | \exists c \in \mathbb{Z^+} 174 | \exists c \in \mathbb{Z^+} 175 | \exists d \in \mathbb{Z^+} 176 | \big( P(a,b,c,d) \big),\\ 177 | \end{align*} 178 | which is logically equivalent to the original $E(a,b,c,d)$ we proposed. 179 | 180 | This shows that if just one of the variables in $E(a,b,c,d)$ cannot be said to exist, then the entire proposition becomes false. 181 | 182 | It turns out, in contrast to \emph{Euler's conjecture}, a solution to $P(a,b,c,d)$ can be found. With the values $a=95800$, $b=217519$, $c=414560$, and $d=422481$, $P(a,b,c,d)$ is true. 183 | \end{ex} 184 | 185 | \section{Order of Quantifiers}\index{quantifiers!order of} 186 | 187 | Assuming a domain of discourse of all real numbers, the quantification 188 | \begin{equation} 189 | \exists y \forall x Q(x, y) 190 | \end{equation} 191 | denotes the proposition 192 | ``There is a real number $y$ such that for every real number $x$, $Q(x, y)$.'' 193 | 194 | By contrast, the quantification 195 | \begin{equation} 196 | \forall x \exists y Q(x, y) 197 | \end{equation} 198 | states that 199 | ``For every real number $x$ there is a real number $y$ such that $Q(x, y)$.'' 200 | 201 | 202 | %%% Local Variables: 203 | %%% mode: latex 204 | %%% TeX-master: "../notes" 205 | %%% End: 206 | -------------------------------------------------------------------------------- /discrete/recursion.tex: -------------------------------------------------------------------------------- 1 | \chapter{Recursion}\index{recursion} 2 | 3 | \section{Recursive Definitions} 4 | 5 | \begin{defn}\index{recursive form} 6 | \textbf{Recursive form} defines a set, an equation, or a process by defining a starting set or a value and giving a rule for continuing to build the set, equation, or process based on previously defined terms. 7 | \end{defn} 8 | 9 | The key for recursion is the \emph{rule for continuing to build} the set, equation, or process. This is what allows us to do the new element, new equation, or new process based on previously defined terms. 10 | 11 | A recursive definition has two parts. 12 | 13 | \begin{defn} 14 | In the \textbf{basis step}\index{basis step}, we must define values for some finite number of 15 | elements. For sets, we state the \emph{basic building blocks}\index{basic 16 | building blocks} of the set. for functions, state the values of the function 17 | on the basic building blocks. 18 | \end{defn} 19 | \begin{defn} 20 | The remaining elements in the recursive definition are defined by the 21 | \textbf{recurrence relation}\index{recurrence relation}. For sets, we show how 22 | to build new things from the old with some basic construction rules. For 23 | functions, we show how to compute the value of a function on the new elements 24 | of that set. 25 | \end{defn} 26 | 27 | \subsection{Recursively Defined Functions}\index{functions, recursively defined} 28 | 29 | Let us create a recursive definition of the function $F$, defined on nonnegative integers. To give a recursive definition of $F$: 30 | \begin{enumerate}\item \emph{Basis}. Specify F(0). 31 | \item \emph{Recursive step}. Give a rule for defining $F(n+1)$ from $F$ evaluated at smaller values. 32 | \end{enumerate} 33 | \begin{ex} 34 | \begin{align*} 35 | f(0) &= 1 \\ 36 | f(n) &= f(n-1) +2 37 | \end{align*} 38 | \end{ex} 39 | \begin{ex} 40 | \begin{align*} 41 | g(0) &= 1 \\ 42 | g(k+1) &= g(k)+2 43 | \end{align*} 44 | \end{ex} 45 | \begin{ex} 46 | \begin{align*} 47 | a_0 &= 1 \\ 48 | a_n &= a_{n-1} 49 | \end{align*} 50 | \end{ex} 51 | \begin{ex} 52 | Find the recursive form of $n!$, the function given by 53 | \begin{equation}\label{eq:nfact} 54 | n!=\prod_{k=1}^n k \ 55 | \end{equation} 56 | \begin{figure}[h] 57 | \begin{center} 58 | \includegraphics{discrete/recursion/nfact.eps} 59 | \end{center} 60 | \caption{A plot of $n!$. Its behavior is much harder to describe in the 61 | negatives, so we normally just treat it as having a domain of $n \geq 0$.} 62 | \label{fig:nfact} 63 | \end{figure} 64 | \begin{sol} 65 | The basis step in either the \emph{closed form} or \emph{recursive form} 66 | definition for $n!$ is that $0!=1$. In equation \eqref{eq:nfact}, it is implied 67 | under the convention that the product of no numbers at all is 68 | one\footnote{This is called the \textbf{empty product} or \textbf{nullary 69 | product}, and is responsible for providing the \emph{multiplicative 70 | identity} $1$.} 71 | 72 | So in order to define a recursive form for $n!$, we must start with the 73 | definition: 74 | \begin{equation} 75 | f(n) = 76 | \begin{cases} 77 | 1 & \text{if }n=0 78 | \end{cases} 79 | \end{equation} 80 | 81 | Now that we have the basis step, to get the \emph{recursive step} we will 82 | look at a few instances of the factorial function: 83 | \begin{align*} 84 | f(0) &= 1 \\ 85 | f(1) &= 1 \\ 86 | f(2) &= 2 \\ 87 | f(3) &= 6 \\ 88 | f(4) &= 24 \\ 89 | f(5) &= 120 \\ 90 | & \vdots 91 | \end{align*} 92 | If we are careful, we'll notice that we can factor a $n$ from our result on 93 | each instance. 94 | \begin{align*} 95 | f(1) &= 1\cdot1 \\ 96 | f(2) &= 2\cdot1 \\ 97 | f(3) &= 3 \cdot 2 \\ 98 | f(4) &= 4 \cdot 6 \\ 99 | f(5) &= 5 \cdot 24 \\ 100 | &\vdots 101 | \end{align*} 102 | We notice that $f(n)$, for any $n > 1$, is given by $n$ times the term 103 | before it. By writing this out, we get our \emph{recursive definition for 104 | factorials}. 105 | \begin{equation} 106 | f(n) = 107 | \begin{cases} 108 | 1 & \text{if }n=0 \\ 109 | n \times f(n-1) & \text{if }n > 0 110 | \end{cases} 111 | \end{equation} 112 | As is the case with factorials, \emph{recursive form} often offers the 113 | advantage that it is very intuitive for humans to understand. Its downside is 114 | that it is very seldom computationally faster than its \emph{closed-form} 115 | alternative. For this reason, we should attempt to find closed-form 116 | solutions to recursive definitions where possible or necessary. 117 | 118 | Generally speaking, given a recursive function on a test, we should be able to find a 119 | closed-form representation and vice-versa. 120 | \end{sol} 121 | \end{ex} 122 | \begin{ex} 123 | Find a recursive definition of the \textbf{Fibonacci sequence}\index{Fibonacci 124 | sequence}: 125 | \[ 1, 1, 2, 3, 5, 8 13, 21, 34, \ldots \] 126 | \begin{figure}[h] 127 | \begin{center} 128 | \includegraphics{discrete/recursion/fibonacci.eps} 129 | \end{center} 130 | \caption{A plot of the Fibonacci sequence.} 131 | \label{fig:fibonacci} 132 | \end{figure} 133 | 134 | The Fibonacci sequence is often explained using the analogy of rabbits on an 135 | island. 