├── Misc ├── Pi:3.14... │ ├── README.md │ ├── pi.pdf │ └── pi.tex └── Periodic-table │ ├── trna.png │ ├── trna2.png │ ├── trna3.png │ ├── trna4.png │ ├── elements.csv │ ├── periodic_table.pdf │ ├── README.md │ ├── psheader.txt │ ├── LICENSE │ └── periodic_table.tex ├── .gitignore ├── MTH102: Real-analysis ├── img │ ├── bolanzo.png │ ├── cauchy_no.png │ ├── geogebra.png │ ├── cauchy_yes.png │ ├── intermediate.png │ ├── lim_sup_prob.png │ ├── sup_example.png │ ├── uniform_no.png │ └── uniform_yes.png ├── MTH102_-_Real_analysis.pdf ├── README.md ├── notes.cls ├── ref.bib ├── color-env.sty └── MTH102_-_Real_analysis.tex ├── PHY102: Electromagnetism ├── PHY102_-_Electromagnetism.pdf ├── README.md └── PHY102_-_Electromagnetism.tex ├── MTH201: Differential-Geometry ├── MTH201_-_Differential_Geometry.pdf ├── README.md ├── notes.cls ├── color-env.sty └── MTH201_-_Differential_Geometry.tex ├── PHY202: Statistical-Mechanics and Thermodynamics ├── Foundations_of_Thermodynamics.pdf ├── PHY202_-_Statistical Mechanics.pdf ├── README.md ├── color-tufte.sty └── PHY202_-_Statistical Mechanics.tex ├── README.md └── LICENSE /Misc/Pi:3.14.../README.md: -------------------------------------------------------------------------------- 1 | ## PI -------------------------------------------------------------------------------- /Misc/Pi:3.14.../pi.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Pi:3.14.../pi.pdf -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | __latexindent_temp.tex 2 | *.aux 3 | *.log 4 | *.out 5 | *.synctex.gz 6 | *.aux 7 | *.fls 8 | *.fdb_latexmk -------------------------------------------------------------------------------- /Misc/Periodic-table/trna.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/trna.png -------------------------------------------------------------------------------- /Misc/Periodic-table/trna2.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/trna2.png -------------------------------------------------------------------------------- /Misc/Periodic-table/trna3.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/trna3.png -------------------------------------------------------------------------------- /Misc/Periodic-table/trna4.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/trna4.png -------------------------------------------------------------------------------- /Misc/Periodic-table/elements.csv: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/elements.csv -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/bolanzo.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/bolanzo.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/cauchy_no.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/cauchy_no.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/geogebra.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/geogebra.png -------------------------------------------------------------------------------- /Misc/Periodic-table/periodic_table.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/Misc/Periodic-table/periodic_table.pdf -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/cauchy_yes.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/cauchy_yes.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/intermediate.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/intermediate.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/lim_sup_prob.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/lim_sup_prob.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/sup_example.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/sup_example.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/uniform_no.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/uniform_no.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/img/uniform_yes.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/img/uniform_yes.png -------------------------------------------------------------------------------- /MTH102: Real-analysis/MTH102_-_Real_analysis.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH102: Real-analysis/MTH102_-_Real_analysis.pdf -------------------------------------------------------------------------------- /PHY102: Electromagnetism/PHY102_-_Electromagnetism.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/PHY102: Electromagnetism/PHY102_-_Electromagnetism.pdf -------------------------------------------------------------------------------- /MTH201: Differential-Geometry/MTH201_-_Differential_Geometry.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/MTH201: Differential-Geometry/MTH201_-_Differential_Geometry.pdf -------------------------------------------------------------------------------- /Misc/Periodic-table/README.md: -------------------------------------------------------------------------------- 1 | # Periodic Table 2 | 3 | Nice periodic table made in LaTeX 4 | ## [Original Repository](https://github.com/PaNDanese/nperiodic_table.git) 5 | 6 | 7 | -------------------------------------------------------------------------------- /PHY202: Statistical-Mechanics and Thermodynamics/Foundations_of_Thermodynamics.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/PHY202: Statistical-Mechanics and Thermodynamics/Foundations_of_Thermodynamics.pdf -------------------------------------------------------------------------------- /PHY202: Statistical-Mechanics and Thermodynamics/PHY202_-_Statistical Mechanics.pdf: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/dev-aditya/course-notes-core/HEAD/PHY202: Statistical-Mechanics and Thermodynamics/PHY202_-_Statistical Mechanics.pdf -------------------------------------------------------------------------------- /MTH201: Differential-Geometry/README.md: -------------------------------------------------------------------------------- 1 | ## Differential Geometry of Curves and Surfaces 2 | 3 | These are the notes made during MTH201. The main text is borrowed from the book 4 | [Elementary Differential Geometry by Andrew Pressley](https://www.springer.com/gp/book/9781848828902). -------------------------------------------------------------------------------- /MTH102: Real-analysis/README.md: -------------------------------------------------------------------------------- 1 | ## Real Analysis 2 | 3 | These lecture notes were made during the course MTH102. The text contains main definitions and theorems. Some of the problems and their solutions are also discussed. Set of references are mentioned in Bibliography. 4 | 5 | The graphics in the text are borrowed from Wikipedia (released in public domain). -------------------------------------------------------------------------------- /PHY102: Electromagnetism/README.md: -------------------------------------------------------------------------------- 1 | ## Electromagnetism 2 | 3 | These are not exactly notes. This is a *go through*, brief set of important results and definitions in Electromagnetism. The section on **Maxwell's Equations** are made after following this beautiful and concise lecture by Prof. V. Balakrishnan on [Summary of classical electromagnetism](https://www.youtube.com/watch?v=bsybS5fZGjY&list=PL5E4E56893588CBA8&index=8) -------------------------------------------------------------------------------- /Misc/Periodic-table/psheader.txt: -------------------------------------------------------------------------------- 1 | %! 2 | /N{def}def/B{bind def}N/S{exch}N/X{SN}B 3 | /uscale 1 N 4 | /chp 10 string N 5 | /acp {currentpoint transform} B /rcp {chp cvs print} B 6 | /Cpos{/ishr X uscale mul /hbox X uscale mul /dbox X uscale mul /wbox X 7 | acp /ycpt X ishr{wbox neg add}{dbox neg hbox add /hbox X 0 /dbox X }ifelse /xcpt X 8 | (x1=) print xcpt rcp ( y1=) print ycpt hbox add rcp 9 | ( x2=) print xcpt wbox add rcp ( y2=) print ycpt dbox add rcp = 10 | } B 11 | 12 | -------------------------------------------------------------------------------- /PHY202: Statistical-Mechanics and Thermodynamics/README.md: -------------------------------------------------------------------------------- 1 | ## Statistical Mechanics and Thermodynamics 2 | 3 | I made these notes while following lectures by [Prof. Leonard Susskind](https://theoreticalminimum.com/about). The video lectures 4 | can be found on [this webpage](https://theoreticalminimum.com/courses/statistical-mechanics/2013/spring) or [here on YouTube](https://www.youtube.com/watch?v=D1RzvXDXyqA&list=PL_IkS0viawhr3HcKH607rXbVqy28W_gB7) 5 | 6 | I've also included brief (precisely written) notes on [Foundations of Thermodynamics](https://github.com/dev-aditya/course-notes-core/blob/master/PHY202:Statistical-Mechanics/Foundations_of_Thermodynamics.pdf). 7 | -------------------------------------------------------------------------------- /MTH102: Real-analysis/notes.cls: -------------------------------------------------------------------------------- 1 | \NeedsTeXFormat{LaTeX2e} 2 | \ProvidesClass{notes}[2020/11/01 Notes Class] 3 | 4 | \LoadClass[a4paper,oneside]{book} 5 | \RequirePackage{hyperref} 6 | \RequirePackage[total={5in, 9in}]{geometry} 7 | \hypersetup{ 8 | colorlinks=true, 9 | linkcolor=blue, 10 | filecolor=magenta, 11 | urlcolor=cyan, 12 | } 13 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 | \makeatletter 15 | \def\@seccntformat#1{\@ifundefined{#1@cntformat}% 16 | {\csname the#1\endcsname\quad}% default 17 | {\csname #1@cntformat\endcsname}}% individual control 18 | \newcommand{\section@cntformat}{\S\thesection\quad} 19 | \newcommand{\subsection@cntformat}{\S\thesubsection\quad} 20 | \newcommand{\subsubsection@cntformat}{\S\thesubsubsection\quad} 21 | \makeatletter 22 | 23 | 24 | 25 | -------------------------------------------------------------------------------- /MTH201: Differential-Geometry/notes.cls: -------------------------------------------------------------------------------- 1 | \NeedsTeXFormat{LaTeX2e} 2 | \ProvidesClass{notes}[2020/11/01 Notes Class] 3 | 4 | \LoadClass[a4paper,oneside]{book} 5 | \RequirePackage{hyperref} 6 | \RequirePackage[total={5in, 9in}]{geometry} 7 | \hypersetup{ 8 | colorlinks=true, 9 | linkcolor=blue, 10 | filecolor=magenta, 11 | urlcolor=cyan, 12 | } 13 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 14 | \makeatletter 15 | \def\@seccntformat#1{\@ifundefined{#1@cntformat}% 16 | {\csname the#1\endcsname\quad}% default 17 | {\csname #1@cntformat\endcsname}}% individual control 18 | \newcommand{\section@cntformat}{\S\thesection\quad} 19 | \newcommand{\subsection@cntformat}{\S\thesubsection\quad} 20 | \newcommand{\subsubsection@cntformat}{\S\thesubsubsection\quad} 21 | \makeatletter 22 | 23 | 24 | 25 | -------------------------------------------------------------------------------- /Misc/Periodic-table/LICENSE: -------------------------------------------------------------------------------- 1 | MIT License 2 | 3 | Copyright (c) 2019 Paul Danese 4 | 5 | Permission is hereby granted, free of charge, to any person obtaining a copy 6 | of this software and associated documentation files (the "Software"), to deal 7 | in the Software without restriction, including without limitation the rights 8 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 9 | copies of the Software, and to permit persons to whom the Software is 10 | furnished to do so, subject to the following conditions: 11 | 12 | The above copyright notice and this permission notice shall be included in all 13 | copies or substantial portions of the Software. 14 | 15 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 16 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 17 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 18 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 19 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 20 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE 21 | SOFTWARE. 22 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Course Notes :notebook_with_decorative_cover: 2 | 3 | This repository contains course notes typesetted in LaTeX, for core year courses at IISER Mohali. They are not complete (or may contain errors), 4 | but might be helpful. So, I keep it here. *The templates used in making these notes can be found [here](https://github.com/dev-aditya/LaTeX-template)* 5 | 6 | **Please feel free to share and edit the notes. Create an issue or PR in case you find any typographical errors or any other problem.** 7 | *Feel free to [email me](mailto:adityadev@tuta.io) if you find these notes/text useful :blush:* 8 | ## Courses 9 | 10 | 1. [MTH102: Real Analysis](https://github.com/dev-aditya/course-notes-core/tree/master/MTH102:%20Real-analysis) 11 | 2. [MTH201: Differential Geometry](https://github.com/dev-aditya/course-notes-core/tree/master/MTH201:%20Differential-Geometry) 12 | 3. [PHY102: Electromagnetism](https://github.com/dev-aditya/course-notes-core/tree/master/PHY102:%20Electromagnetism) 13 | 4. [PHY202: Statistical Mechanics and Thermodynamics](https://github.com/dev-aditya/course-notes-core/tree/master/PHY202:%20Statistical-Mechanics%20and%20Thermodynamics) 14 | 15 | ## Some Cool Stuff 16 | * [Pi in LaTeX](https://github.com/dev-aditya/course-notes-core/tree/master/Misc/Pi:3.14...) 17 | * [Periodic Table](https://github.com/dev-aditya/course-notes-core/tree/master/Misc/Periodic-table) 18 | ___________________________________ 19 | ### LICENSE 20 | The text and figures are licensed under the [**Creative Commons Attribution 4.0 International**](https://github.com/dev-aditya/course-notes-core/blob/master/LICENSE) (CC-BY-4.0). 21 | 22 | -------------------------------------------------------------------------------- /MTH102: Real-analysis/ref.bib: -------------------------------------------------------------------------------- 1 | @book{introduction, 2 | title={Elementary analysis}, 3 | author={Kenneth A. Ross}, 4 | publisher={Springer}, 5 | year = {2013} 6 | } 7 | @book{abb01understanding, 8 | title={Understanding analysis}, 9 | author={Abbott, Stephen}, 10 | volume={2}, 11 | year={2001}, 12 | publisher={Springer} 13 | } 14 | @misc{ wiki, 15 | author = {Wikipedia}, 16 | title = " {W}ikipedia: The Free Encyclopedia", 17 | url = "http://en.wikipedia.org/", 18 | } 19 | @misc{ wikib, 20 | author = "{Wikipedia contributors}", 21 | title = "Bolzano–Weierstrass theorem --- {Wikipedia}{,} The Free Encyclopedia", 22 | year = "2020", 23 | url = "https://en.wikipedia.org/w/index.php?title=Bolzano%E2%80%93Weierstrass_theorem&oldid=947971057", 24 | note = "[Online; accessed 7-May-2020]" 25 | } 26 | @misc{ wikiroot, 27 | author = "{Wikipedia contributors}", 28 | title = "Root test --- {Wikipedia}{,} The Free Encyclopedia", 29 | year = "2020", 30 | url = "{https://en.wikipedia.org/w/index.php?title=Root_test&oldid=948125284}", 31 | note = "[Online; accessed 7-May-2020]" 32 | } 33 | @misc{ wikiratio, 34 | author = "{Wikipedia contributors}", 35 | title = "Ratio test --- {Wikipedia}{,} The Free Encyclopedia", 36 | year = "2020", 37 | url = "{https://en.wikipedia.org/w/index.php?title=Ratio_test&oldid=952648008}", 38 | note = "[Online; accessed 7-May-2020]" 39 | } 40 | @misc{ wikicauchy, 41 | author = "{Wikipedia contributors}", 42 | title = "Cauchy sequence --- {Wikipedia}{,} The Free Encyclopedia", 43 | year = "2019", 44 | url = "https://en.wikipedia.org/w/index.php?title=Cauchy_sequence&oldid=928609654", 45 | note = "[Online; accessed 7-May-2020]" 46 | } 47 | @misc{ wikiuniform, 48 | author = "{Wikipedia contributors}", 49 | title = "Uniform continuity --- {Wikipedia}{,} The Free Encyclopedia", 50 | year = "2020", 51 | url = "https://en.wikipedia.org/w/index.php?title=Uniform_continuity&oldid=954710721", 52 | note = "[Online; accessed 7-May-2020]" 53 | } 54 | @misc{ wiki:integral, 55 | author = "{Wikipedia contributors}", 56 | title = "Riemann sum --- {Wikipedia}{,} The Free Encyclopedia", 57 | year = "2020", 58 | url = "https://en.wikipedia.org/w/index.php?title=Riemann_sum&oldid=951325810", 59 | note = "[Online; accessed 8-May-2020]" 60 | } 61 | @misc{ wiki:appendix, 62 | author = "{Wikipedia contributors}", 63 | title = "Taylor series --- {Wikipedia}{,} The Free Encyclopedia", 64 | year = "2020", 65 | url = "https://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=956280889", 66 | note = "[Online; accessed 19-May-2020]" 67 | } -------------------------------------------------------------------------------- /PHY202: Statistical-Mechanics and Thermodynamics/color-tufte.sty: -------------------------------------------------------------------------------- 1 | \ProvidesPackage{color-tufte}[2021/03/11 Color Tufte] 2 | 3 | %%%%%%%%%%%%%%%% 4 | % Requirements % 5 | %%%%%%%%%%%%%%%% 6 | 7 | \RequirePackage{xcolor} 8 | \RequirePackage{graphicx} 9 | \RequirePackage{framed} 10 | \RequirePackage{amsthm} 11 | \RequirePackage[many]{tcolorbox} 12 | %%%%%%%%%%%%%%%%%% 13 | % Environments % 14 | %%%%%%%%%%%%%%%%%% 15 | 16 | 17 | %%%%%%%%%%%%%%%% THEOREM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 18 | \newtheorem{main_equations}{}[] 19 | \tcolorboxenvironment{main_equations}{ 20 | boxrule=0pt, 21 | boxsep=2pt, 22 | colback={White!90!Dandelion}, 23 | enhanced jigsaw, 24 | %borderline west={2pt}{0pt}{Dandelion}, 25 | sharp corners, 26 | before skip=10pt, 27 | after skip=10pt, 28 | breakable, 29 | } 30 | %%%%%%%%%%%%%%% LEMMA %%%%%%%%%%%%%%% 31 | \newtheorem{lemma}{Lemma}[section] 32 | 33 | \tcolorboxenvironment{lemma}{ 34 | boxrule=0pt, 35 | boxsep=2pt, 36 | colback={White!90!Red}, 37 | enhanced jigsaw, 38 | borderline west={2pt}{0pt}{Red}, 39 | sharp corners, 40 | before skip=10pt, 41 | after skip=10pt, 42 | breakable, 43 | } 44 | %%%%%%%%%%%%% DEFINITION %%%%%%%%%%%%%%% 45 | \newtheorem{assume}{Assumption}[section] 46 | 47 | \tcolorboxenvironment{assume}{ 48 | boxrule=0pt, 49 | boxsep=2pt, 50 | colback={White!90!Cerulean}, 51 | enhanced jigsaw, 52 | borderline west={2pt}{0pt}{Cerulean}, 53 | sharp corners, 54 | before skip=10pt, 55 | after skip=10pt, 56 | breakable, 57 | } 58 | %%%%%%%%%%%%%% COROLLARY %%%%%%%%%%%%%%%%%% 59 | \newtheorem{corollary}{Corollary}[section] 60 | 