136 | \begin{quote} 137 | ``A young pair of rabbits (one for each sex) is placed on an island. After 138 | they are 2 months old, each pair of rabbits produces another pair each month. 139 | The number of pairs of rabbits after $n$ months is $f(n)$.'' 140 | \end{quote} 141 | \begin{sol} 142 | Notice that we need \textbf{two} initial conditions to define this 143 | recurrence relation. 144 | \begin{equation} 145 | f(x) = 146 | \begin{cases} 147 | 1 & \text{for }0 \leq x \leq 1 \\ 148 | f(n) + f(n-1) &\text{for } x > 1 149 | \end{cases} 150 | \end{equation} 151 | \end{sol} 152 | \begin{note} 153 | This definition requires two initial conditions. It is very important in recursive definitions to have the right number of initial conditions. 154 | \end{note} 155 | \end{ex} 156 | %\begin{ex} 157 | % Give a recursive definition of 158 | % \[ F(n) = a^n \] 159 | % 160 | % \begin{tabular}{ll} 161 | % $f(0)=a^0=1$ & basis \\ 162 | % $f(n)=a\cdot f(n-1)$& recursion \\ 163 | % \end{tabular} 164 | % \begin{note} 165 | % \[f(n)=a^n=\underbrace{a \cdot a \cdot a \cdot \dots a}_{n}\] 166 | % 167 | % \[f(n-1)=a^{n-1}=\underbrace{a \cdot a \cdot a \cdot \dots a}_{n-1}\] 168 | % \end{note} 169 | %\end{ex} 170 | %\begin{ex} 171 | % Give a recursive definition of 172 | % \[ F(n) = \sum^{n}_{k=0} a_k \] 173 | % 174 | % \begin{tabular}{ll} 175 | % $f(c)=a_0$ & basis \\ 176 | % $f(n)=f(n-1)+a_n$ & recursion \\ 177 | % \end{tabular} 178 | % \begin{note} 179 | % \[ F(n) = \sum^{n}_{k=0} a_k=a_0+a_1+\dots+a_{n-1}+a_n \] 180 | % \end{note} 181 | %\end{ex} 182 | % 183 | %\begin{comment} 184 | %\begin{ex} 185 | % \begin{tabular}{ll} 186 | % Basis. & $f(0)=100,000=A$ \\ 187 | % Recursion. & $f(k)=f(k-1)+f(k-1)*4\%$ \\ 188 | % & $f(k) = (1+\alpha)(f(k-1))$ \\ 189 | % \end{tabular} 190 | % \begin{tabular}{ll} 191 | % 192 | % \end{tabular}d 193 | % \end{ex} 194 | %\end{comment} 195 | % 196 | % Mathematical induction is a way to varify the correctness of a recursive definition. 197 | % 198 | % \begin{ex} 199 | % \begin{align*} 200 | % a_1&=1 \\ 201 | % a_n&=2a_{n-1}+1 \text{ for all integers n $\geq 2$} 202 | % \intertext{Then prove by induction:} 203 | % a_n&=2^n-1 \text{ for all } n \geq 1 204 | % \end{align*} 205 | % \begin{tabular}{ll} 206 | % & $a_1=2^1-1=1=a_1 \text{ by recursion}$\\ 207 | % & If $a_n=2n-1$, then $a_{n+1}=2^{n+1}-1$. \\ 208 | % & assume $a_n=2^n-1$ \\ 209 | % & $a_{n+1}=2a_n+1=2(2^n-1)+1$ \\ 210 | % & $= 2 \cdot 2^n -2 +1 = 2^{n+1}-1$ 211 | % \end{tabular} 212 | % \end{ex} 213 | % 214 | %\section{Recursive Algorithms} 215 | % 216 | %Recursive algorithms are only used because certain algorithms are recursive in nature. Recursion does not save any computational power. For most algorithms, we can define a non-recursive version. However, sometimes it is inconvenient to find a non-recursive equivalent to a recursive algorithm. 217 | % 218 | %A recursive algorithm is one which calls itself to sove ``smaller'' versions of an input problem. Some algorithms are recursive in nature, like the binary search or Fibonacci sequence. 219 | % 220 | %The current status of the algorithm is placed on the \emph{stack}. A stack is a data structure from which entries can be added and deleted only from one end. 221 | % 222 | %\begin{verbatim} 223 | % procedure factorial(n) 224 | % if n < 0 return 'error' 225 | % if n = 0 the nreturn 1 226 | % else 227 | % return (n*factorial(n-1)) 228 | %\end{verbatim} 229 | % 230 | %Say we want to calculate $f(3)$. 231 | % 232 | %\begin{align*} 233 | % f(3)=3 \cdot &f(2)&&\\ 234 | % &\to f(2) = 2 \cdot f(1)&\\ 235 | % &&\to f(1)=1\cdot &f(0)\\ 236 | % &&&\to f(0)=1 237 | %\end{align*} 238 | % 239 | % 240 | % 241 | -------------------------------------------------------------------------------- /discrete/sets/equal.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 20 | 38 | 40 | 42 | 43 | 45 | image/svg+xml 46 | 48 | 49 | 50 | 51 | 52 | 56 | 65 | 75 | 82 | 84 | 86 | 90 | 95 | 96 | 100 | 105 | 106 | 110 | 115 | 116 | 120 | 125 | 126 | 127 | 128 | 130 | 133 | 140 | 147 | 148 | 151 | 158 | 159 | 160 | 161 | 162 | 163 | -------------------------------------------------------------------------------- /discrete/sets/subset.svg: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | 20 | 38 | 40 | 42 | 43 | 45 | image/svg+xml 46 | 48 | 49 | 50 | 51 | 52 | 56 | 65 | 75 | 84 | 91 | 93 | 95 | 99 | 104 | 105 | 109 | 114 | 115 | 116 | 117 | 119 | 122 | 129 | 130 | 131 | 132 | 139 | 141 | 143 | 147 | 152 | 153 | 157 | 162 | 163 | 164 | 165 | 167 | 170 | 177 | 178 | 179 | 180 | 181 | 182 | -------------------------------------------------------------------------------- /fibonacci/fib.tex: -------------------------------------------------------------------------------- 1 | \section{The Fibonacci Sequence} 2 | 3 | \begin{figure}[h] 4 | \begin{center} 5 | \input{fibonacci/fibgraph1} 6 | \end{center} 7 | \caption{A plot of equation \ref{eq:nint}.} 8 | \end{figure} 9 | The \emph{Fibonacci Sequence} is the first recursive number sequence known in Europe. Its first 10 numbers are 10 | 11 | \[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55\dots \quad \text{.} \] 12 | 13 | A Fibonacci sequence, in general, is any sequence of numbers in which each number is the sum of the two preceeding numbers.\cite{britannica12} 14 | 15 | 16 | \subsection{History} 17 | 18 | French mathematician Edouard Lucas coined the term ``Fibonacci sequence'' in the 19th century. 19 | The sequence is found throughout nature, as in the spirals of sunflower heads, pine cones, snail shells, and animal horns.\cite{britannica12} 20 | Because of this natural prevalence, patterns based on the Fibonacci sequence are considered aesthetically pleasing. 21 | The sequence can be found in Mozart and Beethoven's works as well as in classical art and architecture. \cite[p.~94]{design10} 22 | 23 | 24 | \subsection{Mathematics} 25 | 26 | \begin{defn}The \emph{Fibonacci numbers} are the sequence of numbers \(\{F_n\}^\infty_{n=1}\) defined by the linear recurrence equation 27 | \begin{equation} 28 | F_n=F_{n-1}+F_{n-2}. 29 | \end{equation} 30 | Often, we will see them defined with \(F_0=0\). 31 | \end{defn} 32 | 33 | This can be represented in the \emph{closed form} 34 | \begin{equation} 35 | F_n=\left[ \frac{\Phi^n}{\sqrt{5}}\right] 36 | \label{eq:nint} 37 | \end{equation} 38 | where \([x]\) is the \emph{nearest integer function}. \cite{mwfib} 39 | 40 | 41 | \section{The Golden Ratio} 42 | 43 | \begin{figure}[ht] 44 | \begin{center} 45 | \includegraphics{fibonacci/vitruvian.jpg} 46 | %\includegraphics[width=0.225\textwidth]{vitruvian.jpg} 47 | \end{center} 48 | \caption{The Vitruvian Man, said to depict ideal human proportions, bases its proportions on the golden ratio.}%\cite[p.~115]{design10}} 49 | % citation wasn't working 50 | %\footnote{\url{http://en.wikipedia.org/wiki/File:Da_Vinci_Vitruve_Luc_Viatour.jpg} 51 | \end{figure} 52 | 53 | 54 | The Fibonacci sequence and the golden ratio are closely related. 55 | 56 | \begin{defn} 57 | The \emph{golden ratio}, denoted \( \Phi \), is given by the positive solution to the equation 58 | \begin{equation} 59 | \Phi^2 - \Phi - 1 = 0 60 | \end{equation} 61 | \end{defn} 62 | 63 | Using the quadratic equation 64 | (\ref{app:eq:quadratic}) 65 | we can find that 66 | \begin{align*} 67 | \Phi =& \frac{1 \pm \sqrt{1^2-4(1)(-1)}}{2} \\ 68 | =& \frac{1 \pm \sqrt{1+4}}{2} \\ 69 | =& \frac{1 \pm \sqrt{5}}{2} \\ 70 | \intertext{and taking the positive root} 71 | \Phi =& \frac{1 + \sqrt{5}}{2} \\ 72 | =& 1.6180339887498948\dots 73 | \end{align*} 74 | \cite{mwgolden} 75 | 76 | We will notice that many closed-form representations of the Fibonacci sequence use the golden ratio. 77 | For example, \emph{Binet's Formula} 78 | \begin{equation} 79 | F_n=\frac{\Phi^n-(-\Phi)^{-n}}{\sqrt{5}} 80 | \label{eq:binet} 81 | \end{equation} 82 | derived\footnote{Though not for the first time.} by Binet in 1843 and equation \ref{eq:nint} both write \(F_n\) in terms of \( \Phi \).\cite{mwbinet} 83 | 84 | The ratio of consecutive terms in the Fibonacci sequence approximate the golden ratio: 85 | \begin{align*} 86 | \frac{1}{1} &= 1 \\ 87 | \frac{2}{1} &= 2 \\ 88 | \frac{3}{2} &= 1.5 \\ 89 | \frac{5}{3} &= 1.\overline{6}\dots \\ 90 | \frac{8}{5} &= 1.6 \\ 91 | \frac{13}{8} &= 1.625 \\ 92 | \frac{21}{13} &\approx 1.6153846 93 | \end{align*} 94 | Through this, we can conclude that 95 | \begin{equation} 96 | \lim_{n\to \infty} \frac{F_n}{F_{n-1}}=\Phi 97 | \label{eq:limphi} 98 | \end{equation} 99 | \cite{mwfib} 100 | 101 | \begin{figure}[h] 102 | \begin{center} 103 | \input{fibonacci/grat} 104 | \end{center} 105 | \caption{Equation \ref{eq:limphi} converges to \( \Phi \).} 106 | \end{figure} 107 | 108 | 109 | \section{Culture} 110 | 111 | \subsection{Lateralus} 112 | 113 | The song ``\emph{Lateralus}'' by the American rock band Tool counts out the Fibonacci sequence in its syllables:\footnote{\url{http://en.wikipedia.org/w/index.php?title=Lateralus\%20(song)&oldid=479876017}} 114 | \begin{figure}[H] 115 | \begin{tabular}{r|l} 116 | 1 & Black, \\ 117 | 1 & then, \\ 118 | 2 & white are, \\ 119 | 3 & all I see, \\ 120 | 5 & in my infancy, \\ 121 | 8 & red and yellow then came to be, \\ 122 | 5 & reaching out to me, \\ 123 | 3 & lets me see. \\ 124 | 2 & There is, \\ 125 | 1 & so, \\ 126 | 1 & much, \\ 127 | 2 & more and \\ 128 | 3 & beckons me, \\ 129 | 5 & to look through to these, \\ 130 | 8 & infinite possibilities. \\ 131 | 13 & As below so above and beyond I imagine,\\ 132 | 8 & drawn beyond the lines of reason.\\ 133 | 5 & Push the envelope. \\ 134 | 3 & Watch it bend. \\ 135 | \end{tabular} 136 | \caption{ Maynard James Keenan's vocals.} 137 | \end{figure} 138 | -------------------------------------------------------------------------------- /fibonacci/grat.tex: -------------------------------------------------------------------------------- 1 | % GNUPLOT: LaTeX picture 2 | \setlength{\unitlength}{0.240900pt} 3 | \ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi 4 | 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\raisebox{-1.4ex}[0pt][0pt]{\rule{1.5em}{\arrayrulewidth}}}}% 35 | \hspace*{\fitchindent}} 36 | % Hypothesis, with longer vert line: for >1 hypothesis 37 | \newcommand{\fj}{\vline% 38 | \makebox[0pt][l]{{% 39 | \raisebox{-1.4ex}[0pt][0pt]{\rule{1.5em}{\arrayrulewidth}}}}% 40 | \hspace*{\fitchindent}} 41 | % Modal subproof: takes argument = operator 42 | \newcommand{\fitchmodal}[1]{% 43 | \makebox[0pt][r]{${}^{#1}$\,}\fvline\hspace*{\fitchindent}} 44 | \newcommand{\fn}{\fitchmodal{\Box}}% Box subproof 45 | \newcommand{\fp}{\fitchmodal{\Diamond}}% Diamond subproof 46 | % Modal subproof with hypothesis in first line (as in Fitch) 47 | \newcommand{\fitchmodalh}[1]{% 48 | \makebox[0pt][r]{${}^{#1}$\,}% 49 | \fvline% 50 | \makebox[0pt][l]{{% 51 | \raisebox{-1.4ex}[0pt][0pt]{\rule{1.5em}{\arrayrulewidth}}}}% 52 | \hspace*{\fitchindent}} 53 | % Rule: formula introduction marker. \fr with line, \fs without line 54 | \newcommand{\fr}{% 55 | 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| \newcommand{\ftag}[2]{\multicolumn{1}% 79 | {!{\makebox[\fitchnumwd][r]{#1}\hspace{\fitchindent}}Ml@{\hspace{\fitchcomind}}}% 80 | {#2}} 81 | 82 | \newenvironment{fitchnum}% 83 | {\ifthenelse{\boolean{resetfitchcounter}}{\setcounter{fitchcounter}{0}}{} 84 | \begin{tabular}{!{\makebox[\fitchnumwd][r]{\fitchcounter }\hspace{\fitchindent}}Ml@{\hspace{\fitchcomind}}l}}% 85 | {\end{tabular}} 86 | 87 | \newenvironment{fitchunum}% 88 | {\begin{tabular}{!{\makebox[\fitchnumwd][r]{}\hspace{\fitchindent}}Ml@{\hspace{\fitchcomind}}l}}% 89 | {\end{tabular}} 90 | 91 | \newenvironment{fitch}{\renewcommand{\arraystretch}{1.5} 92 | \begin{fitchnum}}{\end{fitchnum}} 93 | \newenvironment{fitch*}{\renewcommand{\arraystretch}{1.5} 94 | \begin{fitchunum}}{\end{fitchunum}} 95 | 96 | % The following is useful for giving a numbered formula, then the proof. 97 | \newenvironment{flem}[2]% 98 | {\begin{eqnarray} 99 | \label{#2}\\ 100 | &\begin{fitch}}% 101 | {\end{fitch}\notag\end{eqnarray}} 102 | 103 | %To write comment field for two consecutive lines, with brace 104 | \newcommand{\ftwocom}[1]{% 105 | \parbox[t]{3cm}{ 106 | \raisebox{-.6\baselineskip}[\baselineskip][0pt]{% 107 | $\left. 108 | \begin{aligned} 109 | \,\\ \, 110 | \end{aligned} 111 | \right\}$\quad #1} 112 | }} -------------------------------------------------------------------------------- /frontmatter/cc-license.tex: -------------------------------------------------------------------------------- 1 | \begin{center} 2 | \textsc{This work is Copyright} \copyright\ 2012--2018 Nathan Typanski. 3 | \\\noindent\rule{\textwidth}{0.4pt} 4 | \end{center} 5 | 6 | This work is made available under the terms of the \emph{Creative Commons 7 | Attribution-NonCommercial-ShareAlike 3.0 Unported License}. 8 | 9 | \textbf{You are free:} 10 | \begin{itemize} 11 | \item[] \textbf{to Share}\dash{}to copy, distribute and transmit the work 12 | \item[] \textbf{to Remix}\dash{}to adapt the work 13 | \end{itemize} 14 | 15 | \textbf{Under the following conditions:} 16 | \begin{itemize} 17 | \item[]\ccby\textbf{Attribution}\dash{}You must attribute the work in the manner 18 | specified by the author or licensor (but not in any way that suggests that 19 | they endorse you or your use of the work). 