61 | \tcolorboxenvironment{corollary}{ 62 | boxrule=0pt, 63 | boxsep=2pt, 64 | colback={White!90!Yellow}, 65 | enhanced jigsaw, 66 | borderline west={2pt}{0pt}{Yellow}, 67 | sharp corners, 68 | before skip=10pt, 69 | after skip=10pt, 70 | breakable, 71 | } 72 | %%%%%%%%%%%%% PROPOSITIONS %%%%%%%%%%%%%%%%%%% 73 | \newtheorem{definition}[subsection]{Definition} 74 | \tcolorboxenvironment{definition}{ 75 | boxrule=0pt, 76 | boxsep=2pt, 77 | colback={green!10}, 78 | enhanced jigsaw, 79 | borderline west={2pt}{0pt}{Green}, 80 | sharp corners, 81 | before skip=10pt, 82 | after skip=10pt, 83 | breakable, 84 | } 85 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 86 | %\tcolorboxenvironment{proof}{ 87 | %boxrule=0pt, 88 | %boxsep=2pt, 89 | %blanker, 90 | %borderline west={2pt}{0pt}{NavyBlue!80!white}, 91 | %before skip=10pt, 92 | %after skip=10pt, 93 | %left=12pt, 94 | %right=12pt, 95 | %breakable, 96 | %} 97 | 98 | %%%%%%%% Problem %%%%%%%%%%%%% 99 | \newtheorem{prob}{Problem} 100 | \newenvironment{problem} 101 | {\colorlet{shadecolor}{White!90!Orange}\begin{shaded}\begin{prob}} 102 | {\end{prob}\end{shaded}} 103 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 104 | 105 | 106 | 107 | 108 | -------------------------------------------------------------------------------- /Misc/Pi:3.14.../pi.tex: -------------------------------------------------------------------------------- 1 | \documentclass[10pt]{article} 2 | 3 | \usepackage{graphicx} 4 | \usepackage{shapepar} 5 | \usepackage{microtype} 6 | 7 | \def\pipar#1{\shapepar{\pishape}#1\par} 8 | \def\pishape{% 9 | {25.0839}% 10 | {0.0838926}b{14.3456}\\% 11 | {0.0838926}t{14.3456}{33.3054}\\% 12 | {0.503356}t{11.5772}{37.6678}\\% 13 | {1.25839}t{9.98322}{39.6812}\\% 14 | {2.09732}t{8.52614}{41.5578}\\% 15 | {2.85235}t{7.21477}{42.8691}\\% 16 | {3.27181}t{6.7953}{43.2886}\\% 17 | {4.11074}t{5.95638}{43.7081}\\% 18 | {5.28524}t{4.78188}{43.7081}\\% 19 | {5.62081}t{4.44631}{15.1007}st{19.547}{12.6678}st{32.2148}{15.0168}\\% 20 | {5.62081}t{4.44631}{7.9698}t{19.547}{2.34899}t{32.2148}{2.34899}\\% 21 | {6.04027}t{4.18011}{6.22257}t{19.4227}{2.37488}t{32.1424}{2.37047}\\% 22 | {6.87919}t{3.64772}{5.16101}t{19.1741}{2.42667}t{31.9978}{2.41343}\\% 23 | {7.63423}t{3.16856}{4.04621}t{18.9504}{2.47328}t{31.8676}{2.45208}\\% 24 | {8.05369}t{2.90236}{3.80463}t{18.8261}{2.49917}t{31.7953}{2.47356}\\% 25 | {8.38926}t{2.6894}{3.61137}t{18.7267}{2.51989}t{31.7245}{2.50373}\\% 26 | {9.22819}t{2.15701}{3.12823}t{18.4781}{2.57167}t{31.5474}{2.57045}\\% 27 | {9.98322}t{1.67785}{2.96021}t{18.2544}{2.61828}t{31.388}{2.6305}\\% 28 | {11.5772}t{0.415968}{2.85584}t{17.7821}{2.71667}t{31.0515}{2.75727}\\% 29 | {11.9966}t{0.0838926}{2.91826}t{17.6578}{2.74256}t{30.9629}{2.79063}\\% 30 | {12.4161}t{0.0838926}{2.64861}t{17.5336}{2.76846}t{30.8743}{2.82399}\\% 31 | {12.7517}t{0.0838926}{2.43289}t{17.4088}{2.81003}t{30.8035}{2.85068}\\% 32 | {13.1711}t{0.0838926}{2.22315}t{17.2529}{2.862}t{30.715}{2.88404}\\% 33 | {13.5906}t{0.0838926}{2.01342}t{17.097}{2.91397}t{30.6264}{2.9174}\\% 34 | {14.0101}t{0.0838926}{0.838926}t{16.9411}{2.96594}t{30.5378}{2.95076}\\% 35 | {14.0101}e{0.922819}t{16.9411}{2.96594}t{30.5378}{2.95076}\\% 36 | {14.7651}t{16.6605}{3.05948}t{30.3785}{3.01081}\\% 37 | {15.604}t{16.3487}{3.16342}t{30.2013}{3.18792}\\% 38 | {16.7785}t{15.9121}{3.30893}t{30.0039}{3.3854}\\% 39 | {21.896}t{14.0101}{3.94295}t{29.1434}{4.24586}\\% 40 | {25.0839}t{12.7724}{4.3305}t{28.6074}{4.78188}\\% 41 | {25.8389}t{12.4793}{4.42229}t{28.6074}{4.78188}\\% 42 | {29.0268}t{11.2416}{4.80984}t{28.6074}{5.39494}\\% 43 | {29.4463}t{11.0415}{4.8981}t{28.6074}{5.47561}\\% 44 | {30.2013}t{10.6813}{5.04094}t{28.6074}{5.62081}\\% 45 | {30.6208}t{10.4812}{5.12029}t{28.6074}{5.72383}\\% 46 | {34.9832}t{8.40004}{5.9456}t{28.9901}{6.41263}\\% 47 | {35.4027}t{8.19993}{6.00914}t{29.0268}{6.58557}\\% 48 | {37.3322}t{7.27942}{6.30143}t{29.5346}{7.04256}\\% 49 | {38.1711}t{6.87919}{6.42852}t{29.7554}{6.64702}\\% 50 | {38.5906}t{6.87919}{6.29195}t{29.8658}{6.44925}\\% 51 | {39.3456}t{7.09492}{5.59084}t{30.1612}{5.9965}\\% 52 | {39.7651}t{7.21477}{5.20134}t{30.3254}{5.4129}\\% 53 | {40.5201}t{7.63423}{4.24257}t{30.6208}{4.36242}\\% 54 | {40.9396}t{8.01175}{3.56544}t{31.2081}{2.60067}\\% 55 | {41.3591}t{8.38926}{2.43289}t{31.7953}{0.838926}\\% 56 | {41.3591}e{10.8221}e{32.6342}% 57 | } 58 | 59 | \begin{document} 60 | 61 | \pipar{3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1} 62 | 63 | \end{document} -------------------------------------------------------------------------------- /MTH102: Real-analysis/color-env.sty: -------------------------------------------------------------------------------- 1 | \ProvidesPackage{color-env}[2020/10/31 v1.0 color-env] 2 | 3 | %%%%%%%%%%%%%%%% 4 | % Requirements % 5 | %%%%%%%%%%%%%%%% 6 | \RequirePackage{xcolor} 7 | \RequirePackage{graphicx} 8 | \RequirePackage[framemethod=TikZ]{mdframed} %% for colored framed boxes 9 | \RequirePackage{framed} %% for problem 10 | \RequirePackage{amsthm} %% for theorem environment 11 | %%%%%%%%%%%%%%%%%% 12 | % Environments % 13 | %%%%%%%%%%%%%%%%%% 14 | 15 | %%%%%%%% Problem %%%%%%%%%%%%% 16 | \colorlet{shadecolor}{orange!12} 17 | \newtheorem{prob}{Problem} 18 | \newenvironment{problem} 19 | {\begin{shaded}\begin{prob}} 20 | {\end{prob}\end{shaded}} 21 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22 | 23 | %%%%%%%%%%%%%%%% THEOREM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24 | \newcounter{theo}[section] \setcounter{theo}{0} %% counter 25 | \renewcommand{\thetheo}{\arabic{chapter}.\arabic{section}.\arabic{theo}} 26 | \newenvironment{theorem}[2][]{% 27 | \refstepcounter{theo}% 28 | \ifstrempty{#1}% 29 | {\mdfsetup{% 30 | frametitle={% 31 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 32 | \node[anchor=east,rectangle,fill=blue!20] 33 | {\strut Theorem~\thetheo};}} 34 | }% 35 | {\mdfsetup{% 36 | frametitle={% 37 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 38 | \node[anchor=east,rectangle,fill=blue!20] 39 | {\strut Theorem~\thetheo:~#1};}}% 40 | }% 41 | \mdfsetup{innertopmargin=10pt,linecolor=blue!20,% 42 | linewidth=2pt,topline=true,% 43 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 44 | } 45 | \begin{mdframed}[]\relax% 46 | \label{#2}}{\end{mdframed}} 47 | %%%%%%%%%%%%%%% For lemma %%%%%%%%%%%%%%% 48 | \newcounter{lem}[section]\setcounter{lem}{0} 49 | \renewcommand{\thelem}{\arabic{chapter}.\arabic{section}.\arabic{lem}} 50 | \newenvironment{lemma}[2][]{% 51 | \refstepcounter{lem}% 52 | \ifstrempty{#1}% 53 | {\mdfsetup{% 54 | frametitle={% 55 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 56 | \node[anchor=east,rectangle,fill=orange!20] 57 | {\strut Lemma~\thelem};}} 58 | }% 59 | {\mdfsetup{% 60 | frametitle={% 61 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 62 | \node[anchor=east,rectangle,fill=orange!20] 63 | {\strut Lemma~\thelem:~#1};}}% 64 | }% 65 | \mdfsetup{innertopmargin=10pt,linecolor=green!20,% 66 | linewidth=2pt,topline=true,% 67 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 68 | } 69 | \begin{mdframed}[]\relax% 70 | \label{#2}}{\end{mdframed}} 71 | 72 | %%%%%%%%%%%%% definition %%%%%%%%%%%%%%% 73 | \newcounter{def}[subsection]\setcounter{def}{0} 74 | \renewcommand{\thedef}{\arabic{chapter}.\arabic{section}.\arabic{def}} 75 | \newenvironment{definition}[2][]{% 76 | \refstepcounter{def}% 77 | \ifstrempty{#1}% 78 | {\mdfsetup{% 79 | frametitle={% 80 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 81 | \node[anchor=east,rectangle,fill=red!25] 82 | {\strut Definition~\thedef};}} 83 | }% 84 | {\mdfsetup{% 85 | frametitle={% 86 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 87 | \node[anchor=east,rectangle,fill=red!25] 88 | {\strut Definition~\thedef:~#1};}}% 89 | }% 90 | \mdfsetup{innertopmargin=10pt,linecolor=red!25,% 91 | linewidth=2pt,topline=true,% 92 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 93 | } 94 | \begin{mdframed}[]\relax% 95 | \label{#2}}{\end{mdframed}} 96 | %%%%%%%%%%%%%% corollary %%%%%%%%%%%%%%%%%% 97 | \newcounter{cor}[section]\setcounter{cor}{0} 98 | \renewcommand{\thecor}{\arabic{chapter}.\arabic{section}.\arabic{cor}} 99 | \newenvironment{corollary}[2][]{% 100 | \refstepcounter{cor}% 101 | \ifstrempty{#1}% 102 | {\mdfsetup{% 103 | frametitle={% 104 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 105 | \node[anchor=east,rectangle,fill=yellow!20] 106 | {\strut Corollary~\thecor};}} 107 | }% 108 | {\mdfsetup{% 109 | frametitle={% 110 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 111 | \node[anchor=east,rectangle,fill=yellow!20] 112 | {\strut Corollary~\thecor:~#1};}}% 113 | }% 114 | \mdfsetup{innertopmargin=10pt,linecolor=yellow!20,% 115 | linewidth=2pt,topline=true,% 116 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 117 | } 118 | \begin{mdframed}[]\relax% 119 | \label{#2}}{\end{mdframed}} 120 | 121 | %%%%%%%%%%%%%proposition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 122 | \newcounter{prop}[section]\setcounter{prop}{0} 123 | \renewcommand{\theprop}{\arabic{chapter}.\arabic{section}.\arabic{prop}} 124 | \newenvironment{proposition}[2][]{% 125 | \refstepcounter{prop}% 126 | \ifstrempty{#1}% 127 | {\mdfsetup{% 128 | frametitle={% 129 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 130 | \node[anchor=east,rectangle,fill=green!25] 131 | {\strut Proposition~\theprop};}} 132 | }% 133 | {\mdfsetup{% 134 | frametitle={% 135 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 136 | \node[anchor=east,rectangle,fill=green!25] 137 | {\strut Proposition~\theprop:~#1};}}% 138 | }% 139 | \mdfsetup{innertopmargin=10pt,linecolor=green!25,% 140 | linewidth=2pt,topline=true,% 141 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 142 | } 143 | \begin{mdframed}[]\relax% 144 | \label{#2}}{\end{mdframed}} 145 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 146 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 147 | \expandafter\let\expandafter\oldproof\csname\string\proof\endcsname 148 | \let\oldendproof\endproof 149 | \renewenvironment{proof}[1][\proofname]{% 150 | \oldproof[\bf \scshape \large #1]% 151 | }{\oldendproof} 152 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 153 | 154 | 155 | %%%%%%%%%%%%%%%% 156 | %\expandafter\let\expandafter\oldproof\csname\string\proof\endcsname 157 | %\let\oldendproof\endproof 158 | %\renewenvironment{proof}[1][\proofname]{% 159 | % \oldproof[\bf \scshape \large #1]% 160 | %}{\oldendproof} 161 | %%%%%%%%%%%%%%%% 162 | 163 | 164 | 165 | -------------------------------------------------------------------------------- /MTH201: Differential-Geometry/color-env.sty: -------------------------------------------------------------------------------- 1 | \ProvidesPackage{color-env}[2020/10/31 v1.0 color-env] 2 | 3 | %%%%%%%%%%%%%%%% 4 | % Requirements % 5 | %%%%%%%%%%%%%%%% 6 | \RequirePackage{xcolor} 7 | \RequirePackage{graphicx} 8 | \RequirePackage[framemethod=TikZ]{mdframed} %% for colored framed boxes 9 | \RequirePackage{framed} %% for problem 10 | \RequirePackage{amsthm} %% for theorem environment 11 | %%%%%%%%%%%%%%%%%% 12 | % Environments % 13 | %%%%%%%%%%%%%%%%%% 14 | 15 | %%%%%%%% Problem %%%%%%%%%%%%% 16 | \colorlet{shadecolor}{orange!12} 17 | \newtheorem{prob}{Problem} 18 | \newenvironment{problem} 19 | {\begin{shaded}\begin{prob}} 20 | {\end{prob}\end{shaded}} 21 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 22 | 23 | %%%%%%%%%%%%%%%% THEOREM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 24 | \newcounter{theo}[section] \setcounter{theo}{0} %% counter 25 | \renewcommand{\thetheo}{\arabic{chapter}.\arabic{section}.\arabic{theo}} 26 | \newenvironment{theorem}[2][]{% 27 | \refstepcounter{theo}% 28 | \ifstrempty{#1}% 29 | {\mdfsetup{% 30 | frametitle={% 31 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 32 | \node[anchor=east,rectangle,fill=blue!20] 33 | {\strut Theorem~\thetheo};}} 34 | }% 35 | {\mdfsetup{% 36 | frametitle={% 37 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 38 | \node[anchor=east,rectangle,fill=blue!20] 39 | {\strut Theorem~\thetheo:~#1};}}% 40 | }% 41 | \mdfsetup{innertopmargin=10pt,linecolor=blue!20,% 42 | linewidth=2pt,topline=true,% 43 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 44 | } 45 | \begin{mdframed}[]\relax% 46 | \label{#2}}{\end{mdframed}} 47 | %%%%%%%%%%%%%%% For lemma %%%%%%%%%%%%%%% 48 | \newcounter{lem}[section]\setcounter{lem}{0} 49 | \renewcommand{\thelem}{\arabic{chapter}.\arabic{section}.\arabic{lem}} 50 | \newenvironment{lemma}[2][]{% 51 | \refstepcounter{lem}% 52 | \ifstrempty{#1}% 53 | {\mdfsetup{% 54 | frametitle={% 55 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 56 | \node[anchor=east,rectangle,fill=orange!20] 57 | {\strut Lemma~\thelem};}} 58 | }% 59 | {\mdfsetup{% 60 | frametitle={% 61 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 62 | \node[anchor=east,rectangle,fill=orange!20] 63 | {\strut Lemma~\thelem:~#1};}}% 64 | }% 65 | \mdfsetup{innertopmargin=10pt,linecolor=green!20,% 66 | linewidth=2pt,topline=true,% 67 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 68 | } 69 | \begin{mdframed}[]\relax% 70 | \label{#2}}{\end{mdframed}} 71 | 72 | %%%%%%%%%%%%% definition %%%%%%%%%%%%%%% 73 | \newcounter{def}[subsection]\setcounter{def}{0} 74 | \renewcommand{\thedef}{\arabic{chapter}.\arabic{section}.\arabic{def}} 75 | \newenvironment{definition}[2][]{% 76 | \refstepcounter{def}% 77 | \ifstrempty{#1}% 78 | {\mdfsetup{% 79 | frametitle={% 80 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 81 | \node[anchor=east,rectangle,fill=red!25] 82 | {\strut Definition~\thedef};}} 83 | }% 84 | {\mdfsetup{% 85 | frametitle={% 86 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 87 | \node[anchor=east,rectangle,fill=red!25] 88 | {\strut Definition~\thedef:~#1};}}% 89 | }% 90 | \mdfsetup{innertopmargin=10pt,linecolor=red!25,% 91 | linewidth=2pt,topline=true,% 92 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 93 | } 94 | \begin{mdframed}[]\relax% 95 | \label{#2}}{\end{mdframed}} 96 | %%%%%%%%%%%%%% corollary %%%%%%%%%%%%%%%%%% 97 | \newcounter{cor}[section]\setcounter{cor}{0} 98 | \renewcommand{\thecor}{\arabic{chapter}.\arabic{section}.\arabic{cor}} 99 | \newenvironment{corollary}[2][]{% 100 | \refstepcounter{cor}% 101 | \ifstrempty{#1}% 102 | {\mdfsetup{% 103 | frametitle={% 104 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 105 | \node[anchor=east,rectangle,fill=yellow!20] 106 | {\strut Corollary~\thecor};}} 107 | }% 108 | {\mdfsetup{% 109 | frametitle={% 110 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 111 | \node[anchor=east,rectangle,fill=yellow!20] 112 | {\strut Corollary~\thecor:~#1};}}% 113 | }% 114 | \mdfsetup{innertopmargin=10pt,linecolor=yellow!20,% 115 | linewidth=2pt,topline=true,% 116 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 117 | } 118 | \begin{mdframed}[]\relax% 119 | \label{#2}}{\end{mdframed}} 120 | 121 | %%%%%%%%%%%%%proposition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 122 | \newcounter{prop}[section]\setcounter{prop}{0} 123 | \renewcommand{\theprop}{\arabic{chapter}.\arabic{section}.\arabic{prop}} 124 | \newenvironment{proposition}[2][]{% 125 | \refstepcounter{prop}% 126 | \ifstrempty{#1}% 127 | {\mdfsetup{% 128 | frametitle={% 129 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 130 | \node[anchor=east,rectangle,fill=green!25] 131 | {\strut Proposition~\theprop};}} 132 | }% 133 | {\mdfsetup{% 134 | frametitle={% 135 | \tikz[baseline=(current bounding box.east),outer sep=0pt] 136 | \node[anchor=east,rectangle,fill=green!25] 137 | {\strut Proposition~\theprop:~#1};}}% 138 | }% 139 | \mdfsetup{innertopmargin=10pt,linecolor=green!25,% 140 | linewidth=2pt,topline=true,% 141 | frametitleaboveskip=\dimexpr-\ht\strutbox\relax 142 | } 143 | \begin{mdframed}[]\relax% 144 | \label{#2}}{\end{mdframed}} 145 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 146 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 147 | \expandafter\let\expandafter\oldproof\csname\string\proof\endcsname 148 | \let\oldendproof\endproof 149 | \renewenvironment{proof}[1][\proofname]{% 150 | \oldproof[\bf \scshape \large #1]% 151 | }{\oldendproof} 152 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 153 | 154 | 155 | %%%%%%%%%%%%%%%% 156 | %\expandafter\let\expandafter\oldproof\csname\string\proof\endcsname 157 | %\let\oldendproof\endproof 158 | %\renewenvironment{proof}[1][\proofname]{% 159 | % \oldproof[\bf \scshape \large #1]% 160 | %}{\oldendproof} 161 | %%%%%%%%%%%%%%%% 162 | 163 | 164 | 165 | -------------------------------------------------------------------------------- /PHY102: Electromagnetism/PHY102_-_Electromagnetism.tex: -------------------------------------------------------------------------------- 1 | \documentclass[a4paper]{article} 2 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 | \usepackage[english]{babel} 4 | \usepackage{amsthm,amssymb,amsmath} 5 | \usepackage{enumerate} 6 | \usepackage{semantic} 7 | \usepackage{graphicx} %%% \to incluse figures 8 | \usepackage{commath} %%% for \abs command 9 | \usepackage{subfig} %%%% for two figures side by side 10 | \usepackage{todonotes} 11 | \numberwithin{equation}{subsection} %%% to number within section 12 | \usepackage{hyperref} %%% for url 13 | \hypersetup{ 14 | colorlinks=true, 15 | linkcolor=blue, 16 | filecolor=magenta, 17 | urlcolor=cyan, 18 | } 19 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20 | \begin{document} 21 | \title{\bf Electromagnetism\\ 22 | \large PHY102 } 23 | \author{} 24 | \date{\today} 25 | \maketitle 26 | \begin{center} 27 | \subsection*{\S Fundamental Law of Vector analysis} 28 | \end{center} 29 | \paragraph{1.Fundamental Theorem of Gradients} 30 | \begin{equation} 31 | dT = \vec{\nabla} T \cdot d\vec{l 32 | }\end{equation} 33 | %%%%%%%%%%%%%% 34 | \paragraph{2.Gauss Theorem} 35 | \begin{equation} 36 | \int_{\nu} (\nabla \cdot \vec{v})d\tau = \oint_{S} \vec{v}\cdot d \vec{s} 37 | \end{equation} 38 | %%%%%%%%%%%%%% 39 | \paragraph{3.Stoke's Theorem} 40 | \begin{equation} 41 | \int_{S} (\nabla \times \vec{v})\cdot d\vec{s} = \oint_{P} \vec{v}\cdot d \vec{l} 42 | \end{equation} 43 | %%%%%%%%%%%%%% 44 | \paragraph{4.Gradient Operator} 45 | \begin{equation} 46 | \vec{\nabla} = (\frac{d}{dx}\hat{i}+\frac{d}{dy}\hat{j}+\frac{d}{dz}\hat{k}) 47 | \end{equation} 48 | \textbf{Some Important realtions and Identities (let V \& U be a vector field and $\lambda$ be scalar field):} 49 | 50 | \begin{gather*} 51 | \vec{\nabla}\cdot (\vec{\nabla}\times \vec{V}) = 0 \quad \text{divergence of curl is zero} \\ 52 | \vec{\nabla}\times (\vec{\nabla} \lambda) = 0 \quad \text{curl of gradient is zero }\\ 53 | \vec{\nabla}\cdot (\vec{U}\times \vec{V}) = \vec{V}\cdot(\vec{\nabla} \times \vec{U}) - \vec{U}\cdot(\vec{\nabla} \times \vec{V})\\ 54 | \vec{\nabla}\times (\vec{U}\times \vec{V}) = (\vec{V}\cdot\vec{\nabla}) \vec{U} - (\vec{U}\cdot\vec{\nabla}) \vec{V} + (\vec{\nabla}\cdot \vec{V}) \vec{U} - (\vec{\nabla}\cdot \vec{U}) \vec{V} \\ 55 | \vec{\nabla} \times (f(r)\hat{r}) = 0 \quad (\text{curl of central force or field is zero})\\ 56 | \vec{\nabla} \times (\vec{\nabla}\times \vec{V}) = \vec{\nabla} \cdot (\vec{\nabla} \cdot \vec{V}) - \nabla^2 \vec{V}\\ 57 | \vec{\nabla} \times (\lambda \vec{V}) = \lambda \vec{\nabla} \times \vec{A} + \vec{\nabla}\lambda \times \vec{A} 58 | \end{gather*} 59 | 60 | \textbf{How are divergences and curls defined?