20 | \item[] \ccnc\textbf{Noncommercial}\dash{}You may not use this work for 21 | commercial purposes. 22 | \item[] \ccsa\textbf{Share Alike}\dash{}If you alter, transform, or build upon 23 | this work, you may distribute the resulting work only under the same or 24 | similar license to this one. 25 | \end{itemize} 26 | 27 | \textbf{With the understanding that:} 28 | \begin{itemize} 29 | \item[]\textbf{Waiver}\dash{}Any of the above conditions can be \emph{waived} if 30 | you get permission from the copyright holder. 31 | \item[]\textbf{Public Domain}\dash{}Where the work or any of its elements is in 32 | the \emph{public domain} under applicable law, that status is in no way 33 | affected by the license. 34 | \item[]\textbf{Other Rights}\dash{}In no way are any of the following rights 35 | affected by the license: 36 | \begin{itemize} 37 | \item Your fair dealing or \emph{fair use} rights, or other applicable 38 | copyright exceptions and limitations; 39 | \item The author's \emph{moral} rights; 40 | \item Rights other persons may have either in the work itself or in how the 41 | work is used, such as \emph{publicity} or privacy rights. 42 | \end{itemize} 43 | \item[]\textbf{Notice}---For any reuse or distribution, you must make clear 44 | to others the license terms of this work. 45 | \end{itemize} 46 | \begin{center} 47 | \url{http://creativecommons.org/licenses/by-nc-sa/3.0/} 48 | 49 | \cc{} 50 | \end{center} 51 | -------------------------------------------------------------------------------- /frontmatter/main.tex: -------------------------------------------------------------------------------- 1 | \input{frontmatter/cc-license} 2 | \input{frontmatter/preface} 3 | \input{frontmatter/toc} 4 | -------------------------------------------------------------------------------- /frontmatter/preface.tex: -------------------------------------------------------------------------------- 1 | \chapter*{Preface}\epigraph{ 2 | Hold to the now, the here, through which all future plunges to the 3 | past. 4 | } 5 | {Ulysses, James Joyce, Episode 9} 6 | \addcontentsline{toc}{chapter}{Preface} 7 | 8 | I started this notebook my Math 240 (Calculus II) class at Christopher Newport 9 | University on \formatdate{13}{02}{2012}. I had picked up the basics of \LaTeX\ in my free hour 10 | before class because I wanted to learn how to type mathematical documents. Why? 11 | Because \LaTeX\ is cool. 12 | 13 | I started to care about math because it serves as the logical foundation for 14 | physics. I realized quickly in my studies that I did not know enough math, and I 15 | did not know it rigorously enough to truly understand physics. I cared about 16 | physics because it is a prerequisite for understanding the basics of computer 17 | engineering, my major. 18 | % thanks Kyle Martin for correcting a typo here. 19 | 20 | % EDIT: REMOVED 11/6/2012 21 | %%%%%%%%%% 22 | % Discrete mathematics, a subject which has only recently grown to popularity in 23 | % concurrence with computer science, is included first because it describes much 24 | % of the logical foundation for mathematics in ways I had never encountered 25 | % before. In many ways, it involves thinking about the basic thought processes 26 | % that we take for granted in continuous mathematics. When we make claims in 27 | % mathematics such as 28 | % \begin{quote} 29 | % ``\(x=6\)'' 30 | % \end{quote} 31 | % \begin{quote} 32 | % ``A limit of sums is a sum of limits.'' 33 | % \end{quote} 34 | % there is an underlying logical structure that governs the meanings of such 35 | % statements and how we conceptualize and work with them. For this reason, I have 36 | % found discrete mathematics extremely enlightening in my own study of calculus 37 | % and beyond. 38 | 39 | This is now the longest document I've ever written. It has grown to represent 40 | a sizable portion of my college education at this time. It's also the first time I've 41 | developed a sustainable organizational system for my notes. Everything before 42 | this, and everything besides this, lies in stacks of scattered legal pads in at 43 | least four different locations. 44 | 45 | My goal is to finally organize my thoughts and conceptualize this material in a 46 | way I have never even attempted before. 47 | 48 | Whatever it takes. 49 | 50 | \hfill{Nathan Typanski} 51 | 52 | \hfill \date{April 9, 2012} 53 | \newpage 54 | \section*{A note to the reader} 55 | This text is a work in progress. 56 | 57 | Everything in this document is subject to change. 58 | 59 | No claim is made as to the accuracy of any of the information contained herein. 60 | There may be mistakes, inaccuracies, or outright lies included among otherwise relevant and complete content. 61 | Always check with a reputable source (e.g.\ a math book). 62 | 63 | \newpage 64 | \section*{A note on references} 65 | Whenever a number is appended to a sentence in brackets ([11], for example), it means the preceeding section has a citation in the bibliography and can be examined in the original source. 66 | This is very commonly used in simple tables that are reproduced here, or in sections of text where the material is not different enough from its source text. 67 | 68 | The references for this text are not fully established in the official text. 69 | There are still a number of places where citations may be missing, or provided only by name and without a complete entry in the bibliography. 70 | However, I am regularly going back through this text and completely rewriting sections that have been copied verbatim (for example, a couple of proofs in the appendix are not my work, though I am prudent to say so outright). 71 | This is largely because this material is presented in draft form, and many of the citations are provided in the \LaTeX\ source but not yet finalized to the reader. 72 | Over time, all instances of this will be adequately removed. 73 | 74 | %%% Local Variables: 75 | %%% mode: latex 76 | %%% TeX-master: "notes" 77 | %%% End: 78 | -------------------------------------------------------------------------------- /frontmatter/toc.tex: -------------------------------------------------------------------------------- 1 | \setcounter{tocdepth}{3} 2 | \tableofcontents{} 3 | 4 | \listoftables{} 5 | 6 | \listoffigures{} 7 | -------------------------------------------------------------------------------- /graphs/nf2nfp1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/nathantypanski/texnotes/1661b41965c1450b77b13f63073d84eb93c784c8/graphs/nf2nfp1.pdf -------------------------------------------------------------------------------- /graphs/np22nm1.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/nathantypanski/texnotes/1661b41965c1450b77b13f63073d84eb93c784c8/graphs/np22nm1.pdf -------------------------------------------------------------------------------- /graphs/p1sin1x.pdf: 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\part{Appendix} 13 | \include{appendix/main} 14 | %\input{discrete/algorithms} 15 | 16 | \printindex 17 | 18 | % Should we be printing unused bibliography references in these drafts? 19 | % probably not. 20 | %\nocite{*} 21 | \bibliographystyle{plainnat} 22 | \bibliography{Bibliography} 23 | %% ---------------------------------------------------------------- 24 | \end{document} 25 | -------------------------------------------------------------------------------- /notes.tex.latexmain: -------------------------------------------------------------------------------- 1 | 2 | -------------------------------------------------------------------------------- /photos/cauchy.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/nathantypanski/texnotes/1661b41965c1450b77b13f63073d84eb93c784c8/photos/cauchy.jpg -------------------------------------------------------------------------------- /photos/cont2.png: 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showpage 92 | %%Trailer 93 | end restore 94 | %%EOF 95 | -------------------------------------------------------------------------------- /photos/tabletnotes.jpg: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/nathantypanski/texnotes/1661b41965c1450b77b13f63073d84eb93c784c8/photos/tabletnotes.jpg -------------------------------------------------------------------------------- /photos/vim.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/nathantypanski/texnotes/1661b41965c1450b77b13f63073d84eb93c784c8/photos/vim.png -------------------------------------------------------------------------------- /physics/electric.tex: -------------------------------------------------------------------------------- 1 | \chapter{Electric Fields} 2 | 3 | 4 | \section{Electric Charge} 5 | 6 | There are two kinds of electric charges, named \textbf{positive} and 7 | \textbf{negative} by Benjamin Franklin (1706-1790)\cite[p.~643]{serway}. 8 | 9 | Electric charge is quantized and always conserved in an isolated system. 10 | 11 | Electrical \textbf{conductors}\index{conductors} are materials in which some of the electrons are 12 | free electrons that are not bound to atoms and can move relatively freely 13 | through the material; electrical \textbf{insulators}\index{insulators} are materials in which all 14 | electrons are bound to atoms and cannot move freely through the 15 | material.\cite[p.~644]{serway} 16 | 17 | 18 | \section{Coulomb's Law} 19 | 20 | \textbf{Coulomb's law}\index{Coulomb's law} describes the properties of the electric force between two 21 | stationary charged particles. These stationary charged particles are modeled as 22 | \textbf{point charges}\index{point charge}, theoretical charged particles of zero size. Coulomb's 23 | law shows us that the force between two charged objects varies in 24 | proportion with the charge $q$ placed on the objects and inversely with the 25 | square of the distance between them. 26 | \begin{equation} 27 | \label{eq:coulombslaw} 28 | |\vec{F}_q|=k_e\frac{q_1q_2}{r^2} 29 | \end{equation} 30 | Where $k_e$ is a constant called the \textbf{Coulomb constant}\index{Coulomb 31 | constant}. In SI units, $k_e$ has the value 32 | \begin{equation} 33 | \label{coulombconstant} 34 | k_e = 8.9876 \times 10^9 N \cdot m^2 / C^2 35 | \end{equation} 36 | Often, we write the Coulomb constant in terms of the \textbf{permittivity of 37 | free space}, $\varepsilon_0$ 38 | \begin{equation} 39 | k_e = \frac{1}{4 \pi \varepsilon_0} 40 | \end{equation} 41 | Where $\varepsilon_0$ has the value 42 | \begin{equation} 43 | 8.8542 \times 10^{-12} C^2 / N \cdot m^2 44 | \end{equation} 45 | 46 | 47 | \section{Electric Field} 48 | 49 | The \textbf{electric field}\index{electric field} produced by a point charge is given by 50 | \begin{equation} 51 | \label{eq:electricfield} 52 | \vec{E} \equiv \frac{\vec{F_e}}{q_0} 53 | \end{equation} 54 | Where $q_0$ is a test charge, which we take by convention to be a point charge 55 | of positive value. \textbf{An electric field exists at a point if a test charge 56 | at that point experiences an electric force}. 57 | 58 | Electric fields can be visualized by drawing \textbf{electric field lines}, 59 | first conceived by Faraday. The electric field vector $\vec{E}$ is tangent to 60 | the electric field line at each point and with direction the same as the 61 | electric field vector. The number of lines per unit area on a surface 62 | perpendicular to the lines is proportional to the magnitude of the electric 63 | field inthat region. The density of the field lines, therefore, represents 64 | strength of $\vec{E}$. 65 | 66 | 67 | \section{Motion of a Charged Particle in a Uniform Electric Field} 68 | 69 | To get the motion of a charged particle in a uniform electrif field, simply use 70 | $\vec{F}=ma$ to find that 71 | \[ \vec{F_e} = q\vec{E} = m\vec{a} \] 72 | and rearrange this to get 73 | \begin{equation} 74 | \vec{a} = \frac{q\vec{E}}{m}. 75 | \end{equation} 76 | 77 | 78 | \section{Gauss's Law}\index{Gauss's law} 79 | 80 | 81 | \textbf{Electric flux}\index{electric flux} is defined as the product of the 82 | magnitude of the electric field $E$ and the surface area $A$ perpendicular to 83 | the field. 84 | \begin{equation} 85 | \Phi_E = EA 86 | \end{equation} 87 | Electric flux is proportional to the number of electric field lines penetrating 88 | some surface. 