} 61 | \begin{description} 62 | \item[Divergence] The divergence of a vector field $F(x)$ at a point $x_0$ is defined as the limit of the ratio of the surface integral of $F$ out of the surface of a closed volume V enclosing $x_0$ to the volume of V, as V shrinks to zero: 63 | 64 | 65 | $$ \mathrm{div}\ \vec{F} |_{x_0} = \lim\limits_{V \to 0} \frac{1}{|V|} \int_{S(V)} \vec{F} \cdot\hat{n} dS$$ 66 | where $|V|$ is the volume of $V$, $S(V)$ is the boundary of $V$, and $\hat{n}$ is the outward unit normal to that surface.Since this definition is coordinate-free, it shows that the divergence is the same in any coordinate system. 67 | 68 | \item[Curl] 69 | The curl of a vector field F,at a point is defined in terms of its projection onto various lines through the point. If $\hat{n}$ is any unit vector, the projection of the curl of F onto $\hat{n}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{n}$ divided by the area enclosed, as the path of integration is contracted around the point. 70 | 71 | $$ \text{curl}\ \vec{F} \cdot\hat{n} = \lim\limits_{A \to 0} \left(\frac{1}{|A|} \oint_C \vec{F} \cdot d\vec{r}\right)$$ 72 | 73 | where $\oint_C \vec{F} \cdot d\vec{r}$ is a line integral along the boundary of the area in question, and $|A|$ is the magnitude of the area. This equation defines the projection of the curl of F onto $\hat{n}$, where $\hat{n}$ is the normal vector to the surface bounded by C; and C is defined via the right-hand rule 74 | \end{description} 75 | 76 | 77 | 78 | \paragraph{}%%%%%% for extra space 79 | \paragraph{} 80 | \begin{center} 81 | \subsection*{\S Elektrodynamisch} 82 | \end{center} 83 | %%%%%%%%%%%%%% 84 | \paragraph{1.Coulombs Law} 85 | \begin{equation} 86 | \vec{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_1}{r^2} \hat{r} 87 | \end{equation} 88 | %%%%%%%%%%%%%% 89 | \paragraph{2.Electric Field} 90 | \begin{equation} 91 | \vec{E( r)} = \frac{1}{4\pi\epsilon_0} \int_{\nu} \frac{\rho(r')}{r^2} \hat{r}\ d\tau ' 92 | \end{equation} 93 | \paragraph{3.Electric Potential} 94 | \begin{equation} 95 | \phi = - \int_{a}^{b} \vec{E}\cdot d\vec{l} 96 | \end{equation} 97 | \begin{center} 98 | \textbf{Or} 99 | \end{center} 100 | \begin{equation*} 101 | \vec{E} = -\nabla \phi 102 | \end{equation*} 103 | 104 | \paragraph{5.Poission's Equation and Laplace's Equation} 105 | \begin{equation} 106 | \label{poission} 107 | \nabla^2 \phi = -\rho/\epsilon_0\qquad \text{Poission's Equation} 108 | \end{equation} 109 | \begin{equation} 110 | \label{poission} 111 | \nabla^2 \phi = 0\qquad \text{Laplace's Equation} 112 | \end{equation} 113 | \paragraph{6.Lorentz Forces} 114 | \begin{equation} 115 | \vec{F} = q(\vec{E}+\vec{v}\times \vec{B}) 116 | \end{equation} 117 | \paragraph{7.Biot-Savarts Law} 118 | \begin{equation} 119 | \vec{B}(r) = \frac{\mu _0}{4\pi} \frac{q[\vec{v}\times \vec{r}]}{r^3} 120 | \end{equation} 121 | \paragraph{}%%%%%% for extra space 122 | \paragraph{}\begin{center} 123 | \subsection*{\S Electric Current} 124 | \end{center} 125 | 126 | \paragraph{1.Current density and its relations} 127 | \begin{equation} 128 | \vec{j} = \sigma \vec{E} \quad\text{For steady currents} 129 | \end{equation} 130 | \begin{equation} 131 | i = \int_{S} \vec{j}\cdot d\vec{a} 132 | \end{equation} 133 | \begin{equation} 134 | \vec{j} = \rho _e \bar{u_e} 135 | \end{equation} 136 | \paragraph{2.Steady Currents and Equation of contuinity} 137 | \begin{equation} 138 | \vec{\nabla}\cdot \vec{j}(\vec{r},t) = - \frac{\partial \rho(\vec{r},t)}{\partial t} 139 | \end{equation} 140 | And in case of steady currents 141 | \begin{equation} 142 | \vec{\nabla}\cdot \vec{j} =0 143 | \end{equation} 144 | \paragraph{3. Force on a current carrying conductor in a magnetic field} 145 | \begin{equation} 146 | \vec{F} = i \int d\vec{l}\times \vec{B} 147 | \end{equation} 148 | \paragraph{4. Force on a current carrying loop in a magnetic field.} 149 | \begin{equation} 150 | \vec{F} = \vec{\nabla} (\vec{\mu} \cdot \vec{B}) 151 | \end{equation} 152 | 153 | \paragraph{Magnetic field due to any arbitrary moving charge distribution} 154 | 155 | \begin{equation} 156 | \begin{aligned} 157 | B(\vec{r}) = \frac{\mu_0}{4\pi} \int \rho' {[\vec{v}\times \vec{r}] \over {r}^3} dV'\\ 158 | \text{Or} \\ 159 | B(\vec{r}) = \frac{\mu_0}{4\pi} \int {[\vec{j'}\times \vec{r}] \over {r}^3} dV'\\ 160 | \end{aligned} 161 | \end{equation} 162 | 163 | 164 | where primes variables denote the position of charge distribution and the un-primed variables denote the position of the point where field is tu be calculated. 165 | 166 | 167 | \paragraph{4.Maxwell's equation} 168 | \begin{gather} 169 | \vec{\nabla} \cdot \vec{E}(\vec{r},t) = {\rho(\vec{r},t) \over \epsilon_0} \\ 170 | \vec{\nabla} \times \vec{E }(\vec{r},t) + \frac{\partial \vec{B}(\vec{r},t) }{\partial t} = 0 \\ 171 | \vec{\nabla}\cdot \vec{B} (\vec{r},t)= 0\\ 172 | \vec{\nabla} \times \vec{B}(\vec{r},t) - \frac{1}{c^2}\frac{\partial\vec{E}(\vec{r},t)}{\partial t} =\mu_0 \vec{j} (\vec{r},t) 173 | \end{gather} 174 | The above equations are valid for any space-time configuartion. 175 | 176 | 177 | In the above case the most important thing to notice is that there are 8 equations to solve(i.e. two of them are scalar and two are vector equations with 3 components in each.). But the unknowns are 6 (i.e. the three components of Electric and magnetic field each). So, mathematically the solutions are infinite. But is it Ture?.Lets see. 178 | 179 | If we compare the Gass theorem for Electric fields and the Ampere $\tilde{}$ Maxwell's equation for magnetic fields, we can see that the quantities $\vec{j}$ and $\rho$ are related through the equation of contuinity. Also, observe that the gauss law for magnetism and faraday's law are independent of the source and are homogeneous equations. And the othere two are dependent on the source (and non-homogeneous too). 180 | 181 | You may ask a questions. Why specifying the EM Fields as divergences and curls?. What is so special about them? Why are they alone enough or do we need any other realtion for specifying fields.The answer lies in \href{https://en.wikipedia.org/wiki/Helmholtz_decomposition}{Helmholtz Theorem}. The curl free part of the vector (i.e $\vec{\nabla} \phi$) and the divergence free part (i.e. $ \vec{\nabla} \times \vec{A}$ ). Physically, we can interpret it as the div tells us how much 182 | a vector is pointing in specific direction \& curl says how much the vector points in normal direction.\\ Coming back to the Maxwell'ss equations. The gauss law for magnetic fields implies: 183 | 184 | $$ \underbrace{\vec{\nabla}\cdot \vec{B} (\vec{r},t)= 0}_{\text{No monoploes exist}} \Rightarrow \vec{B} = \vec{\nabla}\times \vec{A}$$ 185 | We call $\vec{A}$ as a Vector potential.\\ 186 | If we substitute the above relation in eq(0.0.21) we get to the realtion: 187 | 188 | 189 | $$ \vec{\nabla} \times \left(\vec{E} + \frac{\partial \vec{A}}{\partial t}\right) = 0 \quad \rightarrow \text{it's curl free (Why?)}$$ 190 | 191 | $$ \Rightarrow \vec{E} + \frac{\partial \vec{A}}{\partial t} = - \vec{\nabla} \phi \quad \{ \text{$\phi$ has nothing to do with electrostatic potential} \} $$ 192 | 193 | The minus sign above is purely a convention as to make the equation consistent with electrostatic. We call $\phi$ as scalar potential. 194 | 195 | At the end we get. 196 | 197 | \begin{gather} 198 | \vec{B} = \vec{\nabla} \times \vec{A} \quad \& \quad \vec{E} = - \frac{\partial \vec{A}}{\partial t} - \vec{\nabla} \phi 199 | \label{potentials} 200 | \end{gather} 201 | 202 | Now we need not to solve the 8 equations. But the problem has been reduced to 4 equations (i.e. $\vec{A}$ with three components and $\phi$ scalar field). So, all electric and magnetic fields can be found using four quantities. 203 | 204 | Now, let's see what happens when we substitute the results we got in eq \ref{potentials} in the ``source dependent" ``non-homogeneous" Maxwell equations. 205 | 206 | $$ 207 | \Rightarrow \vec{\nabla} \cdot \vec{E} =\vec{\nabla} \cdot \left(- \frac{\partial \vec{A}}{\partial t} - \vec{\nabla} \phi\right) = {\rho \over \epsilon_0} 208 | $$ 209 | \begin{equation} 210 | \Rightarrow \frac{\partial( \vec{\nabla}\cdot\vec{A})}{\partial t} + \nabla^2 \phi = -{\rho \over \epsilon_0} 211 | \label{laplacian} 212 | \end{equation} 213 | \& 214 | $$ 215 | \Rightarrow \vec{\nabla} \times \vec{B} - \frac{1}{c^2}\frac{\partial\vec{E}}{\partial t} =\mu_0 \vec{j} 216 | $$ 217 | $$ 218 | \Rightarrow \vec{\nabla} \times (\vec{\nabla} \times \vec{A}) - \frac{1}{c^2}( - \vec{\nabla} \phi - \frac{\partial\vec{A}}{\partial t}) =\mu_0 \vec{j} 219 | $$ 220 | It can further be simplified and can be written as: 221 | 222 | \begin{equation} 223 | \vec{\nabla}\left( \vec{\nabla} \cdot \vec{A} + \frac{1}{c^2} \frac{\partial\phi}{\partial t}\right) + {\frac{1}{c^2} \frac{\partial^2\vec{A}}{\partial^2 t} - \nabla^2 \vec{A}}\footnote{$ \text{the operator} (\frac{1}{c^2} \frac{\partial^2}{\partial^2t} - \nabla^2) \text{is also written as} \ \Box\ \text{d'alembert operator} $} = \mu_0 \vec{j}\label{ocillation} 224 | \end{equation}\todo[color= white]{Remember: 2\textsuperscript{nd} derivative means the curvature of the curve} 225 | 226 | Now, you can't solve the two equation that we got above ,because $\vec{A}$ and $\phi$ are inter-dependent. 227 | 228 | Observe that if $\vec{\nabla}\cdot \vec{A} = 0$ (called Coulomb's Gauge) then the Eq \ref{laplacian} becomes a poisson's equation. Or if we put $(\vec{\nabla} \cdot \vec{A} + \frac{1}{c^2} \frac{\partial\phi}{\partial t}) = 0$ (called Lorentz Gauge) then the equation \ref{ocillation} looks like eq of simple harmonic motion in 3D. And why can we do that? Simply because if suppose we find a field $\vec{A}$ which satisfies the relation $\vec{B} = \vec{\nabla} \times \vec{A}$ but is unable to satisfy the Coulomb Gauge, the we can simply add gradient of a scalar field $\chi$ to it such that $\vec{A'} = \vec{A} + \vec{\nabla} \chi$ and $\vec{\nabla} \cdot \vec{A} = - {\nabla}^2 \chi$. 229 | Since, $\vec{A}$ and $\phi$ are auxilary quantities \& are not unique. If we change one i.e $\vec{A'} = \vec{A} + \vec{\nabla} \chi$, then simultaneously we have to change $ \phi' = \phi - \frac{\partial \chi}{\partial t}$. To compensate for the change in $\vec{E}$. It is called Gauge invariance.(Which one is a better choice?). 230 | 231 | Now coming back to the equations of fields in term of vector and scalar potential. For $\phi$ general solution (for coulomb gauge) can be written for the free space (i.e $\phi \to 0$ as $r \to 0$). 232 | 233 | $$ \phi (\vec{r},t) = \int \underbrace{\frac{\rho(\vec{r},t) d V'}{4\pi \epsilon_0 |\vec{r}-\vec{r_0}|}}_ {\text{notice the t dependence.} \atop \text{ It's not electrostatics}} $$ 234 | 235 | But the $\phi(\vec{r},t)$, depends on $\rho(\vec{r},t)$. So, if we change the $\rho$ at some point far away; the $\phi$ should immediately change at every point \& 236 | it violates the S.T.R. But no $\vec{A}$ is also there which counters the violation of S.T.R. 237 | 238 | \paragraph{Gauge Transformations} These are the possibility of changing the potential without changing the EM fields. 239 | 240 | Fundamentally, electric fields and the magnetic fields are A and $\phi$. 241 | \end{document} -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | Attribution 4.0 International 2 | 3 | ======================================================================= 4 | 5 | Creative Commons Corporation ("Creative Commons") is not a law firm and 6 | does not provide legal services or legal advice. Distribution of 7 | Creative Commons public licenses does not create a lawyer-client or 8 | other relationship. Creative Commons makes its licenses and related 9 | information available on an "as-is" basis. 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For 392 | the avoidance of doubt, this paragraph does not form part of the public 393 | licenses. 394 | 395 | Creative Commons may be contacted at creativecommons.org. 396 | 397 | -------------------------------------------------------------------------------- /PHY202: Statistical-Mechanics and Thermodynamics/PHY202_-_Statistical Mechanics.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = pdflatex 2 | \documentclass{tufte-handout} 3 | 4 | \title{\centering PHY202: Statistical Physics} 5 | \author{Aditya Dev} 6 | 7 | 8 | \date{\today} % without \date command, current date is supplied 9 | 10 | %\geometry{showframe} % display margins for debugging page layout 11 | 12 | \usepackage{graphicx} % allow embedded images 13 | \setkeys{Gin}{width=\linewidth,totalheight=\textheight,keepaspectratio} 14 | \usepackage{amsmath} % extended mathematics 15 | \usepackage{booktabs} % book-quality tables 16 | \usepackage{units} % non-stacked fractions and better unit spacing 17 | \usepackage{multicol} % multiple column layout facilities 18 | \usepackage{lipsum} % filler text 19 | \usepackage{fancyvrb} % extended verbatim environments 20 | \fvset{fontsize=\normalsize}% default font size for fancy-verbatim environments 21 | 22 | % Standardize command font styles and environments 23 | \newcommand{\doccmd}[1]{\texttt{\textbackslash#1}}% command name -- adds backslash automatically 24 | \newcommand{\docopt}[1]{\ensuremath{\langle}\textrm{\textit{#1}}\ensuremath{\rangle}}% optional command argument 25 | \newcommand{\docarg}[1]{\textrm{\textit{#1}}}% (required) command argument 26 | \newcommand{\docenv}[1]{\textsf{#1}}% environment name 27 | \newcommand{\docpkg}[1]{\texttt{#1}}% package name 28 | \newcommand{\doccls}[1]{\texttt{#1}}% document class name 29 | \newcommand{\docclsopt}[1]{\texttt{#1}}% document class option name 30 | \newenvironment{docspec}{\begin{quote}\noindent}{\end{quote}}% command specification environment 31 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 32 | % add numbers to chapters, sections, subsections 33 | \setcounter{secnumdepth}{2} 34 | \usepackage{xcolor} 35 | \definecolor{g1}{HTML}{077358} 36 | \definecolor{g2}{HTML}{00b096} 37 | % chapter format %(if you use tufte-book class) 38 | %\titleformat{\chapter}% 39 | %{\huge\rmfamily\itshape\color{red}}% format applied to label+text 40 | %{\llap{\colorbox{red}{\parbox{1.5cm}{\hfill\itshape\huge\color{white}\thechapter}}}}% label 41 | %{2pt}% horizontal separation between label and title body 42 | %{}% before the title body 43 | %[]% after the title body 44 | 45 | % section format 46 | \titleformat{\section}% 47 | {\normalfont\Large\itshape\color{g1}}% format applied to label+text 48 | {\llap{\colorbox{g1}{\parbox{1.5cm}{\hfill\color{white}\thesection}}}}% label 49 | {1em}% horizontal separation between label and title body 50 | {}% before the title body 51 | []% after the title body 52 | 53 | % subsection format 54 | \titleformat{\subsection}% 55 | {\normalfont\large\itshape\color{g2}}% format applied to label+text 56 | {\llap{\colorbox{g2}{\parbox{1.5cm}{\hfill\color{white}\thesubsection}}}}% label 57 | {1em}% horizontal separation between label and title body 58 | {}% before the title body 59 | []% after the title body 60 | 61 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 62 | \usepackage{color-tufte} 63 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 64 | 65 | 66 | \begin{document} 67 | 68 | \maketitle% this prints the handout title, author, and date 69 | 70 | \begin{abstract} 71 | \noindent 72 | I'm made these notes after following the lectures by Prof. Susskind 73 | \end{abstract} 74 | 75 | %\printclassoptions 76 | 77 | 78 | \section{Entropy}\label{sec:page-layout} 79 | 80 | We define Entropy as follows: 81 | 82 | \begin{definition}[Entropy]\label{def:entropy} 83 | Suppose a system can take values from set \(\mathcal{S} = \{x_1, x_2, x_3, \ldots x_n\}\), 84 | with probability of 85 | \(x_i\) being \(\mu(x_i)\). Then the (\textit{Boltzmann}) entropy \(S\) is defined as 86 | \[S = -k_b \sum _{i} \mu(x_i) \log(\mu(x_i))\] 87 | \end{definition} 88 | 89 | Entropy can be thought of as the amount of ignorance/uncertainity that you have. It's obvious that if 90 | probability is 1 or 0 the you are \(100 \%\) sure that a particular result will be observed or not 91 | observed. 92 | 93 | \begin{assume} 94 | If \(S_a\) and \(S_b\) be respective entropies for system A and B. Then 95 | \[S_{\mathrm{net}} = S_a + S_b\] 96 | \label{assum:entropy} 97 | \end{assume} 98 | 99 | Using above assumption, if we have \(N\) identical systems then the net entropy of the system is 100 | given as \(S = -Nk_b \sum \mu_i\log(\mu_i)\) 101 | 102 | \subsection{*On existence of Entropy} 103 | 104 | Does entropy exists? Or is it just a mathematial formalism? 