89 | 90 | In order to calculate it, we take the surface integral of $\vec E \cdot \ud \vec 91 | A$. We are often interested in evaluating this integral across a closed surface, 92 | where the flux would be given by 93 | \begin{equation} 94 | \label{eq:electricflux} 95 | \Phi_E = \oint \vec E \cdot \ud \vec A = \oint E_n \ud A 96 | \end{equation} 97 | 98 | Gauss came up with a way to simplify this integral for \emph{closed surfaces}, 99 | often called \emph{gaussian surfaces}: 100 | \begin{equation} 101 | \label{eq:gausslaw} 102 | \Phi_E = \oint \vec E \cdot \ud A = \frac{q_{in}}{\varepsilon_0} 103 | \end{equation} 104 | 105 | For a \textbf{conductor in electrostatic equilibrium}, Gauss's law tells us that 106 | there must be zero field within it otherwise the motion of electrons would 107 | contradict the notion of equilibrium. Immediately outside the conductor, Gauss's 108 | law tells us 109 | \begin{equation} 110 | E = \frac{\sigma}{\varepsilon_0} 111 | \end{equation} 112 | 113 | \section{Electric Potential} 114 | 115 | In order to talk about potential energy in an electric field, we need to jump 116 | through a few hoops to get there. First, remember our definition of 117 | work\index{work}: 118 | \[ W = \vec{F} \cdot \ud \vec s \] 119 | In order for work to occur, we must be actually moving something. So we move a 120 | theoretical test charge, $q_0$, from one point $a$ to another point $b$ in an 121 | electric field, and then suddenly we are doing work. That is, we are producing a 122 | change in potential energy for the system. This change in potential energy 123 | $\Delta U = U_b - U_a$ is given by the integral of the work from $a$ to $b$. 124 | 125 | So what's our force? By rearranging equation \eqref{eq:electricfield} we can 126 | find that $\vec{F_e} = q_0 \vec{E}$. Since this force is conservative, 127 | Where $\Delta U$ is given by 128 | \begin{equation} 129 | \Delta U = -q_0 \int_{a}^{b} \vec{E} \cdot \ud \vec{s} 130 | \end{equation} 131 | 132 | The \textbf{potential difference} $\Delta V$ describes a change in potential 133 | energy. Between two points $a$ and $b$, $\Delta V = V_b - V_a$. To calculate 134 | this value, we must take a test charge and move it from $a$ to $b$, summing the 135 | change in potential energy in the system. The inverse of this value would give 136 | us its potential difference. 137 | \begin{equation} 138 | \label{eq:potentialdifference} 139 | \Delta V = \frac{\Delta U}{q_0} = -\int_{a}^{b} \vec{E} \cdot \ud \vec{s} 140 | \end{equation} 141 | 142 | -------------------------------------------------------------------------------- /physics/main.tex: -------------------------------------------------------------------------------- 1 | \part{Physics} 2 | \setcounter{section}{0} 3 | \section*{\emph{Syst\'eme International} Prefixes} 4 | \begin{table}[h] 5 | \centering 6 | \begin{tabular}{llr} 7 | \textbf{Power} & \textbf{Prefix} & \textbf{Abbreviation} \\ \hline 8 | $10^{-24}$ & yocto & y \\ 9 | $10^{-21}$ & zepto & z \\ 10 | $10^{-18}$ & atto & a \\ 11 | $10^{-15}$ & femto & f \\ 12 | $10^{-12}$ & pico & p \\ 13 | $10^{-9}$ & nano & n \\ 14 | $10^{-6}$ & micro & $\upmu$ \\ 15 | $10^{-3}$ & milli & m \\ 16 | $10^{-2}$ & centi & c \\ 17 | $10^{-1}$ & deci & d \\ 18 | $10^{3}$ & kilo & k \\ 19 | $10^{6}$ & mega & M \\ 20 | $10^{9}$ & giga & G \\ 21 | $10^{12}$ & tera & T \\ 22 | $10^{15}$ & peta & P \\ 23 | $10^{18}$ & exa & E \\ 24 | $10^{21}$ & zeta & Z \\ 25 | $10^{24}$ & yotta & Y \\ \hline 26 | \end{tabular} 27 | \caption{Prefixes for powers of ten.} 28 | \label{tab:si_prefixes} 29 | \end{table} 30 | 31 | \section*{Significant Figures} 32 | 33 | When computing results from measured numbers, pay careful attention to the 34 | number of \textbf{significant figures}\index{significant figures}. When 35 | multiplying or dividing, the number of significant digits in the answer is the 36 | same as that of the factor with the least significant figures. When adding or 37 | subtracting, the number of decimal places should equal the least number of 38 | decimal places of any term. 39 | 40 | \section*{Coordinate Systems} 41 | 42 | In two dimensions, we place our mathematical description of an object's motion 43 | within a \textbf{Cartesian coordinate system}\index{Cartesian coordinates}, also 44 | called \emph{rectangular coordinates}. 45 | 46 | Other times, we establish a coordinate system using \textbf{plane polar 47 | coordinates}\index{polar coordinates}\index{plane polar coordinates}, given in 48 | the form $(r,\theta)$. Here, $r$ is the distance from the origin to the point 49 | with cartesian coordinates $(x, y)$ and $\theta$ is the angle between a fixed 50 | axis and a line drawn from the origin to that point. By convention, we usually 51 | choose the fixed axis to be positive $\hat x$ and measure $\theta$ counterclockwise 52 | from that axis. This lets us easily switch between the two systems using 53 | \begin{equation} 54 | \label{eq:xcos} 55 | x = r \cos \theta 56 | \end{equation} 57 | \begin{equation} 58 | \label{eq:ysin} 59 | y = r \sin \theta 60 | \end{equation} 61 | From here, we can find that 62 | \begin{equation} 63 | \label{eq:tantheta} 64 | \tan \theta = \frac{y}{x} 65 | \end{equation} 66 | \begin{equation} 67 | \label{eq:rpyth} 68 | r = \sqrt{x^2 + y^2} 69 | \end{equation} 70 | 71 | \section*{Vector and Scalar Quantities} 72 | 73 | A \textbf{scalar quantity}\index{scalar quantity} is completely specified by a single value with an 74 | appropriate unit and has no direction. 75 | 76 | A \textbf{vector quantity}\index{vector quantity} is completely specified by a number and appropriate 77 | units plus a direction. 78 | 79 | \section*{Unit Vectors} 80 | 81 | A \textbf{unit vector}\index{unit vector} is a dimensionless vector having a 82 | magnitude of exactly $1$. 83 | 84 | \input{physics/newton} 85 | %\input{physics/electric} 86 | \input{physics/circuits} 87 | -------------------------------------------------------------------------------- /physics/newton.tex: -------------------------------------------------------------------------------- 1 | \chapter{Newton's Laws of Motion} 2 | 3 | The following is an excerpt from \cite[p.~20]{newton}, written by Sir Isaac Newton himself: 4 | 5 | \begin{itemize} 6 | \item[\textbf{Law I.} ] 7 | \emph{ 8 | Every body perserveres in its state of rest, or of uniform motion in a right 9 | line, unless it is compelled to change that state by forces impres'd thereon. 10 | } 11 | 12 | Projectiles persevere in their motions, so far as they are not retarded by 13 | the resiliance of air, or impell'd downwards by the force of gravity. A 14 | top, whole parts by their cohesion are preptually drown aside from 15 | rectilinear motions, does not cease its rotation, otherwise than it is 16 | retarded by air. The greater bodies of the Planets and Comets, meeting 17 | with less resistance in more free spaces, preserve their motions both 18 | progressive and circular for a much longer time. 19 | %\hfill\cite[p.~19]{newton} 20 | \item[\textbf{Law II.} ] 21 | \emph{ 22 | The alteration of motion is ever proportional to the motive force 23 | impres'd; and is made in the direction of the right line in which that 24 | force is impres'd. 25 | } 26 | 27 | If any force generates a motion, a double force will generate double 28 | motion, a triple force triple the motion, whether that force be impres'd 29 | altogether and at once, or gradually and successively. And this motion 30 | (being always directed the same way with the generating force) if the body 31 | moved before, is added to or subducted from the former motion, according 32 | as they directly conspire with or are directly contrary to each other; or 33 | obliquely joyned, when they are oblique, so as to produce a new motion 34 | compounded from the determination of both. 35 | %\hfill\cite[p.~20]{newton} 36 | \item[\textbf{Law III.} ] 37 | \emph{ 38 | To every Action there is always opposed an equal Reaction: or the mutual 39 | actions of two bodies upon each other are always equal, and directed to 40 | contrary parts. 41 | } 42 | 43 | Whatever draws or presses another is as much drawn or pressed by that 44 | other. If you press a stone with your finger, the finger is also pressed by 45 | the stone. If a horse draws a stone tyed to a rope, the horse (if I may so 46 | say) will be equally drawn back toward the stone: for the distended rope, 47 | by the same endeavour to relax or unbend it self, will draw the horse as 48 | much towards the stone, and will obstruct the progress of the stone as 49 | much as it advances that of the other. If a body impinge upon another, and 50 | by its force change the motion of the other; that body also (because of 51 | the equality of the mutual pressure) will undergo an equal change, in its 52 | own motion, towards the contrary part. The changes made by these actions 53 | are qual, not in the velocities, but in the motions of bodies; that is to 54 | say, if the bodies are not hinder'd by any other impediments. For because 55 | the motions are equally changed, the changes of the velocities made towars 56 | contrary parts, are reciprocally proportional to the bodies. This Law 57 | takes place also in Attractions, as will be proved in the next Scholium. 58 | \end{itemize} 59 | % \section{Kinematic Equations} 60 | % 61 | % \textbf{Average acceleration} is defined as the \emph{change in velocity} over 62 | % divided by the \emph{change in time} during which the change in velocity occurs. 63 | % Average acceleration is not normally accurate enough for our purposes, however, 64 | % so we will speak more frequently about the \textbf{Instantaneous acceleration}, 65 | % given by 66 | % \begin{equation} 67 | % \label{eq:acceleration} 68 | % \vec a(t) = \frac{\ud \vec v}{\ud t} 69 | % \end{equation} 70 | % as a function of time. 71 | % 72 | % For one direction, $\hat x$, we can use simple calculus to derive our basic 73 | % kinematic equations. We can then generalize this to apply to all of kinematics. 74 | % 75 | % Take equation \eqref{eq:acceleration} and rewrite it for just the $\hat x$ 76 | % direction. 77 | % \[ a_x = \frac{\ud v_x}{\ud t} \] 78 | % Rewrite this as $\ud v_x = a_x \ud t$ and take the integral of both sides from 79 | % $0$ to $t$, our final time. 80 | % \[ \int \ud v_x = \int^t_0 a_x \ud t \] 81 | % Assume acceleration is constant, giving us 82 | % \begin{align} 83 | % \nonumber v_x \bigg|^t_0 &= a \int^t_0 \ud t \\ 84 | % v_{xf} - v_{xi} &= a_x(t-0) = a_x t \\ 85 | % \intertext{We can get an equation for velocity if we add $v_{xi}$ to each 86 | % side:} 87 | % v_{xf} &= a_x t + v_{xi} \label{eq:finalvelocity} 88 | % \end{align} 89 | % 90 | % Now we take the definition for \textbf{instantaneous 91 | % velocity}\index{instantaneous velocity} 92 | % \[ v_x = \frac{\ud x}{\ud t} \] 93 | % and rearrange it and write it as an integral, just as we did for acceleration: 94 | % \[ x_f - x_i = \int^t_0 v_x \ud t \] 95 | % Now we substitute equation \ref{eq:finalvelocity} into this integral as $v_x$ 96 | % to get 97 | % \begin{align} 98 | % \nonumber 99 | % x_f - x_i &= \int^t_0 (v_{xi} + a_x t) \ud t \\ 100 | % \intertext{Because an integral of a sum is a sum of integrals, we can say} 101 | % \nonumber &= \int^t_0 v_{xi} \ud t + a_x \int^t_0 t \ud t \\ 102 | % \nonumber &= v_{xi}(t-0)+a_x \left( \frac{t^2}{2}-0 \right) \\ 103 | % x_f - x_i &= v_{xi} t + \frac{1}{2}a_x t^2\\ 104 | % x_f &= \frac{1}{2}a_x t^2 + v_{xi} t + x_i 105 | % \end{align} 106 | -------------------------------------------------------------------------------- /resources/inkscape.tex: -------------------------------------------------------------------------------- 1 | %%%%%%%%%%%% 2 | % packages % 3 | %%%%%%%%%%%% 4 | 5 | \usepackage[letterpaper, twoside]{geometry} % set the margins to 1in on all sides 6 | \usepackage[bookmarks, hidelinks]{hyperref} % generates pdf index 7 | \usepackage{cclicenses} 8 | %\usepackage[letterpaper]{geometry} % forget that 9 | \usepackage{graphicx} % to include figures 10 | \usepackage{mathtools} % great math stuff 11 | \usepackage{amsfonts} % for blackboard bold, etc 12 | \usepackage{amsthm} % better theorem environments 13 | \usepackage{thmtools} % better interaction with theorem environments 14 | \usepackage{color} % color 15 | \usepackage{verbatim} % multiline comments 16 | \usepackage{array} % better table formatting 17 | \usepackage{microtype} % sexier typesetting 18 | \usepackage{url} % urls 19 | \usepackage{pgfplots} % sexier graphs 20 | \usepackage{fancyhdr} % being fancy 21 | \usepackage{float} % more figure control 22 | \usepackage{wrapfig} % wrapping text around figures 23 | \usepackage{subfigure} % side-by-side figures 24 | \usepackage{wrapfig} % figure wrapping 25 | \usepackage{polynom} % polynomial long division 26 | \usepackage{setspace} % used to set line spacing, e.g  \setstretch{1.3} 27 | \usepackage{makeidx} % indexing 28 | \usepackage[square, numbers, comma, sort&compress]{natbib} % Use the ``Natbib'' style for the references in the Bibliography 29 | %\usepackage{lastpage} %\lastpage 30 | \usepackage{caption3} % load caption package kernel first 31 | \DeclareCaptionOption{parskip} % disable ``parskip'' caption option 32 | \usepackage[font=smaller]{caption} 33 | \usepackage[europeancurrents, europeanvoltages, americanresistors, cuteinductors, americanports, nosiunitx, noarrowmos]{circuitikz} % logic gates 34 | \usepackage{tikz} % stuff for circuits and more 35 | %\usepackage{booktabs} % book-like tables 36 | \usepackage{multirow} 37 | \usepackage{cancel} 38 | \usepackage{fitch} % fitch-style proofs 39 | \usepackage{amssymb} 40 | \usepackage{upgreek} 41 | \usepackage{algorithmicx} 42 | \usepackage{algorithm} 43 | \usepackage{algpseudocode} 44 | \usepackage{epigraph} 45 | % \usepackage{showkeys} % show reference keys 46 | \usepackage{xr} %Allows us to cross-reference in external files 47 | 48 | %%%%%%%%%%%% 49 | % settings % 50 | %%%%%%%%%%%% 51 | 52 | % initialize indexing commands 53 | \makeindex 54 | 55 | % spacing between lines 56 | %\setstretch{1.3} 57 | 58 | % number equations by section 59 | \numberwithin{equation}{section} 60 | \numberwithin{figure}{chapter} 61 | \numberwithin{table}{chapter} 62 | \numberwithin{subsection}{section} 63 | 64 | % resize my tikz/pgf plots 65 | \pgfplotsset{scale=0.5} 66 | 67 | % get fancy 68 | \pagestyle{fancy} 69 | % with this we ensure that the chapter and section 70 | % headings are in lowercase. 