105 | 106 | I'll try to answer it it in my own way. But for that I need to make an assumption, and 107 | it's a fairly good assumption. Let me give an example to support my argument, 108 | suppose we are given an experiment of tossing a fair coin, let say \(n\) number 109 | of times, and we are asked to count probabilities of number of getting a head or tails. 110 | 111 | Since, the above experiment follows a Binomial distribution, we know that most number of 112 | times we are getting equal number of heads and tails (\textit{because number of arrangement given 113 | equal number of heads and tails is the most}). So, the assumption that we'll make is: 114 | \begin{assume} 115 | Given a constraint on a system, in equilibrium, the system will exist in the state with maximum 116 | number of configurations (\textit{or microstates}). And state is consistent with the constraint. 117 | \end{assume} 118 | 119 | We'll see in Microcanonical Ensemble, the constrain on the system is energy \(E\), volume \(V\) and \(N\) the 120 | number of particles. 121 | 122 | Let \(\Omega(E, V, N)\) denote the possible number of microstates of the system. Let's sub-divide this 123 | system into two non-interacting parts, call \(A\) and \(B\). So, we have: 124 | \[\begin{gathered} 125 | \Omega(E, V, N) = \Omega_A(E_A, V_A, N_A)\Omega_B(E_B, V_B, N_B)\\ 126 | E = E_A + E_B\\ 127 | V = V_A + V_B\\ 128 | N = N_A + N_B 129 | \end{gathered}\] 130 | 131 | Since, by assumption 1.2 we need to maximize the \(\Omega\). So, 132 | \[\begin{gathered} 133 | \dfrac{\partial \Omega}{\partial E} = 0\\ 134 | \dfrac{\partial (\Omega_A \Omega_B)}{\partial E} = 0 135 | \end{gathered}\] 136 | 137 | Since, \(E\) is constant, we can formulate the whole argument in terms of either \(E_A\) or \(E_B\). 138 | 139 | \[\begin{gathered} 140 | \dfrac{\partial (\Omega_A(E_A) \Omega_B(E - E_A))}{\partial E_A} = 0\\ 141 | \dfrac{\partial (\Omega_A(E_A))}{\partial E_A}\Omega_B(E - E_A) - 142 | \dfrac{\partial (\Omega_B(E - E_A))}{\partial E_A}\Omega_A(E_A) = 0\\ 143 | \frac{1}{\Omega_A(E_A)}\dfrac{\partial (\Omega_A(E_A))}{\partial E_A} = 144 | \frac{1}{\Omega_B(E - E_A)}\dfrac{\partial (\Omega_B(E - E_A))}{\partial E_A}\\ 145 | \dfrac{\partial \log(\Omega_A(E_A))}{\partial E_A} = \dfrac{\partial 146 | \log(\Omega_B(E - E_A))}{\partial E_A} 147 | \end{gathered}\] 148 | 149 | We define the quantity \(k_b\log(\Omega)\) \footnote{the factor \(k_b\) has something to do 150 | with units}and ``Entropy'' and denote it be \(S\). 151 | \[S = k_b\log(\Omega)\] 152 | 153 | Hence, the entropy comes before any of the physical quantities like pressure, temperature etc. 154 | We'll see in the next section, that how the two definition of Entropy are related. 155 | 156 | \section{Microcanonical Ensemble} 157 | 158 | 159 | 160 | In \textit{Microcanonical Ensemble} \footnote{\textit{An \textbf{ensemble} is a large number of points in the phase space that can be described 161 | by a density function 162 | \(\mu (x, p)\)}} we consider an isolated system with \(N\) 163 | particles and energy \(E\) in a volume 164 | \(V\). By definition, such a system exchanges neither particles nor energy with the surroundings. 165 | 166 | Let a system consist of {\(N\) particles}, for which each particle can have \(M\) configurations 167 | \footnote{Configurations can be the rotation angle about some axis or spin being up or down etc}. 168 | Also, if a particle is in \(i^{th}\) state then 169 | let the energy associated with that particle be \(\epsilon _i\). 170 | 171 | For this system the total energy 172 | \(E = \sum n_i \epsilon_i\), where \(n_i\) denotes the number of particles in state (\textit{ 173 | or configuration i 174 | }). Hence the constrain equations on the system are: 175 | \[\begin{gathered} 176 | E = \sum n_i \epsilon_i\\ 177 | \sum n_i = N 178 | \end{gathered}\] 179 | 180 | In the thermodynamic limit of Statistical mechanics we usually assume \(N\) and \(V\) to be very large 181 | (i.e \(N, V \to \infty\)) 182 | such that \(\rho = \frac{N}{V}\) is constant. In such case, we can assume that the number of particle 183 | in each state increases proportional to \(N\). Let probability 184 | of finding particle in \(i^{th}\) configuration being \(\mu_i\) and by the definition of probability 185 | we have \(\mu_i = \lim_{N \to \infty} \frac{n_i}{N}\) or \(n_i = \mu_i N\). 186 | 187 | Now, we can reformulate the above constrain equations in terms of probability. And we have 188 | \[\begin{gathered} 189 | \sum \mu_i = 1 \Longleftrightarrow \sum n_i = N\tag*{1}\\ 190 | \sum_i N\mu_i \epsilon_i = E \Longleftrightarrow \sum_i \mu_i \epsilon_i = \epsilon \\ 191 | \end{gathered}\] 192 | 193 | We define \(\epsilon = \frac{E}{N} = \langle E \rangle\) as average energy per particle. 194 | 195 | Let \(\Omega\) be the number of possible microstates of the system 196 | \footnote{in our case it's the possible arrangements N particles, with exactly \(n_i\) number of particles 197 | in \(i^{th}\)configuration}. 198 | If you know some basics combinatorics, then for our case \(\Omega\) is give as 199 | \[\begin{gathered} 200 | \Omega = \frac{N!}{\Pi n_i !}\\ 201 | \log(\Omega) = \log(N!) - \sum \log(n_i!) 202 | \end{gathered}\] 203 | Now we need to introduce \textit{Stirling's Approximation}, it says for large \(N\) 204 | \[\begin{gathered} 205 | N! \sim e^{-N} N^N\\ 206 | log(N!) \sim N \log(N) - N 207 | \end{gathered}\] 208 | 209 | Therefore we have, 210 | \[\begin{gathered} 211 | \log(\Omega) = N \log(N) - N - \sum n_i \log(n_i) + \sum n_i \\ 212 | \log(\Omega) = N \log(N) - N - \sum n_i \log(n_i) + N \\ 213 | \log(\Omega) = N \log(N) - \sum N \mu _i\log(N) - \sum N \mu _i\log(\mu_i)\\ 214 | \log(\Omega) = N \log(N) - N \log(N) \sum \mu _i - \sum N \mu_i \log(\mu_i) \\ 215 | \log(\Omega) = -N\sum \mu_i \log(\mu_i) 216 | \end{gathered}\] 217 | 218 | Buy assumption 1.1 observe that \(\boxed{S = k_b\log(\Omega) = 219 | -N k_b \sum \mu_i\log(\mu_i)}\). Hence, entropy of the system can also be defined as 220 | \(S = k_b \log(\Omega)\)\footnote{\(\Omega\) is the total number of accessible 221 | microstates of the system}. Also, note that \(-k_b \sum \mu_i\log(\mu_i)\) is the entropy associated 222 | with each particle. 223 | 224 | \section{Boltzmann Distribution} 225 | \textit{In statistical mechanics and mathematics, a Boltzmann distribution 226 | (also called Gibbs distribution) is a probability distribution or probability 227 | measure that gives the probability that a system will be in a 228 | certain state as a function of that state's energy and the temperature of the system.} 229 | 230 | We'll try to derive the Boltzmann distribution from Microcanonical ensemble. The Boltzmann 231 | distribution is the distribution that maximizes the entropy \(S = -k_b 232 | \sum \mu_i\log(\mu_i)\) or equivalently \(S = -\sum \mu_i\log(\mu_i)\) 233 | 234 | We need to find the probability distribution of the system under the given constrains, such that 235 | the entropy is maximum Hence, we would use \href{https://en.wikipedia.org/wiki/Lagrange_multiplier}{Lagrange 236 | multiplier}. 237 | 238 | Let 239 | \[\begin{gathered} 240 | F(\{\mu_i\}) = - \sum \mu_i \log(\mu_i) + \alpha (1 - 241 | \sum \mu_i) + \beta (\epsilon - \sum \mu_i \epsilon_i) \\ 242 | = - \sum \mu_i \log(\mu_i) + \alpha G(\{\mu_i\}) + \beta G'(\{\mu_i\}) \tag*{2} 243 | \end{gathered}\] 244 | where \(G'(\{\mu_i\}) = 0 \ \& \ G(\{\mu_i\}) = 0\) are constrain equations\footnote{ 245 | there was no need for them but, it looks trippy, and that's how mathematicians do it. 246 | }. 247 | 248 | To maximize, differentiate (2) w.r.t \(\mu_i\) and equate to \(0\), we get 249 | \[\begin{gathered} 250 | \dfrac{d F}{d \mu_i}= - \log(\mu_i) - 1 - \alpha - \beta \epsilon_i =0 \\ 251 | \mu_i = e^{-1 - \alpha} e^{-\beta \epsilon}\\ 252 | \boxed{\mu_i(\vec{r}, \vec{p}) = \frac{e^{-\beta \epsilon_i(\vec{r}, \vec{p})}}{z}} 253 | \end{gathered}\] 254 | 255 | where \(z = e^{1 + \alpha}\). \(z\) is called the \textit{partition function}. \textbf{NOTICE, THE DEPENDENCE OF 256 | ENERGY ON PHASE SPACE} 257 | 258 | Substitute \(\mu_i\) in above equations and you will see that 259 | \[\begin{gathered} 260 | \sum \mu_i = 1 \Longleftrightarrow \boxed{z(\beta) = \sum e^{-\beta \epsilon_i}}\\ 261 | \sum \epsilon_i \mu_i = \epsilon \Longleftrightarrow \sum \epsilon_i 262 | e^{-\beta \epsilon_i} = z \epsilon 263 | \end{gathered}\] 264 | 265 | Also, observe that 266 | \[\begin{gathered} 267 | \dfrac{\partial z(\beta)}{\partial \beta} = - \sum \epsilon_i e^{-\beta \epsilon_i}= -z\epsilon\\ 268 | \boxed{\dfrac{\partial \log(z)}{\partial \beta} = - \epsilon} 269 | \end{gathered}\] 270 | 271 | Let's look at how does the Entropy looks like. We had 272 | \[\begin{gathered} 273 | S = -k_b \sum \mu_i \log(\mu_i)\\ 274 | = -\frac{k_b}{z} \sum \{ e^{-\beta \epsilon_i} \log(e^{-\beta \epsilon_i}) 275 | - e^{-\beta \epsilon_i} \log(z)\}\\ 276 | = \frac{k_b}{z} \sum \{ e^{-\beta \epsilon_i}{\beta \epsilon_i}) 277 | + e^{-\beta \epsilon_i} \log(z)\}\\ 278 | = \beta k_b \sum \frac{e^{-\beta \epsilon_i}}{z} \epsilon_i + k_b \log(z) 279 | \sum \frac{e^{-\beta \epsilon_i}}{z}\\ 280 | \boxed{S = k_b\beta \epsilon + k_b \log(z)} 281 | \end{gathered}\] 282 | 283 | We define (\textit{stastical definition of temperature}) temperature as 284 | \footnote{don't question why is that so, it is what it is} 285 | \[\dfrac{\partial S}{\partial E} = \frac{1}{T}\] 286 | So, 287 | 288 | \[\begin{gathered} 289 | \frac{1}{T} = \dfrac{\partial S}{\partial \epsilon} = k_b \beta\\ 290 | \beta = \frac{1}{k_b T} 291 | \end{gathered}\] 292 | 293 | So, finally everything unveils it self and we get 294 | \begin{main_equations} 295 | \[ 296 | \begin{gathered} 297 | \mu_i = \frac{e^{\frac{-\epsilon_i}{k_b T}}}{z}\\ 298 | z(T) = \sum_{i} e^{\frac{-\epsilon_i}{k_b T}}\\ 299 | \epsilon = -\frac{\partial \log(z)}{\partial \beta} = 300 | k_B T^2 \frac{\partial \log(z)}{\partial T}\\ 301 | S = \frac{\epsilon}{T} + k_b \log(z) 302 | \end{gathered} 303 | \] 304 | \end{main_equations} 305 | 306 | \subsection{Relation to thermodynamic variables} 307 | 308 | We know variance is defined as : 309 | \[\begin{gathered} 310 | \langle (\Delta X)^2 \rangle= \left\langle (X - \langle X\rangle)^2\right \rangle\\ 311 | = \langle X^2 \rangle - (\langle X \rangle)^2 312 | \end{gathered}\] 313 | 314 | We calculate: 315 | \[\begin{gathered} 316 | \langle E \rangle = \epsilon = -\frac{\partial \log(z)}{\partial \beta}\\ 317 | \langle E \rangle = \frac{1}{z}\sum_i e^{-\beta E_i} E_i ^2 \\ 318 | = \frac{1}{z} \frac{\partial^2 z}{\partial^2 \beta} 319 | \end{gathered}\] 320 | 321 | So, \(\langle (\Delta E)^2 \rangle\) is: 322 | \[\begin{gathered} 323 | \langle (\Delta E)^2 \rangle = \frac{1}{z} \frac{\partial^2 z}{\partial^2 \beta} 324 | - \left(-\frac{1}{z} \frac{\partial z}{\partial\beta}\right)^2\\ 325 | = \frac{\partial^2 \log(z)}{\partial^2 \beta} + \left(\frac{1}{z} 326 | \frac{\partial z}{\partial\beta}\right)^2 327 | -\left(\frac{1}{z} \frac{\partial z}{\partial\beta}\right)^2\\ 328 | = \frac{\partial^2 \log(z)}{\partial^2 \beta}\\ 329 | = - \dfrac{\partial \langle E \rangle}{\partial \beta} 330 | \end{gathered}\] 331 | or we can write it as 332 | \[\begin{gathered} 333 | \langle (\Delta E)^2 \rangle = - \dfrac{\partial \langle E \rangle}{\partial \beta}\\ 334 | = - \dfrac{\partial \langle E \rangle}{\partial T}\dfrac{\partial T}{\partial \beta}\\ 335 | = k_b T^2\dfrac{\partial \langle E \rangle}{\partial T}\\ 336 | = k_b T^2 C_v\\ 337 | \text{or}\\ 338 | \boxed{C_v = \frac{1}{k_b T^2} \langle (\Delta E)^2 \rangle} 339 | \end{gathered}\] 340 | where \(C_v\) is the heat capacity at constant volume\footnote{it's heat capacity at ``constant volume" 341 | because we are talking about Microcanonical 342 | ensemble.}. 343 | 344 | Observe that: 345 | \[A = E - TS= -k_b T \log(z)\] 346 | 347 | where \(A\) is the helmholtz free energy\footnote{it's the definition}. 348 | 349 | We define pressure \(P\) to be: 350 | \[\begin{gathered} 351 | P = \left. -\dfrac{\partial E}{\partial V} \right|_{S} 352 | = \left. -\dfrac{\partial A}{\partial V} \right|_{T}\\ 353 | \boxed{P = k_b T \left (\dfrac{\partial \log(z)}{\partial V} \right ) _{T} } \tag*{1} 354 | \end{gathered}\] 355 | \subsection{*Meaning of Partition Function} 356 | 357 | \textit{Source: Wikipedia} 358 | 359 | 360 | It may not be obvious why the partition function, as we have defined it above, is an important 361 | quantity. First, consider what goes into it. The partition function is a function of the temperature 362 | \(T\) and the microstate energies \(\epsilon_1, \epsilon_2, \epsilon_3, \) etc. 363 | The microstate energies are determined by other 364 | thermodynamic variables, such as the number of particles and the volume, as well as microscopic 365 | quantities like the mass of the constituent particles. This dependence on microscopic variables 366 | is the central point of statistical mechanics. With a model of the microscopic constituents of 367 | a system, one can calculate the microstate energies, and thus the partition function, which will 368 | then allow us to calculate all the other thermodynamic properties of the system. 369 | 370 | The partition function can be related to thermodynamic properties because it has a very important 371 | statistical meaning. The probability \(\mu_i\) that the system occupies microstate \(i\) is 372 | 373 | \[\mu_i = \frac{1}{z} \mathrm{e}^{- \beta \epsilon_i}\] 374 | 375 | Thus, as shown above, the partition function plays the role of a normalizing constant 376 | (note that it does not depend on i), ensuring that the probabilities sum up to one: 377 | 378 | \[\sum_i \mu_i = \frac{1}{z} \sum_i \mathrm{e}^{- \beta \epsilon_i} = \frac{1}{z} z = 1\] 379 | 380 | This is the reason for calling \(z\) the ``partition function": it encodes how the probabilities 381 | are partitioned among the different microstates, based on their individual energies. 382 | The letter \(z\) stands for the German word \textit{Zustandssumme}, ``sum over states". The usefulness 383 | of the partition function stems from the fact that it can be used to relate macroscopic 384 | thermodynamic quantities to the microscopic details of a system through the derivatives 385 | of its partition function. Finding the partition function is also equivalent to performing a 386 | Laplace transform of the density of states function from the energy domain to the \(\beta\) domain, 387 | and the inverse Laplace transform of the partition function reclaims the state density function 388 | of energies. 389 | 390 | \section{The Ideal Gas} 391 | 392 | \subsection{Introduction} 393 | Suppose a system is made of N sub-systems (\textit{particles}) with negligible interaction energy, that is, 394 | we can assume the particles are essentially non-interacting. If the partition functions of the 395 | sub-systems are \(\zeta_1, \zeta_2, \ldots \zeta _N,\) respectively then the partition function 396 | of the entire system is the product 397 | of the individual partition functions: 398 | 399 | \[z =\prod_{j=1}^{N} \zeta_j.\] 400 | 401 | If the sub-systems have the same physical properties, then their partition functions 402 | are equal, \(\zeta_1, \zeta_2, \ldots \zeta _N,\) in which case 403 | 404 | \[z = \zeta^N\] 405 | 406 | However, there is a well-known exception to this rule. If the sub-systems 407 | are actually identical particles, in the quantum mechanical sense that they 408 | are impossible to distinguish even in principle, the total partition function 409 | must be divided by a \(N!\)\footnote{This is to ensure that we do not ``over-count" 410 | the number of microstates. }: 411 | 412 | \[z = \frac{\zeta^N}{N!}\] 413 | 414 | While this may seem like a strange requirement, it is actually necessary 415 | to preserve the existence of a thermodynamic limit for such systems. 416 | This is known as the Gibbs paradox. 417 | \subsection{The partition function for ideal gas} 418 | 419 | If you have taken an introductory probability theory course, then you may know in continuous 420 | case, PMF (probability mass function) is replaced by PDF (probability density function). 421 | 422 | In classical mechanics, the position and momentum variables of a particle can vary continuously, 423 | so the set of microstates is actually uncountable. In classical statistical mechanics, it is 424 | rather inaccurate to express the partition function as a sum of discrete terms. 425 | In this case we must describe the partition function using an integral rather than a sum. 426 | 427 | We'll assume the systen to be non-interacting. 428 | The partion function for single particle (assuming it to a subsystem) is defined as\footnote{ 429 | To make it into a dimensionless quantity, we must divide it by h, it also has something to do with 430 | the precision with which we can measure the position and momenta in phase space. 431 | }: 432 | \[\begin{gathered} 433 | \zeta = \frac{1}{h^3}\int_p \int_x e^{-\beta E(x, p)} d^3x d^3p 434 | \end{gathered}\] 435 | 436 | For non-interating particles energy is due to momentum only. So, it implies: 437 | \[\begin{gathered} 438 | \zeta = \frac{1}{h^3}\int_p \int_x e^{-\beta \frac{p^2}{2m}} d^3x d^3p\\ 439 | = \frac{1}{h^3} \int_x d^3 x \int_p e^{-\beta \frac{p^2}{2m}}d^3p\\ 440 | = \frac{V}{h^3} \int_p e^{-\beta \frac{p^2}{2m}}d^3p \tag*{a} 441 | %%\left(\frac{2\pi m}{\beta}\right)^{3/2} 442 | \end{gathered}\] 443 | 444 | observe that \(p^2 = p_x ^2 + p_y ^2 + p_z ^2\) and \(d^3 p= d p_x dp_y dp_z\). 445 | We'll make an assumption 446 | 447 | \begin{assume}[Equipartition theorem] 448 | Energy is distributed equally among all the degrees of freedom. 449 | In other words if total energy is \(E\) and there are \(d\) degree of freedom. Then each 450 | degree of freedom contains \(\frac{E}{d}\) amount of energy. 