71 | \renewcommand{\chaptermark}[1]{% 72 | \markboth{#1}{}} 73 | \renewcommand{\sectionmark}[1]{% 74 | \markright{\thesection\ #1}} 75 | \fancyhf{} % delete current header and footer 76 | \fancyhead[LE]{{\bfseries\thepage}\quad\rightmark} 77 | \fancyhead[RO]{\leftmark\quad\bfseries\thepage} 78 | \renewcommand{\headrulewidth}{0pt} 79 | \renewcommand{\footrulewidth}{0pt} 80 | %\addtolength{\headheight}{0.5pt} % space for the rule 81 | \addtolength{\headheight}{1em} % space for the rule 82 | \fancypagestyle{plain}{% 83 | \fancyhead{} % get rid of headers on plain pages 84 | \renewcommand{\headrulewidth}{0pt} % and the line 85 | } 86 | 87 | \NeedsTeXFormat{LaTeX2e} 88 | \ProvidesPackage{Commons} 89 | 90 | % draw a thin border around figures 91 | %\floatstyle{boxed} 92 | %\restylefloat{figure} 93 | 94 | %%%%%%%%%%%%%%%%%%%% ABBREVIATIONS %%%%%%%%%%%%%%%%%%%% 95 | \newcommand{\pha}{{}_\bullet} 96 | \newcommand{\phb}{{}_\blacktriangle} 97 | \newcommand{\phc}{\blacktriangle} 98 | 99 | \newcommand{\dotleq}{\mathrel{\raisebox{1.2ex}{$⋅$}\mkern-13.5mu \leq}} 100 | \renewcommand{\projlim}{\varprojlim} 101 | \renewcommand{\injlim}{\varinjlim} 102 | \newcommand{\dottimes}{\stackrel[\cdot]{\times}} 103 | 104 | %\newcommand{\xra}[1]{\xrightarrow{#1}} 105 | \newcommand{\xra}[1]{\, \tikz[baseline] \draw (0pt,3.5pt) -- (7pt,3.5pt); {\raisebox{1.5pt}{\ensuremath{\scriptstyle #1}}} \tikz[baseline] \draw [->] (0pt,3.5pt) -- (7pt,3.5pt); \, } 106 | 107 | %\renewcommand{\obar}[1]{\overline{#1}} 108 | \newcommand{\ubar}[1]{\underline{#1}} 109 | 110 | \newcommand{\set}[1]{\left\{#1\right\}} 111 | \newcommand{\pa}[1]{\left(#1\right)} 112 | \newcommand{\ang}[1]{\left<#1\right>} 113 | \newcommand{\bra}[1]{\left[#1\right]} 114 | \newcommand{\abs}[1]{\left|#1\right|} 115 | \newcommand{\norm}[1]{\left\|#1\right\|} 116 | 117 | \newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}} 118 | \newcommand{\pmat}[1]{\pa{\mat{#1}}} 119 | \newcommand{\bmat}[1]{\bra{\mat{#1}}} 120 | \newcommand{\sismat}[1]{\left\{\mat{#1}\right.} 121 | \newcommand{\sm}[1]{\begin{smallmatrix}#1\end{smallmatrix}} 122 | \newcommand{\psm}[1]{\pa{\sm{#1}}} 123 | \newcommand{\bsm}[1]{\bra{\sm{#1}}} 124 | \newcommand{\tci}[2]{\set{\,#1 \mid{} #2\,}} 125 | \newcommand{\tcia}[2]{\ang{\,#1 \mid{} #2\,}} 126 | \newcommand{\pfrac}[2]{\pa{\frac{#1}{#2}}} 127 | \newcommand{\bfrac}[2]{\bra{\frac{#1}{#2}}} 128 | \newcommand{\psfrac}[2]{\pa{\sfrac{#1}{#2}}} 129 | \newcommand{\bsfrac}[2]{\bra{\sfrac{#1}{#2}}} 130 | \newcommand{\der}[2]{\frac{\partial #1}{\partial #2}} 131 | \newcommand{\pder}[2]{\pfrac{\partial #1}{\partial #2}} 132 | \newcommand{\sder}[2]{\sfrac{\partial #1}{\partial #2}} 133 | \newcommand{\psder}[2]{\psfrac{\partial #1}{\partial #2}} 134 | 135 | \newcommand{\leg}[2]{\pfrac{#1}{#2}} 136 | \newcommand{\estdim}[2]{\left[#1 : #2\right]} 137 | \newcommand{\abel}[2]{\left[#1, #2\right]} 138 | \newcommand{\brk}[2]{\left[#1,#2\right]} 139 | 140 | %%%%%%%%%%%%%%%%%%%% OPERATORS %%%%%%%%%%%%%%%%%%%% 141 | \DeclareMathOperator{\Schemes}{\mathfrak{Sch}} 142 | \DeclareMathOperator{\AffineSchemes}{\mathfrak{AffSch}} 143 | \DeclareMathOperator{\Sets}{\mathfrak{Sets}} 144 | \DeclareMathOperator{\Groupoids}{\mathfrak{Groupoids}} 145 | \DeclareMathOperator{\TopologicalSpaces}{\mathfrak{Top}} 146 | 147 | \DeclareMathOperator{\Alb}{Alb} 148 | \DeclareMathOperator{\Aut}{Aut} 149 | \DeclareMathOperator{\B}{B} 150 | \DeclareMathOperator{\cod}{cod} 151 | \DeclareMathOperator{\de}{d} 152 | \DeclareMathOperator{\diag}{diag} 153 | \DeclareMathOperator{\Div}{Div} 154 | \DeclareMathOperator{\Ext}{Ext} 155 | \DeclareMathOperator{\Fix}{Fix} 156 | \DeclareMathOperator{\gen}{g} 157 | \DeclareMathOperator{\GL}{GL} 158 | \DeclareMathOperator{\Hilb}{Hilb} 159 | \DeclareMathOperator{\Ho}{H} 160 | \DeclareMathOperator{\ho}{h} 161 | \DeclareMathOperator{\Hom}{Hom} 162 | \DeclareMathOperator{\id}{id} 163 | \DeclareMathOperator{\Image}{Im} 164 | \DeclareMathOperator{\Isom}{Isom} 165 | \DeclareMathOperator{\Mod}{Mod} 166 | \DeclareMathOperator{\Mor}{Mor} 167 | \DeclareMathOperator{\argen}{p_a} 168 | \DeclareMathOperator{\geomgen}{p_g} 169 | \DeclareMathOperator{\Pic}{Pic} 170 | \DeclareMathOperator{\Proj}{Proj} 171 | \DeclareMathOperator{\irr}{q} 172 | \DeclareMathOperator{\res}{res} 173 | \DeclareMathOperator{\sgn}{sgn} 174 | \DeclareMathOperator{\Sing}{Sing} 175 | \DeclareMathOperator{\Spec}{Spec} 176 | \DeclareMathOperator{\Stab}{Stab} 177 | \DeclareMathOperator{\tr}{tr} 178 | \DeclareMathOperator{\Tors}{Tors} 179 | \DeclareMathOperator{\vp}{v.p.} 180 | 181 | %%%%%%%%%%%%%%%%%%%% LETTERS %%%%%%%%%%%%%%%%%%%% 182 | \newcommand{\frakM}{\mathfrak{M}} 183 | \newcommand{\frakm}{\mathfrak{m}} 184 | 185 | %%%%%%%%%%%%%%%%%%% 186 | % custom commands % 187 | %%%%%%%%%%%%%%%%%%% 188 | 189 | \newcommand{\margin}{\marginpar} 190 | \newcommand{\ud}{\,\mathrm{d}} 191 | \newcommand{\lets}{\text{let }} 192 | \DeclareMathOperator{\=H}{\stackrel{\text{H}}{=}} 193 | \newcommand{\ddx}{\frac{\ud}{\ud x}} 194 | \newcommand{\szinfty}{\sum_{n=0}^{\infty}} 195 | 196 | %%%%%%%%%%%% 197 | % theorems % 198 | %%%%%%%%%%%% 199 | 200 | \theoremstyle{definition} \newtheorem{ex}{Example}[section] 201 | \theoremstyle{definition} \newtheorem*{sol}{Solution} 202 | \theoremstyle{plain} \newtheorem{theorem}{Theorem} 203 | \theoremstyle{plain} \newtheorem{corollary}{Corollary} 204 | \theoremstyle{definition} \newtheorem*{defn}{Definition} 205 | \theoremstyle{definition} \newtheorem{homework}{Homework} 206 | \theoremstyle{remark} \newtheorem*{remark}{Remark} 207 | \theoremstyle{remark} \newtheorem*{note}{Note} 208 | 209 | 210 | -------------------------------------------------------------------------------- /titlepage.tex: -------------------------------------------------------------------------------- 1 | \title{\textbf{\ 2 | Mathematics 3 | }\\An undergraduate notebook} 4 | \author {Nathan Typanski} 5 | \date {\today} 6 | \maketitle 7 | 8 | \newpage 9 | 10 | \cleardoublepage 11 | --------------------------------------------------------------------------------