451 | \end{assume} 452 | 453 | The above form of \(p^2\) becomes \(p^2 = 3 \bar{p} ^2\) and \(d^3 p = (d \bar{p})^3\). And equation 454 | \ref{a} becomes 455 | \[\begin{gathered} 456 | \zeta = \frac{V}{h^3} \int_p e^{-\beta \frac{p^2}{2m}}d^3p \\ 457 | = \frac{V}{h^3} \left(\int_p e^{-\beta \frac{\bar{p}^2}{2m}}d \bar{p}\right)^3\\ 458 | \zeta = \frac{V}{h^3}\left(\frac{2\pi m}{\beta}\right)^{3/2} 459 | \end{gathered}\] 460 | 461 | Hence, we have taken an ideal gas consisting of identical \(N\) particles. 462 | The partition function for the whaole system is 463 | \[\begin{gathered} 464 | z = \frac{(\zeta)^N}{N!}\\ 465 | z = \frac{V^N}{h^{3N}N!}\left(\frac{2\pi m}{\beta}\right)^{3N/2}\\ 466 | \log(z) = N \log(V) + \frac{3N}{2} 467 | \log(2\pi m)- \frac{3N}{2}\log(\beta) - \log(N!) - 3N\log(h)\end{gathered}\] 468 | Energy for the system is 469 | \[\begin{gathered} 470 | E = - \frac{\partial \log(z)}{\partial \beta} \\ 471 | = \frac{3N}{2\beta} \\ 472 | \boxed{E = \frac{3N}{2} k_b T} 473 | \end{gathered}\] 474 | 475 | Don't forget the Pressure from equation (i): 476 | \[\begin{gathered} 477 | P = k_b T \left (\dfrac{\partial \log(z)}{\partial V} \right ) _{T}\\ 478 | P = k_b T \frac{N}{V} \implies \boxed{PV = N k_b T} 479 | \end{gathered}\] 480 | \end{document} 481 | -------------------------------------------------------------------------------- /Misc/Periodic-table/periodic_table.tex: -------------------------------------------------------------------------------- 1 | % !TEX program = xelatex 2 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3 | % This was inspired by Ivan Griffin's work on a periodic table 4 | % (http://www.texample.net/tikz/examples/author/ivan-griffin/) 5 | % Back in 2015, I adapted/modified Ivan's periodic table to suit 6 | % my needs for a chemistry class that I taught. 7 | % What you find here is a complete 're-design.' so I am releasing 8 | % the re-designed code with a new copyright license. 9 | % Specifically, this code is distributed under the MIT open source 10 | % license. 11 | % Copyright 2019 Paul N. Danese 12 | % Permission is hereby granted, free of charge, to any person 13 | % obtaining a copy of this software and associated documentation 14 | % files (the "Software"), to deal in the Software without 15 | % restriction, including without limitation the rights to use, 16 | % copy, modify, merge, publish, distribute, sublicense, and/or 17 | % sell copies of the Software, and to permit persons to whom the 18 | % Software is furnished to do so, subject to the following 19 | % conditions: 20 | % The above copyright notice and this permission notice shall 21 | % be included in all copies or substantial portions of the Software. 22 | % THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, 23 | % EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF 24 | % MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. 25 | % IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR 26 | % ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF 27 | % CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION 28 | % WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. 29 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 30 | \documentclass[9pt]{article} 31 | \usepackage{geometry} 32 | \usepackage{ccicons} 33 | \usepackage{chemfig} 34 | \setchemfig{atom sep=2.4em, double bond sep=3.5pt, bond style={line width=1.5pt}} 35 | 36 | \usepackage{chemmacros} 37 | \usepackage{multicol} 38 | \usepackage{siunitx} 39 | \geometry{ 40 | letterpaper, 41 | left=2mm, 42 | right=2mm, 43 | bottom=2mm, 44 | top=2mm 45 | } 46 | \usepackage{graphicx} 47 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 48 | % url stuff %% 49 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 50 | \usepackage{hyperref} 51 | \hypersetup{% remove ugly boxes around url 52 | colorlinks, 53 | urlcolor={blue!80!black} 54 | } 55 | \urlstyle{same} % make url use the standard font 56 | 57 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 58 | % tikz stuff %% 59 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 60 | \usepackage{tikz} % the workhorse package here 61 | \usepackage{pgfornament} % for the fancy corners and such 62 | \usetikzlibrary{positioning,shapes} 63 | 64 | 65 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 66 | % font stuff %% 67 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 68 | \usepackage{fontspec} % I like fonts. 69 | \setmainfont{Scala Pro}[Numbers={Proportional,Lining}] 70 | \setsansfont{Scala Sans OT}[Numbers={Proportional,Lining}] 71 | 72 | 73 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 74 | % color stuff %% 75 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 76 | \definecolor{maroon}{HTML}{800000} 77 | \definecolor{fancyyellow}{HTML}{FFFDE9} 78 | \usepackage{pagecolor} % to color the page background (for extra fanciness) 79 | 80 | 81 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 82 | % tikz styles %% 83 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 84 | \tikzstyle{ebox} = [draw=black, thick,rectangle, minimum width=25mm,minimum height=25mm, node distance=25mm] % declares the features of each element box 85 | \tikzstyle{Offsetter} = [ minimum width=24mm, minimum height=25mm, node distance=25mm] % declares features of an unseen box similar to ebox to provide a seed-point for the hydrogen box 86 | 87 | % colors for different atomic blocks 88 | \tikzstyle{Sblock} = [fill=cyan!10] 89 | \tikzstyle{Pblock} = [fill=blue!10] 90 | \tikzstyle{Dblock} = [fill=orange!10] 91 | \tikzstyle{Fblock} = [fill=green!10] 92 | \tikzstyle{Noblock} = [fill=gray!10] 93 | 94 | % styles for the key and for details found at the bottom of the screen 95 | \tikzstyle{KeysLabel} = [minimum height = 2.5cm, inner sep=0pt, text width = 7.5cm] 96 | \tikzstyle{DetailsLabel} = [minimum height = 2.5cm, inner sep=0pt, text width = 35cm] 97 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 98 | \newcommand{\moleculep}{\chemfig{ 99 | O% 10 100 | =[:90]% 9 101 | ( 102 | -[:30,,,1]OH% 11 103 | ) 104 | -[:150]% 8 105 | ( 106 | <:[:90,,,1]NH_2% 12 107 | ) 108 | -[:210]% 7 109 | -[:150]% 4 110 | =^[:90]% 3 111 | -[:150]% 2 112 | =^[:210]% 1 113 | -[:270]% 6 114 | =^[:330]% 5 115 | ( 116 | -[:30]% -> 4 117 | ) 118 | }} 119 | 120 | \newcommand{\moleculeq}{\chemfig{ 121 | O% 10 122 | =[:90]% 9 123 | ( 124 | -[:150,,,2]HO% 11 125 | ) 126 | -[:30]% 8 127 | ( 128 | <[:90,,,1]NH_2% 12 129 | ) 130 | -[:330]% 7 131 | -[:30]% 4 132 | =_[:90]% 3 133 | -[:30]% 2 134 | =_[:330]% 1 135 | -[:270]% 6 136 | =_[:210]% 5 137 | ( 138 | -[:150]% -> 4 139 | ) 140 | }} 141 | 142 | % this macro is used to add contents to each 'box' for each element 143 | % parameter list is as follows: 144 | % #1: symbol 145 | % #2: element name 146 | % #3: atomic_number 147 | % #4: atomic_mass 148 | % #5: electronegativity 149 | % #6: shell occupied by 'last' ground-state electron 150 | % #7: subshell occupied by 'last' ground-state electron 151 | \newcommand\periodicelement[7]{% 152 | \begin{minipage}{22mm}% 153 | \centering% 154 | \par\noindent\parbox[t]{.333\textwidth}{\raggedright\textsf{#3}}% 155 | \parbox[t]{.333\textwidth}{\centering\textsf{#5}}% 156 | \parbox[t]{.333\textwidth}{\raggedleft \textsf{#6\textit{#7}}}\par% 157 | \vspace*{0.5em}% 158 | {\Huge #1}\\% 159 | \vspace*{0.14em}% 160 | \textsf{#2}\\% 161 | #4\\% 162 | \end{minipage}}% 163 | 164 | % create a phantom endash to go in certain boxes that don't have or need a number 165 | \newcommand{\pendash}{\phantom{\textendash}} 166 | % This macro is primarily used to create asterisks that delineate the 167 | % lanthanide and actinide series at the bottom of the table. 168 | \newcommand{\offsetterbox}[9]{\node[name=#1, Offsetter, #8, #9] {\periodicelement{#1}{#2}{#3}{#4}{#5}{#6}{#7}};} 169 | 170 | % name text position shading 171 | \newcommand{\groupbox}[4]{\node[name=#1, Offsetter, #3, #4,yshift=1mm] {\periodicelement{\pendash}{\pendash}{\pendash}{\textsf{#2}}{\pendash}{\pendash}{\pendash}};} 172 | 173 | % for showing the period number 174 | \newcommand{\periodbox}[3]{\node[name=#1,left of=#2,xshift=-16pt]{\textsf{\textit{#3}}};} 175 | 176 | % this macro is used to add element-specific data to each box. 177 | % The first 7 parameters are identical to that of \periodicelement 178 | % parameter 8 indicates the position of the box relative to 179 | % a previously laid element 180 | % parameter 9 is used to set the background color of the box 181 | \newcommand{\periodicbox}[9]{\node[name=#1, ebox, #8, #9] {\periodicelement{#1}{#2}{#3}{#4}{#5}{#6}{#7}};} 182 | 183 | 184 | \begin{document} 185 | %\pagecolor{fancyyellow} % comment this line if you want a white background 186 | \begin{tikzpicture}[scale=0.606, transform shape, rotate=90,overlay] 187 | % Group 1 188 | \node[name=blank1, Offsetter,anchor=south] at (-43.3,-4) {}; 189 | \node[name=blank2, Offsetter, below of=blank1] {}; 190 | \periodicbox{H}{hydrogen}{1}{1.008}{2.20}{1}{s}{below of=blank2}{Sblock} 191 | \periodicbox{Li}{lithium}{3}{6.9675}{0.98}{2}{s}{below of=H}{Sblock} 192 | \periodicbox{Na}{sodium}{11}{22.99}{0.93}{3}{s}{below of=Li}{Sblock} 193 | \periodicbox{K}{potassium}{19}{39.098}{0.82}{4}{s}{below of=Na}{Sblock} 194 | \periodicbox{Rb}{rubidium}{37}{85.468}{0.82}{5}{s}{below of=K}{Sblock} 195 | \periodicbox{Cs}{caesium}{55}{132.91}{0.79}{6}{s}{below of=Rb}{Sblock} 196 | \periodicbox{Fr}{francium}{87}{(223)}{\pendash}{7}{s}{below of=Cs}{Sblock} 197 | 198 | \periodbox{p1}{H}{1} 199 | \periodbox{p2}{Li}{2} 200 | \periodbox{p3}{Na}{3} 201 | \periodbox{p4}{K}{4} 202 | \periodbox{p5}{Rb}{5} 203 | \periodbox{p6}{Cs}{6} 204 | \periodbox{p7}{Fr}{7} 205 | 206 | % Here we set the title of the document with some cute ornaments 207 | \node[anchor=west, name=phewest,scale=1,right of=H, xshift=7.62cm, yshift=5.2cm] {\moleculep}; 208 | 209 | Here we set the title of the document with some cute ornaments 210 | \node[anchor=west, name=pheeast,scale=1,right of=H, xshift=32.62cm, yshift=5.2cm] {\moleculeq}; 211 | 212 | \node[anchor=west, name=diagramTitle,scale=1.75,right of=H,xshift=11.14cm,yshift=3cm] {\Huge \textbf{\textit{Periodic Table of the Elements}} }; 213 | 214 | % Group 2 215 | \periodicbox{Be}{beryllium}{4}{9.0122}{1.57}{2}{s}{right of=Li}{Sblock} 216 | \periodicbox{Mg}{magnesium}{12}{24.3055}{1.31}{3}{s}{below of=Be}{Sblock} 217 | \periodicbox{Ca}{calcium}{20}{40.078}{1.00}{4}{s}{below of=Mg}{Sblock} 218 | \periodicbox{Sr}{strontium}{38}{87.62}{0.95}{5}{s}{below of=Ca}{Sblock} 219 | \periodicbox{Ba}{barium}{56}{137.33}{0.89}{6}{s}{below of=Sr}{Sblock} 220 | \periodicbox{Ra}{radium}{88}{(226)}{0.9}{7}{s}{below of=Ba}{Sblock} 221 | 222 | 223 | % Group 3 224 | \periodicbox{Sc}{scandium}{21}{44.956}{1.36}{3}{d}{right of=Ca}{Dblock} 225 | \periodicbox{Y}{yttrium}{39}{88.906}{1.22}{4}{d}{below of=Sc}{Dblock} 226 | \periodicbox{*}{lanthanides}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{below of=Y}{Noblock} 227 | \periodicbox{**}{actinides}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{\textcolor{gray!10}{-}}{below of=*}{Noblock} 228 | 229 | 230 | % Group 4 Elements 231 | \periodicbox{Ti}{titanium}{22}{47.867}{1.54}{3}{d}{right of=Sc}{Dblock} 232 | \periodicbox{Zr}{zirconium}{40}{91.224}{1.33}{4}{d}{below of=Ti}{Dblock} 233 | \periodicbox{Hf}{hafnium}{72}{178.49}{1.3}{5}{d}{below of=Zr}{Dblock} 234 | \periodicbox{Rf}{rutherfordium}{104}{(267)}{\pendash}{6}{d}{below of=Hf}{Dblock} 235 | 236 | 237 | 238 | % Group 5 Elements 239 | \periodicbox{V}{vanadium}{23}{50.942}{1.63}{3}{d}{right of=Ti}{Dblock} 240 | \periodicbox{Nb}{niobium}{41}{92.906}{1.6}{4}{d\char"02DA}{below of=V}{Dblock} 241 | \periodicbox{Ta}{tantalum}{73}{180.95}{1.5}{5}{d}{below of=Nb}{Dblock} 242 | \periodicbox{Db}{dubnium}{105}{(268)}{\pendash}{6}{d}{below of=Ta}{Dblock} 243 | 244 | 245 | 246 | 247 | % Group 6 Elements 248 | \periodicbox{Cr}{chromium}{24}{51.996}{1.66}{3}{d\char"02DA}{right of=V}{Dblock} 249 | \periodicbox{Mo}{molybdenum}{42}{95.95}{2.16}{4}{d\char"02DA}{below of=Cr}{Dblock} 250 | \periodicbox{W}{tungsten}{74}{183.84}{2.36}{5}{d}{below of=Mo}{Dblock} 251 | \periodicbox{Sg}{seaborgium}{106}{(269)}{\pendash}{6}{d}{below of=W}{Dblock} 252 | 253 | 254 | 255 | % Group 7 Elements 256 | \periodicbox{Mn}{manganese}{25}{54.938}{1.55}{3}{d}{right of=Cr}{Dblock} 257 | \periodicbox{Tc}{technetium}{43}{(97)}{1.9}{4}{d}{below of=Mn}{Dblock} 258 | \periodicbox{Re}{rhenium}{75}{186.21}{1.9}{5}{d}{below of=Tc}{Dblock} 259 | \periodicbox{Bh}{bohrium}{107}{(270)}{\pendash}{6}{d}{below of=Re}{Dblock} 260 | 261 | 262 | 263 | % Group 8 Elements 264 | \periodicbox{Fe}{iron}{26}{55.845}{1.83}{3}{d}{right of=Mn}{Dblock} 265 | \periodicbox{Ru}{ruthenium}{44}{101.07}{2.2}{4}{d\char"02DA}{below of=Fe}{Dblock} 266 | \periodicbox{Os}{osmium}{76}{190.23}{2.2}{5}{d}{below of=Ru}{Dblock} 267 | \periodicbox{Hs}{hassium}{108}{(269)}{\pendash}{6}{d}{below of=Os}{Dblock} 268 | 269 | 270 | 271 | % Group 9 Elements 272 | \periodicbox{Co}{cobalt}{27}{58.933}{1.88}{3}{d}{right of=Fe}{Dblock} 273 | \periodicbox{Rh}{rhodium}{45}{102.91}{2.28}{4}{d\char"02DA}{below of=Co}{Dblock} 274 | \periodicbox{Ir}{iridium}{77}{192.22}{2.2}{5}{d}{below of=Rh}{Dblock} 275 | \periodicbox{Mt}{meitnerium}{109}{(278)}{\pendash}{6}{d}{below of=Ir}{Dblock} 276 | 277 | 278 | % Group 10 Elements 279 | \periodicbox{Ni}{nickel}{28}{58.693}{1.91}{3}{d}{right of=Co}{Dblock} 280 | \periodicbox{Pd}{palladium}{46}{106.42}{2.20}{4}{d\char"02DA}{below of=Ni}{Dblock} 281 | \periodicbox{Pt}{platinum}{78}{195.08}{2.28}{5}{d\char"02DA}{below of=Pd}{Dblock} 282 | \periodicbox{Ds}{darmstadtium}{110}{(281)}{\pendash}{6}{d}{below of=Pt}{Dblock} 283 | 284 | 285 | % Group 11 Elements 286 | \periodicbox{Cu}{copper}{29}{63.546}{1.90}{3}{d\char"02DA}{right of=Ni}{Dblock} 287 | \periodicbox{Ag}{silver}{47}{107.87}{1.93}{4}{d\char"02DA}{below of=Cu}{Dblock} 288 | \periodicbox{Au}{gold}{79}{196.97}{2.54}{5}{d\char"02DA}{below of=Ag}{Dblock} 289 | \periodicbox{Rg}{roentgenium}{111}{(282)}{\pendash}{6}{d}{below of=Au}{Dblock} 290 | 291 | 292 | % Group 12 Elements 293 | \periodicbox{Zn}{zinc}{30}{65.38}{1.65}{3}{d}{right of=Cu}{Dblock} 294 | \periodicbox{Cd}{cadmium}{48}{112.41}{1.69}{4}{d}{below of=Zn}{Dblock} 295 | \periodicbox{Hg}{mercury}{80}{200.59}{1.9}{5}{d}{below of=Cd}{Dblock} 296 | \periodicbox{Cn}{copernicium}{112}{(285)}{\pendash}{6}{d}{below of=Hg}{Dblock} 297 | 298 | 299 | % Group 13 Elements 300 | \periodicbox{Ga}{gallium}{31}{69.723}{1.81}{4}{p}{right of=Zn}{Pblock} 301 | \periodicbox{Al}{aluminium}{13}{26.982}{1.61}{3}{p}{above of=Ga}{Pblock} 302 | \periodicbox{B}{boron}{5}{10.8135}{2.04}{2}{p}{above of=Al}{Pblock} 303 | \periodicbox{In}{indium}{49}{114.82}{1.78}{5}{p}{below of=Ga}{Pblock} 304 | \periodicbox{Tl}{thallium}{81}{204.385}{1.62}{6}{p}{below of=In}{Pblock} 305 | \periodicbox{Nh}{nihonium}{113}{(286)}{\pendash}{7}{p}{below of=Tl}{Pblock} 306 | 307 | 308 | % Group 14 Elements 309 | \periodicbox{Ge}{germanium}{32}{72.63}{2.01}{4}{p}{right of=Ga}{Pblock} 310 | \periodicbox{Si}{silicon}{14}{28.085}{1.90}{3}{p}{above of=Ge}{Pblock} 311 | \periodicbox{C}{carbon}{6}{12.0105}{2.55}{2}{p}{above of=Si}{Pblock} 312 | \periodicbox{Sn}{tin}{50}{118.71}{1.96}{5}{p}{below of=Ge}{Pblock} 313 | \periodicbox{Pb}{lead}{82}{207.2}{1.8}{6}{p}{below of=Sn}{Pblock} 314 | \periodicbox{Fl}{flerovium}{114}{(289)}{\pendash}{7}{p}{below of=Pb}{Pblock} 315 | 316 | 317 | % Group 15 Elements 318 | \periodicbox{As}{arsenic}{33}{74.922}{2.18}{4}{p}{right of=Ge}{Pblock} 319 | \periodicbox{P}{phosphorus}{15}{30.974}{2.19}{3}{p}{above of=As}{Pblock} 320 | \periodicbox{N}{nitrogen}{7}{14.007}{3.04}{2}{p}{above of=P}{Pblock} 321 | \periodicbox{Sb}{antimony}{51}{121.76}{2.05}{5}{p}{below of=As}{Pblock} 322 | \periodicbox{Bi}{bismuth}{83}{208.98}{2.02}{6}{p}{below of=Sb}{Pblock} 323 | \periodicbox{Mc}{moscovium}{115}{(290)}{\pendash}{7}{p}{below of=Bi}{Pblock} 324 | 325 | 326 | % Group 16 Elements 327 | \periodicbox{Se}{selenium}{34}{78.971}{2.55}{4}{p}{right of=As}{Pblock} 328 | \periodicbox{S}{sulfur}{16}{32.0675}{2.58}{3}{p}{above of=Se}{Pblock} 329 | \periodicbox{O}{oxygen}{8}{15.9995}{3.44}{2}{p}{above of=S}{Pblock} 330 | \periodicbox{Te}{tellurium}{52}{127.6}{2.1}{5}{p}{below of=Se}{Pblock} 331 | \periodicbox{Po}{polonium}{84}{(209)}{2.0}{6}{p}{below of=Te}{Pblock} 332 | \periodicbox{Lv}{livermorium}{116}{(293)}{\pendash}{7}{p}{below of=Po}{Pblock} 333 | 334 | 335 | % Group 17 Elements 336 | \periodicbox{Br}{bromine}{35}{79.904}{2.96}{4}{p}{right of=Se}{Pblock} 337 | \periodicbox{Cl}{chlorine}{17}{35.4515}{3.16}{3}{p}{above of=Br}{Pblock} 338 | \periodicbox{F}{fluorine}{9}{18.998}{3.98}{2}{p}{above of=Cl}{Pblock} 339 | \periodicbox{I}{iodine}{53}{126.9}{2.66}{5}{p}{below of=Br}{Pblock} 340 | \periodicbox{At}{astatine}{85}{(210)}{2.2}{6}{p}{below of=I}{Pblock} 341 | \periodicbox{Ts}{tennessine}{117}{(294)}{\pendash}{7}{p}{below of=At}{Pblock} 342 | 343 | 344 | % Group 18 Elements (the snobs) 345 | \periodicbox{Kr}{krypton}{36}{83.798}{\pendash}{4}{p}{right of=Br}{Pblock} 346 | \periodicbox{Ar}{argon}{18}{39.8775}{\pendash}{3}{p}{above of=Kr}{Pblock} 347 | \periodicbox{Ne}{neon}{10}{20.18}{\pendash}{2}{p}{above of=Ar}{Pblock} 348 | \periodicbox{He}{helium}{2}{4.0026}{\pendash}{1}{s}{above of=Ne}{Sblock} 349 | \periodicbox{Xe}{xenon}{54}{131.29}{2.60}{5}{p}{below of=Kr}{Pblock} 350 | \periodicbox{Rn}{radon}{86}{(222)}{\pendash}{6}{p}{below of=Xe}{Pblock} 351 | \periodicbox{Og}{oganesson}{118}{(294)}{\pendash}{7}{p}{below of=Rn}{Pblock} 352 | 353 | 354 | % lanthanides 355 | \periodicbox{La}{lanthanum}{57}{138.91}{1.1}{5}{d\char"02DA}{below of=**,yshift=-1.3cm}{Dblock} 356 | \periodicbox{Ce}{cerium}{58}{140.12}{1.12}{4}{f\char"02DA}{right of=La}{Fblock} 357 | \periodicbox{Pr}{praseodymium}{59}{140.91}{1.13}{4}{f}{right of=Ce}{Fblock} 358 | \periodicbox{Nd}{neodymium}{60}{144.24}{1.14}{4}{f}{right of=Pr}{Fblock} 359 | \periodicbox{Pm}{promethium}{61}{(145)}{\pendash}{4}{f}{right of=Nd}{Fblock} 360 | \periodicbox{Sm}{samarium}{62}{150.36}{1.17}{4}{f}{right of=Pm}{Fblock} 361 | \periodicbox{Eu}{europium}{63}{151.96}{\pendash}{4}{f}{right of=Sm}{Fblock} 362 | \periodicbox{Gd}{gadolinium}{64}{157.25}{1.2}{4}{f\char"02DA}{right of=Eu}{Fblock} 363 | \periodicbox{Tb}{terbium}{65}{158.93}{\pendash}{4}{f}{right of=Gd}{Fblock} 364 | \periodicbox{Dy}{dysprosium}{66}{162.5}{1.22}{4}{f}{right of=Tb}{Fblock} 365 | \periodicbox{Ho}{holmium}{67}{164.93}{1.23}{4}{f}{right of=Dy}{Fblock} 366 | \periodicbox{Er}{erbium}{68}{167.26}{1.24}{4}{f}{right of=Ho}{Fblock} 367 | \periodicbox{Tm}{thulium}{69}{168.93}{1.25}{4}{f}{right of=Er}{Fblock} 368 | \periodicbox{Yb}{ytterbium}{70}{173.05}{\pendash}{4}{f}{right of=Tm}{Fblock} 369 | \periodicbox{Lu}{lutetium}{71}{174.97}{1.27}{4}{f}{right of=Yb}{Fblock} 370 | 371 | 372 | % actinides 373 | \periodicbox{Ac}{actinium}{89}{(227)}{1.1}{6}{d\char"02DA}{below of=La}{Dblock} 374 | \periodicbox{Th}{thorium}{90}{232.04}{1.3}{5}{f\char"02DA}{right of=Ac}{Fblock} 375 | \periodicbox{Pa}{protactinium}{91}{231.04}{1.5}{5}{f\char"02DA}{right of=Th}{Fblock} 376 | \periodicbox{U}{uranium}{92}{238.03}{1.38}{5}{f\char"02DA}{right of=Pa}{Fblock} 377 | \periodicbox{Np}{neptunium}{93}{(237)}{1.36}{5}{f\char"02DA}{right of=U}{Fblock} 378 | \periodicbox{Pu}{plutonium}{94}{(244)}{1.28}{5}{f}{right of=Np}{Fblock} 379 | \periodicbox{Am}{americium}{95}{(243)}{\pendash}{5}{f}{right of=Pu}{Fblock} 380 | \periodicbox{Cm}{curium}{96}{(247)}{\pendash}{5}{f\char"02DA}{right of=Am}{Fblock} 381 | \periodicbox{Bk}{berkelium}{97}{(247)}{1.3}{5}{f}{right of=Cm}{Fblock} 382 | \periodicbox{Cf}{californium}{98}{(251)}{1.3}{5}{f}{right of=Bk}{Fblock} 383 | \periodicbox{Es}{einsteinium}{99}{(252)}{1.3}{5}{f}{right of=Cf}{Fblock} 384 | \periodicbox{Fm}{fermium}{100}{(257)}{1.3}{5}{f}{right of=Es}{Fblock} 385 | \periodicbox{Md}{mendelevium}{101}{(258)}{1.3}{5}{f}{right of=Fm}{Fblock} 386 | \periodicbox{No}{nobelium}{102}{(259)}{1.3}{5}{f}{right of=Md}{Fblock} 387 | \periodicbox{Lr}{lawrencium}{103}{(266)}{\pendash}{6}{d}{right of=No}{Fblock} 388 | 389 | % Key explaining contents of each box 390 | \periodicbox{Sy}{element}{Z}{saw}{$\chi$}{s}{s}{above of=V,yshift=2.4cm}{} 391 | \node[right of=Sy, name=keyvalues, KeysLabel, xshift=4.5cm]{ 392 | \textsf{Z: atomic number}\newline 393 | \(\chi\): \textsf{Pauling electronegativity}\newline 394 | \textsf{s\textit{s}: last occupied subshell} \newline 395 | Sy: \textsf{symbol} \newline 396 | \textsf{element: element name}\newline 397 | \textsf{saw: standard atomic weight}\char"2020}; 398 | % manicules calling attention to the key 399 | \node[anchor=west,left of=Sy,xshift=-40,scale=5]{\char"EA2D }; 400 | \node[anchor=east,right of=keyvalues,xshift=18,scale=5]{\char"EA2E }; 401 | 402 | % group numbers 403 | \groupbox{gp1}{Group 1}{above of=H}{} 404 | \groupbox{gp2}{2}{above of=Be}{} 405 | \groupbox{gp3}{3}{above of=Sc}{} 406 | \groupbox{gp4}{4}{above of=Ti}{} 407 | \groupbox{gp5}{5}{above of=V}{} 408 | \groupbox{gp6}{6}{above of=Cr}{} 409 | \groupbox{gp7}{7}{above of=Mn}{} 410 | \groupbox{gp8}{8}{above of=Fe}{} 411 | \groupbox{gp9}{9}{above of=Co}{} 412 | \groupbox{gp10}{10}{above of=Ni}{} 413 | \groupbox{gp11}{11}{above of=Cu}{} 414 | \groupbox{gp12}{12}{above of=Zn}{} 415 | \groupbox{gp13}{13}{above of=B}{} 416 | \groupbox{gp14}{14}{above of=C}{} 417 | \groupbox{gp15}{15}{above of=N}{} 418 | \groupbox{gp16}{16}{above of=O}{} 419 | \groupbox{gp17}{17}{above of=F}{} 420 | \groupbox{gp18}{18}{above of=He}{} 421 | 422 | 423 | 424 | % asterisks 425 | \offsetterbox{*}{\pendash}{\pendash}{\pendash}{\pendash}{\pendash}{\pendash}{left of=La}{} 426 | 427 | \offsetterbox{**}{\pendash}{\pendash}{\pendash}{\pendash}{\pendash}{\pendash}{left of=Ac}{} 428 | 429 | 430 | % details 431 | \node[below of=Ac, anchor=west, name=details,DetailsLabel,yshift=-1.7cm]{ 432 | \char"2020 Standard atomic weights (average terrestrial atomic weight) taken from the Commission on Isotopic Abundances and Atomic Weights (\url{http://www.ciaaw.org/abridged-atomic-weights.htm}). If CIAAW indicates a range for the standard atomic weight of an element, I used the arithmetic mean of the boundaries of the range. Elements with atomic weight in parentheses (e.g., Francium (223)) have no known stable isotopes and it is therefore impossible to provide a standard atomic weight. For these elements, the mass of a representative isotope is provided. \\ 433 | \char"02DA Indicates an anomalous (Aufbau rule-breaking) ground state electron configuration.\\ 434 | Inspired by Ivan Griffin's \LaTeX\space Periodic Table. \LaTeX code is released under the MIT open source license. \\ 435 | Final product (this Table) is released under creative commons attribution/share-alike copyright terms. \ccLogo \ccAttribution \ccShareAlike\space \the\year. Paul N. Danese 436 | }; 437 | 438 | \end{tikzpicture} 439 | 440 | % 4 corner angels 441 | \begin{tikzpicture}[overlay] 442 | %\node[name=tr,anchor=north west] at (-0.55,.85){\pgfornament[width=2cm,color=maroon]{131}}; 443 | \node[name=tr,anchor=north west] at (-0.75,1.0){\includegraphics[scale=0.07]{trna}}; 444 | %\node[name=tr,anchor=north west] at (16.3,9.50){\includegraphics[angle=90,origin=c,scale=.4]{crystal}}; 445 | % 446 | \node[name=tr,anchor=north west] at (17.8,1.0){\includegraphics[scale=0.07]{trna2}}; %\node[name=tr,anchor=north west] at (18.5,0.850){\pgfornament[width=2cm,color=maroon,]{trna}}; 447 | %\node[name=tr,anchor=north west] at (-7.9,-22.5){\includegraphics[scale=0.4]{sun}}; 448 | %\node[name=tr,anchor=north west] at (17.5,-23.99){\includegraphics[angle=72,origin=c,scale=0.4]{05}}; 449 | 450 | % \node[name=tr,anchor=north west] at (-0.55,-24.65){\pgfornament[width=2cm,color=maroon,symmetry=h]{131}}; 451 | 452 | \node[name=tr,anchor=north west] at (-0.75,-23.95){\includegraphics[scale=0.07]{trna4}}; 453 | 454 | \node[name=tr,anchor=north west] at (17.8,-23.95){\includegraphics[scale=0.07]{trna3}}; 455 | 456 | % \node[name=tr,anchor=north west] at (18.5,-24.65){\pgfornament[width=2cm,color=maroon,symmetry=h]{132}}; 457 | \end{tikzpicture} 458 | 459 | \end{document} 460 | 461 | 462 | -------------------------------------------------------------------------------- /MTH201: Differential-Geometry/MTH201_-_Differential_Geometry.tex: -------------------------------------------------------------------------------- 1 | 2 | 3 | 4 | \documentclass{notes} 5 | \usepackage{color-env} 6 | \usepackage[english]{babel} 7 | \usepackage{amssymb,amsmath,amsfonts} %%% for maths 8 | \usepackage{bm} 9 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 10 | \usepackage[object=vectorian]{pgfornament} 11 | \usepackage{background} 12 | \usepackage{calligra} 13 | \usepackage{enumerate} %% it make lists 14 | %\usepackage{semantic} 15 | %\usepackage{graphicx} %%% \to include figures 16 | %\usepackage{subfig} %%%% for two figures side by side 17 | \renewcommand\qedsymbol{$\blacksquare$} 18 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 19 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20 | \newcommand{\norm}[1]{\left\lVert #1 \right \rVert} 21 | \newcommand{\abs}[1]{\left| #1 \right|} 22 | \newcommand{\bgamma}{\bm{\gamma}} 23 | \newcommand{\R}[1]{\mathbb{R}^{#1} } 24 | \newcommand{\bsigma}{\bm{\sigma}} 25 | 26 | 27 | \begin{document} 28 | 29 | \begin{titlepage} % Suppresses headers and footers on the title page 30 | \backgroundsetup{ 31 | scale=1, 32 | opacity=1, 33 | angle=0, 34 | color=black, 35 | contents={ 36 | \begin{tikzpicture}[color=black, every node/.style={inner sep= 15pt}] 37 | \node (NW) [anchor=north west] at (current page.north west){\pgfornament[width=2.5cm] {131}}; 38 | \node (NE) [anchor=north east] at (current page.north east){\pgfornament[width=2.5cm, symmetry=v]{131}}; 39 | \node (SW) [anchor=south west] at (current page.south west){\pgfornament[width=2.5cm, symmetry=h]{131}}; 40 | \node (SE) [anchor=south east] at (current page.south east){\pgfornament[width=2.5cm, symmetry=c]{131}}; 41 | \foreach \i in {-4,0,4} 42 | \node[anchor=north,xshift=\i cm] at (current page.north){\pgfornament[scale=0.25,symmetry=v]{71}}; 43 | \foreach \i in {-4,0,4} 44 | \node[xshift=\i cm, yshift=32.25 pt] at (current page.south){\pgfornament[scale=0.25,symmetry=v]{71}}; 45 | \foreach \i in {-8,-4,0,4,8} 46 | \node[yshift=\i cm, xshift=32.25pt, rotate=90] at (current page.west){\pgfornament[scale=0.25,symmetry=v]{71}}; 47 | \foreach \i in {-8,-4,0,4,8} 48 | \node[yshift=\i cm, xshift=-32.25pt, rotate=90] at (current page.east){\pgfornament[scale=0.25,symmetry=v]{71}}; 49 | \foreach \i in {-11,-9,...,7,9} 50 | \node[anchor=west, yshift=\i cm, xshift=52.25pt, rotate=90] at (current page.west){\pgfornament[scale=0.1]{80}}; 51 | \foreach \i in {-11,-9,...,7,9} 52 | \node[anchor=east, yshift=\i cm, xshift=-52.25pt, rotate=-90] at (current page.east){\pgfornament[scale=0.1]{80}}; 53 | \end{tikzpicture} 54 | }} 55 | 56 | \centering % Centre everything on the title page 57 | 58 | \scshape % Use small caps for all text on the title page 59 | 60 | \vspace*{\baselineskip} % White space at the top of the page 61 | 62 | %------------------------------------------------ 63 | % Title 64 | %------------------------------------------------ 65 | 66 | \rule{\textwidth}{1.6pt}\vspace*{-\baselineskip}\vspace*{2pt} % Thick horizontal rule 67 | \rule{\textwidth}{0.4pt} % Thin horizontal rule 68 | 69 | \vspace{0.75\baselineskip} % Whitespace above the title 70 | 71 | {\huge \calligra{Differential Geometry}\\} % Title 72 | 73 | \vspace{0.75\baselineskip} % Whitespace below the title 74 | 75 | \rule{\textwidth}{0.4pt}\vspace*{-\baselineskip}\vspace{3.2pt} % Thin horizontal rule 76 | \rule{\textwidth}{1.6pt} % Thick horizontal rule 77 | 78 | \vspace{2\baselineskip} % Whitespace after the title block 79 | 80 | %------------------------------------------------ 81 | % Subtitle 82 | %------------------------------------------------ 83 | 84 | \LARGE{MTH201} 85 | 86 | \vspace*{3\baselineskip} % Whitespace under the subtitle 87 | 88 | %------------------------------------------------ 89 | % Editor(s) 90 | %------------------------------------------------ 91 | 92 | 93 | \vspace{0.5\baselineskip} % Whitespace before the editors 94 | 95 | %{\scshape \LARGE Prof. Vaibhav Vaish\\ } % Editor list 96 | 97 | \vspace{0.5\baselineskip} % Whitespace below the editor list 98 | 99 | %\textit{\Large IISER, Mohali} % affiliation 100 | 101 | \vfill % Whitespace between editor names and publisher logo 102 | 103 | %------------------------------------------------ 104 | % Author 105 | %------------------------------------------------ 106 | 107 | 108 | \vspace{0.3\baselineskip} % Whitespace under the publisher logo 109 | 110 | 111 | {\large Edited by\\ Aditya Dev} 112 | 113 | \end{titlepage} 114 | \backgroundsetup{contents={}} 115 | \tableofcontents 116 | %\newpage 117 | \chapter{Curves in the plane and in space} 118 | 119 | \section{What is a curve?} 120 | 121 | \begin{definition}[Parametrized curve]{def:curve} 122 | A \textit{parametrized curve} in \(\mathbb{R}^n\) is a map \(\bm{\gamma}: (\alpha, \beta) \to \mathbb{R}^n\), for some \((\alpha, \beta) \subseteq \mathbb{R}\) 123 | 124 | \paragraph{Example}: \(\bm{\gamma}(t): (-\infty, \infty) \to (t, t^2)\) 125 | \end{definition} 126 | 127 | \begin{description} 128 | \item[Note] There can be different parametrizations for the same curve; but it's not mandatory that they have same properties. 129 | \item[Smooth Function] 130 | A function \(f: (\alpha, \beta) \to \mathbb{R}\) is said to be 131 | smooth if the derivative \(\frac{d^n f}{dt^n}\) exists for all $n \geq 1$ and all $t \in (\alpha, \beta)$. 132 | \end{description} 133 | \begin{definition}[Tangent Vector]{def:tangent} 134 | If \(\bm{\gamma}\) is a parametrized curve, its first derivative \(\dot{\bm{\gamma}}(t)\) is called the tangent vector 135 | of \(\bm{\gamma}\) at the point \(\bm{\gamma}\)$(t)$. 136 | \end{definition} 137 | 138 | \begin{proposition}{prop:staright-line} 139 | If the tangent vector of a parametrized curve is constant, the image of the curve 140 | is (part of) a straight line. 141 | \end{proposition} 142 | 143 | 144 | \section{Arc-Length} 145 | \paragraph{Recall that} if \(\mathbf{v} = (v_1, v_2, \ldots v_n) \in \mathbb{R}^n\), then it's length is: 146 | \[ \norm{\mathbf{v}} = \sqrt{v_1 ^2 + v_2 ^2 + \cdots v_n ^2}\] 147 | If \(\mathbf{u}\) is another vector in \(\mathbb{R}^n\) , \(\norm{\mathbf{u} - \mathbf{v}}\) is the length of the straight line segment joining the points \(\mathbf{u}\) and \(\mathbf{v}\) in \(\mathbb{R}^n\). 148 | 149 | \begin{definition}[Arc-length]{def:arc-length} 150 | The \textit{arc-length} of a curve \(\bm{\gamma}\) starting at the point \(\bm{\gamma}(t_0)\) is the function \(s(t)\) given by: 151 | \[s(t) = \int_{t_0}^{t} \norm{\dot{\bm{\gamma}}(x)} dx\] 152 | 153 | Note that if we choose a different starting point, then the new arc-length differs from the previous one (\textit{but how much?}) 154 | \end{definition} 155 | 156 | \begin{definition}{def:unit-speed} 157 | If \(\bm{\gamma}: (\alpha, \beta) \to \mathbb{R}^2\) is a parametrized curve, it's speed at point \(\mathbf{\bm{\gamma}}(t)\) is \(\norm{\dot{\bm{\gamma}}(t)}\), and \(\bm{\gamma}\) is said to be a unit-speed curve if \(\dot{\bm{\gamma}}(t)\) is a unit-vector \(\forall t\in (\alpha, \beta)\). 158 | \end{definition} 159 | 160 | \begin{proposition}{prop:unit-zero} 161 | Let \(\mathbf{n}(t)\) be a unit vector that is a smooth function of a parameter \(t\). Then, the dot product 162 | \[\mathbf{n}(t) \cdot \dot{\mathbf{n}}(t) = 0 \quad \forall t\] 163 | so, either \(\dot{\mathbf{n}}(t)\) is zero or perpendicular to \({\mathbf{n}}(t)\) 164 | 165 | \end{proposition} 166 | 167 | \section{Reparametrization of a curve} 168 | 169 | \begin{definition}[Reparametrization]{def:repar} 170 | A parametrized curve \(\tilde{\bm{\gamma}}: (\tilde{\alpha}, \tilde{\beta}) \to \mathbb{R}^n\) is a reparametrization of a parametrized curve \(\bm{\gamma}: (\alpha, \beta) \to \mathbb{R}^n\) if \(\exists\) a smooth bijective map \(\phi: (\tilde{\alpha}, \tilde{\beta}) \to (\alpha, \beta)\) (the reparametrization map) such that the inverse map \(\phi^{-1}: (\alpha, \beta) \to (\tilde{\alpha}, \tilde{\beta})\) is also smooth and 171 | \[\tilde{\bm{\gamma}}(\tilde{t}) = \bm{\gamma}(\phi(\tilde{t}))\] 172 | \end{definition} 173 | 174 | \begin{definition}[Regular Curve]{:def:regular-curve} 175 | A point \(\bm{\gamma}(t)\) of a parametrized curve \(\bm{\gamma}\) is called a regular point if \(\dot{\bm{\gamma}}(t) \not = 0\) 176 | otherwise \(\bm{\gamma}(t)\) is a singular point of \(\bm{\gamma}\). A curve is regular if all of its points are regular 177 | \end{definition} 178 | 179 | \begin{proposition}{} 180 | Any reparametrization of a regular curve is regular. 181 | \end{proposition} 182 | \begin{proposition}{prop:1.3.2} 183 | If \(\bm{\gamma}(t)\) is a regular curve, its arc-length \(s\), starting at any point of \(\bm{\gamma}\), is smooth function of \(t\). 184 | \end{proposition} 185 | 186 | 187 | \begin{theorem}[Unit-speed reparametrization]{def:unit-repar} 188 | A parametrized curve has a unit-speed reparametrization if and only if it is 189 | regular. 190 | \end{theorem} 191 | \begin{corollary}{1} 192 | Let \(\bm{\gamma}\) be a regular curve and let \(\tilde{\bm{\gamma}}\) be a unit-speed reparametrization of \(\bm{\gamma}\): 193 | \[\tilde{\bm{\gamma}}(u(t)) = \bm{\gamma}(t) \quad \forall t\] 194 | where \(u\) is a smooth function of \(t\). Then, if \(s\) is the arc-length of \(\bm{\gamma}\) (starting at any point), we have: 195 | \[u = \pm s + c \quad \text{for some } c \in \mathbb{R} \tag*{(1.1)}\] 196 | Conversely, if \(u\) is given by Eq. \(1.1\) for some value of \(c\) 197 | and with either sign, then \(\tilde{\bm{\gamma}}\) is a unit-speed reparametrization of \(\bm{\gamma}\). 198 | \end{corollary} 199 | \section{Closed Curves} 200 | 201 | \begin{definition}[Periodic Curve]{def:periodic} 202 | Let \(\bm{\gamma}: \mathbb{R} \to \mathbb{R}^n\) be a smooth curve and let \(T \in \mathbb{R}\). We say that \(\bm{\gamma}\) is 203 | \(T\) -periodic if: 204 | \[\bm{\gamma}(t +T) = \bm{\gamma}(t) \quad \forall t\in \mathbb{R}\] 205 | If \(\bm{\gamma}\) is not constant and is \(T\)-periodic for some \(T \not = 0\), then \(\bm{\gamma}\) is said to be closed. 206 | 207 | \textbf{Note} if \(\bm{\gamma}\) is \(T\)-periodic the it is \(-T\)-periodic too because 208 | \[\bm{\gamma}(t - T) = \bm{\gamma}(t -T + T) = \bm{\gamma}(t)\] 209 | It follows that if \(\bm{\gamma}\) is \(T\)-periodic for some \(T \not = 0\), then it is \(T\)-periodic for some 210 | $(T > 0)$. 211 | \end{definition} 212 | 213 | \begin{definition}[Self-intersection]{def:self-intersection} 214 | A curve \(\bm{\gamma}\) is said to have a self-intersection at a point \(\mathbf{p}\) of the curve if there 215 | exist parameter values \(a \not = b\) such that 216 | \begin{itemize} 217 | \item \(\bm{\gamma}(a) =\bm{\gamma}(b) = \mathbf{p} \) 218 | \item if \(\bm{\gamma}\) is closed with period \(T\) , then \(a - b\) is not an integer multiple of \(T\) . 219 | \end{itemize} 220 | \end{definition} 221 | 222 | \begin{proof} 223 | Assume there exists no lower bond for the period of curve. 224 | Then if \(T_1\) is the period of \(\bgamma\) then \(\exists\ T_2\) such that \(T_2\) is also the period of the curve and by iteration we can show: 225 | \[T_1 > T_2 > T_3 \ldots >0\] 226 | is a sequence of the periods for curve \(\bgamma\) 227 | 228 | Since, the sequence is bounded and monotonic, therefore \(\lim\limits_{r \to \infty} T_r = T\) i.e the sequence is convergent \(\implies\) the sequence is cauchy. 229 | 230 | And by definition of cauchy sequence: 231 | \[\forall \epsilon > 0\ \exists N : \forall m, n> N \implies \abs{T_m -T_n} < \epsilon\] 232 | 233 | Let, \(T_m > T_n\) (won't change the definition).Also, we know that if \(T_m \ \& \ T_n\) are periods of \(\bgamma\) \(\implies T_m - T_n \implies T_m = T_n + \epsilon\) is also the period of gamma (trivial to prove!). 234 | \[ 235 | \begin{gathered} 236 | \bgamma(t + T_n + \epsilon) = \bgamma(t) \\ 237 | \bgamma(t + \epsilon) = \bgamma(t) 238 | \end{gathered} 239 | \] 240 | 241 | Since, it's true \(\forall \epsilon > 0 \implies \bgamma\) is constant. 242 | %\[\implies \dfrac{d \bgamma}{dt} =\lim\limits_{\epsilon \to 0} \dfrac{\bgamma(t + \epsilon) - \bgamma(t)}{\epsilon} = 0 \] 243 | Hence, a contradiction (\(\bgamma\) is non-constant.). So, our assumption was false. 244 | \end{proof} 245 | 246 | \chapter{Curvature} 247 | 248 | \section{What is curvature?} 249 | \begin{definition}[Curvature]{def:curvature} 250 | If \(\bm{\gamma}\) is a unit-speed curve with parameter \(t\), its curvature \(\kappa(t)\) at the point \(\bm{\gamma}(t)\) 251 | is defined to be \(\norm{\ddot{\bm{\gamma}}(t)}\). 252 | 253 | \textbf{Note} we have defined curvature for a unit-speed parametric ony. 254 | \end{definition} 255 | \begin{theorem}{1} 256 | The curvature for any regular curve \(\bm{\gamma}\) is given as 257 | \[\kappa = \dfrac{\norm{ (\dot{\bm{\gamma}}\cdot\dot{\bm{\gamma}}) \ddot{\bm{\gamma}} - (\dot{\bm{\gamma}}\cdot\ddot{\bm{\gamma}}) \dot{\bm{\gamma}}}}{\norm{\dot{\bm{\gamma}}}^4} \] 258 | \end{theorem} 259 | \begin{proposition}{1} 260 | Let \(\bm{\gamma}(t)\) be a regular curve in \(\mathbb{R}^3\). Then its curvature is 261 | \[\kappa = \dfrac{\norm{\ddot{\bm{\gamma}} \times \dot{\bm{\gamma}}}}{\norm{\dot{\bm{\gamma}}}^3}\] 262 | where \(\times\) is our usual vector cross product. 263 | \end{proposition} 264 | 265 | \begin{problem} 266 | Show that, if the curvature \(\kappa(t)\) of a regular curve \(\bm{\gamma}(t) \) is \(>0\) every- 267 | where, then \(\kappa(t)\) is a smooth function of \(t\). Give an example to show 268 | that this may not be the case without the assumption that \(\kappa(t >0)\). 269 | \end{problem} 270 | 271 | \section{2D and 3D curves} 272 | \subsection{Plane curve} 273 | Let \(\bm{\gamma}\) be a unit-speed curve in a plane. And let the tangent vector be 274 | \[\mathbf{t} = \dot{\bm{\gamma}}\] 275 | Note, \(\mathbf{t}\) is a unit-vector. There are two vectors perpendicular to \(\mathbf{t}\); we make a choice by defining \(\mathbf{n}_s\), the \textit{signed unit normal} of \(\bm{\gamma}\), to be the unit vector obtained by rotating \(\mathbf{t}\) anti-clockwise by \(\pi/2 \) 276 | 277 | So, by Proposition \ref{prop:unit-zero}, \(\dot{\mathbf{t}} = \ddot{\bm{\gamma}}\) is perpendicular to \(\mathbf{t}\), and hence parallel to \(\mathbf{n}_s\). Thus, there is a scalar \(\kappa_s\) such that 278 | \[\ddot{\bm{\gamma}}= \kappa_s \mathbf{n}_s\] 279 | \(\kappa_s\) is called the signed curvature of \(\bm{\gamma}\). And since \(\norm{\mathbf{n}_s} = 1\), we have 280 | \[\kappa = \norm{\kappa_s \mathbf{n}_s} = \abs{\kappa_s}\] 281 | 282 | Note, we have defined the signed curvature for unit-speed curve. If \(\bm{\gamma}\) is any regular curve , the we define the above defined parameters to be those of it's unit speed parametrization. 283 | 284 | 285 | Intuitively, since \(\bm{\gamma}(t)\) is assumed to be a unit-speed curve on a plane, then \(\dot{\bm{\gamma}}(t)\) can be measured by angle \(\phi(s)\) such that: 286 | \[\dot{\bm{\gamma}}(s) = (\cos \phi (s), \sin \phi(s)) \tag*{(2)}\] 287 | Also, we can think curvature as rate at which the angle of tangent vector is changing, so if we find the derivative of the above curve then it simply represents a changing angle parameter. 288 | 289 | \begin{proposition}{prop:gg} 290 | Let \(\bm{\gamma}: (\alpha, \beta) \to \mathbb{R}^2\) be a unit speed curve, let \(s_0 \in (\alpha, \beta)\) and let \(\phi _0 \) be such that 291 | \[\dot{\bm{\gamma}}(s_0) = (\cos \phi _0, \sin \phi_0) \] 292 | Then \(\exists!\) smooth function: \(\phi: (\alpha, \beta) \to \mathbb{R}\) such that \(\phi(s_0) = \phi _ 0\) and that Eq. (2) holds for all \(s \in (\alpha, \beta)\) 293 | \end{proposition} 294 | 295 | \begin{definition}[Turning Angle]{def:turning-angle} 296 | The smooth function \(\phi\) in Proposition \ref{prop:gg} is called the turning angle of \(\bm{\gamma}\) 297 | determined by the condition \(\phi(s_0) = \phi_0\) . 298 | \end{definition} 299 | 300 | \begin{proposition}{def:intut-signed-curvature} 301 | Let \(\bm{\gamma}(s)\) be a unit-speed plane curve, and let \(\phi(s)\) be a turning angle for \(\bm{\gamma}\). 302 | Then, 303 | \[\kappa _s = \dfrac{d\phi}{ds}\] 304 | Thus, \textbf{the signed curvature is the rate at which the tangent vector of the 305 | curve} rotates. 306 | \end{proposition} 307 | 308 | \begin{corollary}{} 309 | The total signed curvature of a closed plane curve is an integer multiple of \(2\pi\) 310 | \end{corollary} 311 | \par 312 | 313 | The next result shows that a unit-speed plane curve is essentially determined 314 | once we know its signed curvature at each point of the curve. The meaning of 315 | `essentially’ here is `up to a direct isometry of \(\mathbb{R}^2\) ’, i.e., a map \(M: \mathbb{R}^2 \to \mathbb{R}^2\) of 316 | the form 317 | \[M = T_a \circ \rho_\theta\] 318 | where \(\rho _\theta\) is an anti-clockwise rotation by angle \(\theta\) about the origin, 319 | \[\rho _\theta = (x\cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)\] 320 | and \(T_a\) is a translation by vector \textbf{a} 321 | \[T_a (\mathbf{v}) = \mathbf{v}+ \mathbf{a}\] 322 | for any vector \((x, y) \) and \(\mathbf{v}\) in \(\mathbb{R}^2\) 323 | 324 | 325 | \begin{theorem}{theo:1} 326 | Let \(k : (\alpha, \beta) \to \mathbb{R}\) be any smooth function. Then, there is a unit-speed curve 327 | \(\bm{\gamma}:(\alpha, \beta)\to \mathbb{R}^2\) whose signed curvature is k. 328 | 329 | Further, if \(\tilde{\bm{\gamma}}: (\alpha, \beta)\to \mathbb{R}^2\) is any other unit-speed curve 330 | whose signed curvature is \(k\), there is a direct isometry \(M\) of \(\mathbb{R}^2\) such that 331 | \[\tilde{\bm{\gamma}}(s) = M(\bm{\gamma}(s)) \ \forall s \in (\alpha, \beta) \] 332 | \end{theorem} 333 | 334 | \subsection{Space curve} 335 | 336 | \begin{definition}{def:princ-normal} 337 | Let \(\bm{\gamma}(s)\) be a unit-speed curve in \(\mathbb{R}^3\) , and let \(\mathbf{t} = \dot{\bm{\gamma}}\) be its unit tangent 338 | vector. If the curvature \(\kappa_s\) is non-zero, we define the principal normal of \(\bm{\gamma}\) 339 | at the point \(\bm{\gamma}(s)\) to be the vector 340 | \[\mathbf{n}(s) = \frac{1}{\kappa(s)} \mathbf{\dot{t}}(s)\] 341 | Further, since \(\lVert\mathbf{\dot{t}}\rVert = \kappa\), so \(\mathbf{n}\) is a unit-vector. Therefore by Proposition 342 | \ref{prop:unit-zero}, so \(\mathbf{t}\) and \(\mathbf{n}\) are perpendicular.So 343 | \[\mathbf{b} = \mathbf{t} \times \mathbf{n}\] 344 | is a unit-vector perpendicular to both \(\mathbf{t}\) and \(\mathbf{n}\). The vector \(\mathbf{b}(s)\) is called the 345 | \textit{binormal} vector of \(\bm{\gamma}\) at point \(\bm{\gamma}(s)\). Thus, \(\{ \mathbf{t}, \mathbf{n}, \mathbf{b}\}\) is an 346 | orthonormal basis of \(\mathbb{R}^3\) 347 | , and is \textit{right-handed}. 348 | \end{definition} 349 | 350 | \begin{definition}{def:torison} 351 | From Definition \ref{def:princ-normal} we have 352 | \[\mathbf{b} = \mathbf{t} \times \mathbf{n}\] 353 | Differentiating both sides gives 354 | \[\dot{\mathbf{b}} = \dot{\mathbf{t}} \times \mathbf{n} + \mathbf{t} \times \dot{\mathbf{n}} = \mathbf{t} \times \dot{\mathbf{n}} \tag{(3)}\] 355 | Equation (3) shows 356 | that \(\dot{\mathbf{b}}\) is perpendicular to \(t\). Being perpendicular to both \(\mathbf{t}\) and \(\mathbf{b}\), \(\dot{\mathbf{b}}\) must be 357 | parallel to \(\mathbf{n}\), so 358 | \[\dot{\mathbf{b}} = -\tau \times \mathbf{n}\] 359 | for some scalar \(\tau\) , which is called the \textit{torsion} of \(\bm{\gamma}\). 360 | 361 | \textbf{Note} that the torsion 362 | is only defined if the curvature is non-zero. 363 | \end{definition} 364 | 365 | \begin{definition}{prop:torison} 366 | Let \(\bm{\gamma}(t)\) be a regular curve in \(\mathbb{R}^3\) with nowhere-vanishing curvature. Then, 367 | denoting \(\frac{d}{dt}\) by a dot, its torsion is given by 368 | \[\tau = \dfrac{(\dot{\bm{\gamma}} \times \ddot{\bm{\gamma}})\cdot\dddot{\bm{\gamma}}}{\norm{\dot{\bm{\gamma}} \times \ddot{\bm{\gamma}}}^2}\] 369 | \end{definition} 370 | 371 | \begin{proposition}{1} 372 | Let \(\bm{\gamma}\) be a regular curve in \(\mathbb{R}\) with nowhere vanishing curvature (so that the 373 | torsion \(\tau\) of \(\bm{\gamma}\) is defined). Then, the image of \(\bm{\gamma}\) is contained in a plane if and 374 | only if \(\tau\) is zero at every point of the curve. 375 | \end{proposition} 376 | \begin{theorem}{theo:frenet-serret} 377 | 378 | Let \(\bm{\gamma}\) be a unit-speed curve in \(\mathbb{R}\) with nowhere vainshing curvature. Then, 379 | \[ 380 | \begin{gathered} 381 | \mathbf{\dot{t}} = \kappa \mathbf{n}\\ 382 | \mathbf{\dot{n}} = -\kappa \mathbf{t} + \tau \mathbf{b}\\ 383 | \mathbf{\dot{b}} = \tau \mathbf{n} 384 | \end{gathered} 385 | \] 386 | 387 | The above equation is called as \textit{Frenet-Serret equations}. Notice that the matrix 388 | \[ 389 | \left( 390 | \begin{array}{c c c} 391 | 0 & \kappa & 0 \\ 392 | -\kappa & 0 & \tau\\ 393 | 0 & -\tau & 0 394 | \end{array} 395 | \right) 396 | \] 397 | which is the matrix of linear transformation is a \textit{skew-symmetric}. 398 | \end{theorem} 399 | 400 | \begin{proposition}{1} 401 | Let \(\bm{\gamma}\) be a unit-speed curve in \(\mathbb{R}^3\) with constant curvature and zero torsion. 402 | Then, \(\bm{\gamma}\) is a parametrization of (part of) a circle. 403 | \end{proposition} 404 | 405 | \begin{theorem}{1} 406 | Let \(\bm{\gamma}(s)\) and \(\tilde{\bm{\gamma}}(s)\) be two unit-speed curves in \(\mathbb{R}^3\) with the same curvature 407 | \(\kappa(s) >0 \) and the same torsion \(\tau(s)\) for all \(s\). Then, there is a direct isometry 408 | \(M\) of \(\mathbb{R}^3\) such that 409 | \[\tilde{\bm{\gamma}}(s) = M(\bm{\gamma}(s)) \quad \forall s\] 410 | Further, if \(k\) and \(t\) are smooth functions with \(k > 0\) everywhere, there is a 411 | unit-speed curve in \(\mathbb{R}^3\) whose curvature is \(k\) and whose torsion is \(t\). 412 | \end{theorem} 413 | 414 | \chapter{Global properties of curves} 415 | 416 | \section{Simple closed curves} 417 | 418 | \begin{definition}{def:closed-curves} 419 | A \textit{simple closed curve} in \(\mathbb{R}^2\) is a closed curve in \(\mathbb{R}^2\) that has no self-intersections. 420 | \end{definition} 421 | 422 | \begin{theorem}[Jordan Curve Theorem]{thm:jordan-curve-theorem} 423 | The complement of the image of \(\bgamma\) 424 | (i.e., the set of points of \(\mathbb{R}^2\) that are not in the image of $\bgamma$) is the disjoint 425 | union of two subsets of \(\mathbb{R}^2\), denoted by \(\mathrm{int}( \bgamma)\) and \(\mathrm{ext}(\bgamma)\), with the following 426 | properties: 427 | 428 | \begin{itemize} 429 | \item \(\mathrm{int}(\bgamma)\) is bounded, i.e. it is contained in the circle of sufficiently large radius. 430 | \item \(\mathrm{ext}(\bgamma)\) is unbounded. 431 | \item Both the regions \(\mathrm{int}(\bgamma)\) and \(\mathrm{ext}(\bgamma)\) are connected, i.e they have the property that any two points in the same region can be joined by a curve contained entirely in the region. 432 | \end{itemize} 433 | 434 | \end{theorem} 435 | 436 | \begin{theorem}[Hopf’s Umlaufsatz]{def:Hopf-Umlaufsatz} 437 | The total signed curvature of a simple closed curve in \(\mathbb{R}^2\) is \(\pm 2\pi\). 438 | \end{theorem} 439 | 440 | 441 | \section{The isoperimetric inequality} 442 | 443 | 444 | \begin{definition}[Area of a curve]{} 445 | The area contained by a simple closed curve \(\bgamma\) is 446 | 447 | \[\mathcal{A}(\bgamma) =\int_{\mathrm{int}(\bgamma)} dx dy\] 448 | 449 | \end{definition} 450 | 451 | \begin{theorem}[Green's Theorem]{thm:green-func} 452 | \textit{Let $f (x, y)$ and $g(x, y)$ be smooth functions (i.e., functions 453 | with continuous partial derivatives of all orders), and let \(\bgamma\) be a positively- 454 | oriented simple closed curve. Then,} 455 | \[\int_{\mathrm{int}(\bgamma)} \left(\dfrac{\partial g}{\partial x} -\dfrac{\partial f}{\partial y} \right)dxdy = \int_{\bgamma} f(x, y) dx + g(x, y)dy\] 456 | \end{theorem} 457 | 458 | \begin{proposition}{} 459 | If \(\bgamma(t) = (x(t), y(t))\) is a positively-oriented simple closed curve in \(\mathbb{R}^2\) with 460 | period \(T\), then 461 | \[\mathcal{A}(\bgamma) = \frac{1}{2} \int_{0}^{T} (x \dot{y} - y \dot{x}) dt\] 462 | \end{proposition} 463 | 464 | \begin{theorem}[Isoperimetric Inequality]{} 465 | Let \(\bgamma\) be a simple closed curve, let \(l(\bgamma)\) be its length and let \(\mathcal{A}(\bgamma)\) be the area 466 | contained by it. Then, 467 | \[\mathcal{A}(\bgamma) \leq \frac{1}{4\pi} l(\bgamma) ^2\] 468 | and equality holds if and only if \(\bgamma\) is a circle. 469 | \end{theorem} 470 | \section{The four vertex Theorem} 471 | 472 | \begin{definition}[Vertex]{} 473 | A \textit{vertex }of a curve \(\bgamma(t)\) in \(\mathbb{R}^2\) is a point where its signed curvature \(\kappa_s\) has a stationary point, i.e., where 474 | \(\frac{d \kappa_s}{d t} = 0\). 475 | \end{definition} 476 | 477 | \begin{theorem}[Four Vertex Theorem]{} 478 | Every convex simple closed curve in \(\mathbb{R}^2\) has at least four vertices. 479 | \end{theorem} 480 | 481 | 482 | \chapter{3D Surfaces} 483 | \section{What is a surface?} 484 | \paragraph{} 485 | As in the case of curves, we make two definitions of the concept of surface. One of them (regular 486 | surface) emphasizes the fact that a surface, as we think of it, is a set of points. 487 | The other (parametrised surface) emphasizes the parametrization of the surface. While these two concepts 488 | were similar in the case of curves (every regular curve can be covered with a single parametrization, 489 | so it is a parametrised regular curve), they are different for surfaces: a sphere, for example, is a 490 | regular surface, but not a parametrised regular surface. We will further show that we need two parametric maps 491 | to describe the whole surface of a sphere and to keep it consistent with the other properties such as tangents amd normals etc. 492 | 493 | 494 | \begin{definition}[Surface]{def:surface} 495 | A subset \(S\) of \(\mathbb{R}^3\) is a surface if, for every point \(\mathbf{p} \in S\), there is an open set \(U 496 | \subseteq \mathbb{R}^2\) 497 | and an open set \(W \subseteq \mathbb{R}^3\) containing \(\mathbf{p}\) such that \(S\cap W\) is homeomorphic 498 | to \(U\) i.e. 499 | \[\bsigma: U\subset \R{2} \to S\cap W \] 500 | such that \(\exists \ (u, v) \in U : \bsigma(u, v) = \mathbf{p}\). 501 | \end{definition} 502 | \begin{description} 503 | \item[1] A subset of \(S\) of the form \(S\cap W\), where \(W\) is an open subset 504 | of \(\mathbb{R}^3\) , is called an open subset of \(S\). 505 | \item[2] A continuous bijective function between two topological space (i.e. shapes here) is termed 506 | as \textit{homeomorphism} 507 | \end{description} 508 | 509 | \begin{definition}[Surface Patch]{def:patch} 510 | A homeomorphism \(\bsigma : U \to S \cap W\) as in 511 | previous definition is called a \textit{surface patch or parametrization} of the open subset 512 | \(S\cap W\) of \(S\). 513 | \end{definition} 514 | \begin{description} 515 | \item[] A surface is some subset of R3 516 | that can be covered by surface patches. Each surface patch looks like a (maybe deformed) piece 517 | of \(\R{2}\). 518 | \end{description} 519 | 520 | 521 | \section{Smooth Surfaces} 522 | \begin{description} 523 | \item[Smooth Functions] If \(U\) is an open subset of \(\mathbb{R}^m\) , we say that a map $f : U \to \mathbb{R}^n$ is smooth 524 | if each of the \(n\) components of \(f\), which are functions \(U \to \mathbb{R}\), have continuous 525 | partial derivatives of all orders. 526 | \end{description} 527 | 528 | \begin{definition}[Regular Surface Patch]{def:regular-patch} 529 | A surface patch \(\bsigma : U \to \mathbb{R}^3\) is called regular if it is smooth and the vectors 530 | \(\bsigma_u\) and \(\bsigma_v\) are linearly independent at all points \((u, v)\in \mathbb{R}^2\). Equivalently, \(\bsigma\) 531 | should be smooth and the vector product \(\bsigma_s \times \bsigma_v\) should be non-zero at every 532 | point of \(U\) . 533 | \end{definition} 534 | \begin{definition}[Allowable Patch]{def:allow-patch} 535 | If \(S\) is a surface, an allowable surface patch for \(S\) is a regular surface patch 536 | \(\bsigma : U \to \mathbb{R}^3\) such that \(\bsigma\) is a homeomorphism from \(U\) to an open subset of \(S\). 537 | \end{definition} 538 | 539 | \begin{definition}[Smooth Surface]{def:smooth-surface} 540 | A smooth surface is a surface \(S\) such that, for any point \(\mathbf{p}\in S\) there is an 541 | allowable surface patch \(\bsigma\) as above such that \(\mathbf{p}\in \bsigma(U)\). 542 | \end{definition} 543 | \begin{definition}[Atlas]{def:atlas} 544 | A collection \(\mathcal{A}\) of 545 | allowable surface patches for a surface \(S\) such that every point of \(S\) is in the 546 | image of at least one patch in \(\mathcal{A}\) is called an atlas for the smooth surface \(\mathcal{A}\). 547 | \end{definition} 548 | \begin{proposition}{} 549 | The transition maps of a smooth surface are smooth. 550 | \end{proposition} 551 | 552 | \begin{proposition}{prop:regular-surface-patch} 553 | Let \(U\) and $\tilde{U}$ be open subsets of \(\mathbb{R}^2\) and let \(\bsigma : U \to \mathbb{R}^3\) be a regular surface 554 | patch. Let \(\Phi : \tilde{U} \to U\) be a bijective smooth map with smooth inverse map 555 | \(\Phi^{-1} : U\to \tilde{U} \). Then, \(\tilde{\sigma} = \sigma\circ \Phi : \tilde{U}\to \mathbb{R}^3\) is a regular 556 | surface patch. 557 | \end{proposition} 558 | 559 | \section{Smooth Map} 560 | 561 | In this section, will define the notion of smooth map \(f: S_1 \to S_2\), where \(S_1\) and \(S_2\) are smooth surfaces. 562 | \begin{definition}[Diffeomorphisms]{def:diffeomorphisms} 563 | Smooth maps \(f: S_1 \to S_2\), which are bijective and whose inverse map 564 | \(f^{-1}: S_2 \to S_2\) is smooth are called diffeomorphisms. 565 | \end{definition} 566 | 567 | \begin{proposition}{} 568 | Let \(f: S_1 \to S_2\) be a diffeomorphisms. If \(\bsigma_1\) is a allowable 569 | surface patch on \(S_1\), then \(f\circ \bsigma_1\) is an allowable surface 570 | patch on \(S_2\). 571 | \end{proposition} 572 | 573 | \section{Tangents and derivatives} 574 | 575 | \begin{definition}[Tangent]{def:tangent-3d} 576 | A tangent vector to a surface \(S\) at a point \(\bm{p}\in S\) is the tangent vector 577 | at \(\bm{p}\) of a curve in \(S\) passing through \(\bm{p}\). The tangent space 578 | \(T_{\mathbf{p}} S\) of \(S\) at \(\mathbf{p}\) is the set of 579 | all tangent vectors to \(S\) at \(\mathbf{p}\). 580 | \end{definition} 581 | 582 | \begin{proposition}{} 583 | Let \(\bsigma: U \to \mathbb{R}^3\) be a patch of a surface \(S\) containing a point 584 | \(\mathbf{p}\in S\), let \((u, v)\) be coordinates in \(U\). The tangent space to \(S\) 585 | at \(\mathbf{p}\) is a vector subspace or \(\mathbb{R}^3\) spanned by vector \(\bsigma_u\) 586 | and \(\bsigma_v\) ((the derivatives are evaluated at the point 587 | $(u_0, v_0 ) \in U$ such that $\bsigma(u_0 , v_0) = \mathbf{p}$) 588 | \end{proposition} 589 | Since, by above proposition we can see that the tangent space is 2D and will be called 590 | \textit{tangent plane} form now on.\par 591 | 592 | 593 | \begin{description} 594 | \item[Remember] This text only contains important theorems and definitions from the textbook \textbf{Elementary Differential Geometry}. 595 | And some of the problems for the book are also discussed (non-trivial problems). 596 | \end{description} 597 | \end{document} 598 | 599 | 600 | -------------------------------------------------------------------------------- /MTH102: Real-analysis/MTH102_-_Real_analysis.tex: -------------------------------------------------------------------------------- 1 | \documentclass{notes} 2 | 3 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 | 5 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6 | \usepackage{color-env} 7 | \usepackage[english]{babel} 8 | \usepackage{amssymb,amsmath,amsfonts,commath} %%% for maths 9 | \usepackage{thmtools,framed} 10 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11 | \usepackage{enumerate} %% it make lists 12 | %\usepackage{semantic} 13 | \usepackage{subfig} %%%% for two figures side by side 14 | \usepackage[colorinlistoftodos,prependcaption]{todonotes} %% to write notes 15 | \usepackage[toc,page]{appendix} 16 | \renewcommand\qedsymbol{$\blacksquare$} 17 | \graphicspath{{img/}} 18 | 19 | \begin{document} 20 | 21 | \begin{titlepage} % Suppresses headers and footers on the title page 22 | 23 | \centering % Centre everything on the title page 24 | 25 | \scshape % Use small caps for all text on the title page 26 | 27 | \vspace*{\baselineskip} % White space at the top of the page 28 | 29 | %------------------------------------------------ 30 | % Title 31 | %------------------------------------------------ 32 | 33 | \rule{\textwidth}{1.6pt}\vspace*{-\baselineskip}\vspace*{2pt} % Thick horizontal rule 34 | \rule{\textwidth}{0.4pt} % Thin horizontal rule 35 | 36 | \vspace{0.75\baselineskip} % Whitespace above the title 37 | 38 | {\huge REAL\\ ANALYSIS\\} % Title 39 | 40 | \vspace{0.75\baselineskip} % Whitespace below the title 41 | 42 | \rule{\textwidth}{0.4pt}\vspace*{-\baselineskip}\vspace{3.2pt} % Thin horizontal rule 43 | \rule{\textwidth}{1.6pt} % Thick horizontal rule 44 | 45 | \vspace{2\baselineskip} % Whitespace after the title block 46 | 47 | %------------------------------------------------ 48 | % Subtitle 49 | %------------------------------------------------ 50 | 51 | \LARGE{MTH102~\nocite{abb01understanding}\nocite{introduction}\nocite{wiki}} 52 | 53 | \vspace*{3\baselineskip} % Whitespace under the subtitle 54 | 55 | %------------------------------------------------ 56 | % Editor(s) 57 | %------------------------------------------------ 58 | 59 | 60 | \vspace{0.5\baselineskip} % Whitespace before the editors 61 | 62 | %{\scshape \LARGE Prof. Chandrakant Aribam\\ } % Editor list 63 | 64 | \vspace{0.5\baselineskip} % Whitespace below the editor list 65 | 66 | %\textit{\Large IISER, Mohali} % affiliation 67 | 68 | \vfill % Whitespace between editor names and publisher logo 69 | 70 | %------------------------------------------------ 71 | % Author 72 | %------------------------------------------------ 73 | 74 | 75 | \vspace{0.3\baselineskip} % Whitespace under the publisher logo 76 | 77 | 78 | {\large Edited by\\ Aditya Dev} 79 | 80 | \end{titlepage} 81 | \tableofcontents 82 | %\newpage 83 | \chapter{The set of $\mathbb{R}$ and some of it's properties} 84 | 85 | \section{The set of $\mathbb{R}$} 86 | 87 | \begin{definition}[Completeness Axiom]{} 88 | Every nonempty subset $S$ of $\mathbb{R}$ that is bounded above has a least upper bound. In other words, $sup\ $S exists and is a real number 89 | \end{definition} 90 | \begin{corollary}{cor:inf} 91 | Every nonempty subset S of $\mathbb{R}$ that is bounded below has a greatest lower bound. In other words, inf S exists and is a real number. 92 | 93 | \end{corollary} 94 | % Proof for above corollary 95 | \begin{proof} 96 | Let S be a set which is bounded below i.e. $\forall s \in S$ , $s \geq m$ for some $m \in \mathbb{R}$ .Take the set $ - S = \{ - s\mid s \in S\}$ . 97 | Therefor $ \forall u \in - S\ \Rightarrow \{u\leq - m\} $ or 98 | $$ \Rightarrow\ u\leq sup(- S) \leq - m\ \text{ \{completeness axiom\}}$$ 99 | $$\Rightarrow\ - s \leq sup(- S) \leq - m $$ 100 | $$ \Rightarrow\ m \leq - sup(- S) \leq s $$ 101 | Let $ \exists\ \lambda \in \mathbb{R}$ such that $-sup(- S) \leq \lambda \leq s$ which implies $- s \leq - \lambda \leq sup(- S) $. Since, $ - \lambda $ cannot be the supremum of S. 102 | 103 | Therefore,$\framebox{inf(S) = -sup(-S)} .$ 104 | 105 | \end{proof} 106 | \begin{theorem}[Archemedian Property]{thm:archemedian} 107 | If a $\geq$ 0 and b $\geq$ 0, then $\exists \ n\in \mathbb{N} $ such that $n a \geq b$ . Or in other words we can say that the set of natural numbers is not bounded above. 108 | \end{theorem} 109 | \begin{proof} 110 | 111 | Assume the Archimedean property fails. Then there exist a $\geq$ 0 and 112 | b $\geq$ 0 such that na $\leq$ b $\forall\ n \in \mathbb{N}$. In particular, b is an upper 113 | bound for the set $S= \{ na \mid n \in \mathbb{N} \}$. Let $s_0 = sup S$; this is where we 114 | are using the completeness axiom. Since a $\geq$ 0, we have $s_0 \leq s_0 + a$, 115 | so $s_0 $-$ a\leq s_0$. Since 116 | $s_0$ is the least upper bound for S, $s_0-a$ cannot be an upper bound 117 | for S. It follows that $s_0 - a\leq n_0a$ for some $n_0 \in \mathbb{N}$ This implies 118 | $s_0 \leq (n_0 + 1)a$. Since $(n_0 + 1)a$ is in S, $s_0$ is not an upper bound 119 | for S and we have reached a contradiction. Our 120 | assumption that the 121 | Archimedean property fails was wrong. 122 | \end{proof} 123 | \chapter{Sequences} 124 | \section{Convergence} 125 | 126 | \begin{definition}[Convergence]{def:convergence} 127 | A sequence $\{ s_n \}$ is said to $converge$ to a real number ``s'' provided that 128 | 129 | for each $\epsilon > 0$ there exist as number N such that 130 | \begin{equation} 131 | n > N\ \mathrm{implies} \ |s_n - s|<\epsilon 132 | \end{equation} 133 | 134 | If $(s_n)$ converges to s, we will write $\lim\limits_{n \to \infty} s_n = s$ , or $s_n \to s$. The 135 | number s is called the limit of the sequence $(s_n)$ . A sequence that 136 | does not converge to some real number is said to diverge (Is it?) 137 | \end{definition} 138 | \begin{problem} 139 | 140 | Let $(s_n)$ be a sequence of nonnegative real numbers and suppose 141 | $s = \lim s_n$. Note $s \geq 0$ . Prove $\lim \sqrt[2]{s_n} = \sqrt[2]{s}$ 142 | \end{problem} 143 | \begin{proof}[Solution] 144 | 145 | \textbf{Case 1:}($s > 0$) Let $\epsilon >0 $. Since $\lim s_n = s\ \Rightarrow\ \exists\ N\ \mathrm{such\ that}$ 146 | \begin{equation} 147 | n>N\ \mathrm{implies}\ |s_n - s|<\epsilon 148 | \end{equation} 149 | 150 | \hangindent=1.5cm Now, $n>N$ implies 151 | 152 | $$ 153 | |\sqrt{s_n} - \sqrt{s}| = \frac{|s_n - s|}{\sqrt{s_n} + \sqrt{s}} \leq \frac{|s_n - s|}{\sqrt{s}} < \frac{\sqrt{s}\epsilon }{\sqrt{s}} = \epsilon 154 | $$ 155 | 156 | \hangindent=1.5cm \textbf{Case 2:}($s=0$) Since $s_n > 0$ $\Rightarrow$ $|s_n - 0|<\epsilon\ \forall\ \epsilon>0$ . 157 | Take $\epsilon\ \text{to be}\ \epsilon^2$ . 158 | 159 | $$\Rightarrow n>N\ \mathrm{implies}\ |s_n|<\epsilon^2 \Rightarrow |\sqrt{s_n}|<\epsilon$$ 160 | $$ \Rightarrow |\sqrt{s_n} - 0|<\epsilon $$ 161 | \hangindent=1.5cm So, $\lim \sqrt{s_n} = 0$ 162 | \end{proof} 163 | \begin{problem} 164 | Prove that: 165 | 166 | \begin{enumerate} %%% for horizontal alignmeint in enumerate 167 | \item 168 | $\lim [\sqrt{n^2 +1} - n] = 0$ 169 | \item 170 | $\lim [\sqrt{4n^2 +n}- 2n] = 1/4$ 171 | \end{enumerate} 172 | \end{problem} 173 | \begin{proof}[Solution] 174 | I think it's enough to discuss the strategy because the reader should be able to proceed backwards. 175 | \begin{enumerate} 176 | 177 | \item 178 | Since, $\sqrt{n^2 +1} -n>0$ we can simply remove the modulus sign and write 179 | $ \sqrt{n^2+1} - n <\epsilon$ 180 | $$ \Rightarrow n^2 +1< (\epsilon+n)^2$$ 181 | $$ \Rightarrow n^2 +1< \epsilon^2 + n^2 + 2n\epsilon$$ 182 | $$\Rightarrow \frac{1-\epsilon^2}{2\epsilon}\sqrt{4n^2 +n}- 2n$ by simply assuming the inequality and we'll get the result that $1/4>0$ which is indeed true or if we assume other inequality it'll lead to the contradiction. 186 | 187 | Since, $\frac{1}{4}-(\sqrt{4n^2 +n}- 2n)>0$ we can write 188 | $$\Rightarrow \frac{1}{4}-(\sqrt{4n^2 +n}- 2n) < \epsilon$$ 189 | $$\Rightarrow \frac{1}{4}-\epsilon + 2n < \sqrt{4n^2 +n} $$ 190 | To square both the sides $\frac{1}{4}-\epsilon + 2n >0,\ \forall n\in\mathbb{N}$ which is if $\epsilon<9 / 4$. Squaring both sides and cancelling the similar terms we get 191 | $$ \Rightarrow (\epsilon-1 / 4)^20$ or $\epsilon<3/4$ which also satisfies the above condition of $\epsilon<9/4$. 193 | Since,we have to prove it for small enough $\epsilon$. So, for bigger epsilon it's automatically true i.e. if we prove for $\epsilon<3 / 4$ then it is true for $\epsilon>9 / 4$. So, 194 | $$\Rightarrow \frac{(\epsilon-1 / 4)^2}{(1-4(\epsilon-1 / 4)^2)}N\ \text{implies}\ |s_n - s|<1 $$ 218 | From triangular inequality it implies $|s_n|<|s|+1$ for $n > N$. 219 | Take M = $max\{ |s_1|,|s_2|,|s_3|,\ldots|s|+1\}$ .Then we have $|s_n|\leq M$ for all $n\in N$, so $(s_n)$ is bounded. 220 | 221 | The choice of $\epsilon$ is arbitrary 222 | \end{proof} 223 | 224 | \begin{theorem}{} 225 | If the sequence $(s_n)$ converges to s and k is in $\mathbb{R}$, then the sequence 226 | $(ks_n)$ converges to $(ks)$. That is, $\lim (ks_n) = k \cdot \lim s_n$. 227 | \end{theorem} 228 | 229 | \begin{theorem}{} 230 | If $s_n$ converges to s and $t_n$ converges to t, then $(s_n +t_n)$ converges to $(s+t)$ 231 | $$\lim (s_n + t_n ) = (s+t)$$ 232 | \end{theorem} 233 | \begin{theorem}{} 234 | If $s_n$ converges to s and $t_n$ converges to t, then $(s_n \cdot t_n)$ converges to $(s\cdot t)$ 235 | $$\lim (s_n \cdot t_n ) = (s \cdot t)$$ 236 | 237 | The theorem can be proved using the identity $(a+b)^2 = a^2 + b^2 +2ab$ take $(s_n+t_n)\ and\ (s_n-t_n)$ and proceed. (Wait did I prove $\lim (a_n)^2 = a^2$ if $a_n \to a$. Well, if not then it's easy to prove.) 238 | 239 | \end{theorem} 240 | 241 | \begin{theorem}{} 242 | If $s_n$ converges to s. Then $1/s_n$ converges to $1/s$ for $(s \not = 0)$ 243 | \end{theorem} 244 | \begin{proof} 245 | We begin by observing that 246 | $$ \abs{\frac{1}{s_n}- \frac{1}{s}} = \frac{\abs{s_n- s}}{\abs{ s_n s}}$$ 247 | Because $(s_n) \rightarrow s$, we can make the preceding numerator as small as we like by 248 | choosing n large. The problem comes in that we need a worst-case estimate on 249 | the size of $1/(|s||s_n|)$. Because the $s_n$ terms are in the denominator, we are no 250 | longer interested in an upper bound on $|s_n|$ but rather in an inequality of the 251 | form $|s_n| \geq \delta > 0$. This will then lead to a bound on the size of $1/(|s||s_n|)$. 252 | The trick is to look far enough out into the sequence $(s_n)$ so that the terms 253 | are closer to s than they are to 0. Consider the particular value $\epsilon = |s|/2$. 254 | Because $(s_n) \rightarrow s$, there exists an $N_1$ such that $|s_n - s| < |s|/2$ for all $n \geq N_1$. 255 | This implies $|s_n| > |s|/2$. 256 | Next, choose $N_2$ so that $n \geq N_2$ implies 257 | $ |s_n- s| < |s| $ 258 | $$ |s_n - s| < \frac{\epsilon \cdot s^2}{2}$$ 259 | Finally, if we let $N = max\{N_1, N_2\}$, then $n \geq N$ implies. 260 | $$ \abs{\frac{1}{s_n}- \frac{1}{s}} = \abs{s_n - s}\frac{1}{\abs{s_n s}}< \frac{\epsilon s^2}{2} \frac{1}{|s|\frac{|s|}{2}} = \epsilon$$ 261 | \end{proof} 262 | 263 | \begin{theorem}{} 264 | If $s_n$ converges to s and $t_n$ converges to t (t $\not = 0$), then $\frac{s_n}{t_n}$ converges to $(\frac{s}{t})$ 265 | $$\lim \frac{s_n}{t_n} = \frac{s}{t}$$ 266 | \end{theorem} 267 | \begin{proof} 268 | The proof is trivial and is left as an exercise for the readers 269 | \end{proof} 270 | \begin{theorem}{} 271 | If $s_n < t_n$ then $\lim s_n \leq \lim t_n$. 272 | \end{theorem} 273 | \begin{proof} 274 | Let $s = \lim s_n $ and $t = \lim t_n$. Let $h_n = t_n -s_n > 0 $ . So, $h = \lim h_n \geq 0$ (Why?). Let's take $h < 0 \Rightarrow - h > 0$. So, there exists a $N$ such that $n >N$ implies $ |h_n -h | < -h \Rightarrow h_n < 0$. Contradiction! So, $h \geq 0 \Rightarrow t - s \geq 0$. 275 | \end{proof} 276 | \begin{theorem}[$($\textbf{Basic Examples}$)$]{def:basic-example} 277 | 278 | \begin{enumerate}[\bf a.] %%\usepackage{enumerate} 279 | \item 280 | $ \lim_{n \to \infty} \frac{1}{n^p} = 0\ \mathrm{for} \ p>0 $. 281 | \item 282 | $\lim_{n\to\infty} a^n = 0\ \mathrm{if}\ |a|<1$. 283 | \item 284 | $\lim_{n\to\infty } n^{\frac{1}{n}} = 1$. 285 | \item 286 | $\lim_{n \to \infty} a^{\frac{1}{n}} = 1,\ a>0 $. 287 | \end{enumerate} 288 | \end{theorem} 289 | 290 | %\begin{proof} 291 | % \begin{enumerate}[\bf a.] %%\usepackage{enumerate} 292 | 293 | % \item 294 | % Take $N = (\frac{1}{\epsilon})^{\frac{1}{p}}$, it implies for all $n>N\Rightarrow$ 295 | % $n>(\frac{1}{\epsilon})^{\frac{1}{p}}$ \par 296 | % or $\frac{1}{n^p}<\epsilon$. Since, $\frac{1}{n^p}>0$ implies $|\frac{1}{n^p}|<\epsilon$ 297 | 298 | % \item 299 | % Poof left 300 | % \item 301 | % Poof left 302 | % \item 303 | % Poof left 304 | % \end{enumerate} 305 | 306 | %\end{proof} 307 | \begin{problem} 308 | For a sequence $(s_n)$ of positive real numbers, we have $\lim s_n = +\infty $ 309 | if and only if $ \lim ( \frac{1}{s_n})=0$. 310 | 311 | \end{problem} 312 | 313 | \begin{proof} 314 | We need to prove that 315 | $$\lim s_n =+\infty \ \Rightarrow\ \lim (1/s_n)=0$$ 316 | and 317 | $$\lim (1/s_n)=0 \ \Rightarrow\ \lim s_n =+\infty$$ 318 | \begin{enumerate} 319 | \item 320 | Since $\lim s_n = +\infty$ so for every $n>N$ there exist a M such that $s_n >M$. Take $M= 1/\epsilon,\ \epsilon >0$ 321 | $$\Rightarrow\quad s_n>M=1/\epsilon $$ 322 | $$\Rightarrow\quad \epsilon>\frac{1}{s_n}>0$$ 323 | $$\Rightarrow\quad\abs{\frac{1}{s_n}-0} <\epsilon$$ 324 | \item 325 | Workout the above proof backwards by assuming $\epsilon=1/M$ 326 | \end{enumerate} 327 | \end{proof} 328 | 329 | \begin{problem} 330 | Let $s_1 = 1$ and $s_{n+1} = \sqrt{1+s_n}$ .\\ 331 | Assume that the sequence converge. Prove that the sequence converges to $\frac{1}{2} (1+\sqrt{5})$ 332 | \end{problem} 333 | \begin{proof} 334 | Let the sequence converge to s.So, as $n\to \infty$ $s_{n+1} = s_n = s$ therefore\todo[color=white]{In this problem we have assumed that the limit exist} 335 | $$\Rightarrow s_{n+1}=\sqrt{1+s_n}$$ 336 | $$\Rightarrow s_{n+1} ^2 - s_n -1 = 0$$ 337 | $$\Rightarrow s^2 -s-1=0\ \mathrm{as}\ n\to \infty$$ 338 | 339 | So, ${s = \frac{1}{2}(1+\sqrt{5})}$ 340 | \end{proof} 341 | \paragraph{} 342 | \begin{theorem}{} 343 | Let $\{a_n\}$ be a sequence of positive numbers such that $\lim\limits_{n \to \infty} a_n = L$. Prove that $\lim\limits_{n \to \infty} \sqrt[n]{(a_1a_2\cdots a_n)}= L$ 344 | \end{theorem} 345 | %\begin{proof} 346 | % Let $s_n = (a_1 a_2 a_3 \cdots a_n)^{1 \over n}$. So, we need to prove that $ \lim\limits_{n \to \infty} s_n = L$ (observe that L is positive). So, let's make a strategy. 347 | 348 | % Strategy: 349 | % We need to prove that for every $\epsilon>0$ there exists a $N$ such that $n>N$ implies: 350 | % $$ |s_n -L | <\epsilon$$. 351 | % Using triangular inequality we can show that 352 | % $$ |s_n -L | = |(a_1 a_2 a_3 \cdots a_n)^{1 \over n} - L| \leq |(a_1 a_2 a_3 \cdots a_n)|^{1 \over n} + L $$ 353 | 354 | % From theorem \ref{bounded sequences} we also have $ |a_n| < M$ for some $ M \in \mathbb{R}^{ + }$, which implies $s_n = |(a_1 a_2 a_3 \cdots a_n)|^{1 \over n} \leq (M^n)^{1 \over n} = M$. This , shows that $s_n $ is a bounded sequence. 355 | % So,$$ |(a_1 a_2 a_3 \cdots a_n)^{1 \over n} - L| \leq |(a_1 a_2 a_3 \cdots a_n)|^{1 \over n} + |L| \leq M + |L| $$ 356 | % Since, $ |s_n -L | $ is always less than $M+L$. So, proving it for $ \epsilon < M+L$ will be sufficient. 357 | %\end{proof} 358 | 359 | \begin{theorem}{} 360 | If $\lim \abs{{a_{n+1} \over a_n}}$ exists $[$and equals L$]$, then $\lim (a_n)^ {1 \over n}$ exists $[$ and 361 | equals L $] $. 362 | 363 | Also, deduce $\lim_{n \to \infty} {n \over (n!)^{1 \over n}}$ 364 | \end{theorem} 365 | \begin{proof} 366 | Define the sequence $\{b_n\}$ by $b_1 = a_1$ and $b_n ={a_n \over a_{n-1}}$ 367 | for $n\geq 2$. Since $ \lim_{n \to \infty}{a_n \over a_{n-1}} = L$ , 368 | we have $\lim_{n\to \infty} b_n = L$. Note that $a_n = (b_1b_2\ldots b_n)$. Applying the above Theorem to the sequence ${b_n}$, 369 | we get 370 | : 371 | 372 | $$ \lim_{n \to \infty } (a_n)^ {1 \over n} = \lim\limits_{n \to \infty}(b_1b_2\ldots b_n)^{1 \over n} = L $$ 373 | 374 | Now, let $a_n = {n^n \over n!} $. Note that 375 | 376 | $$ \lim_{n \to \infty } {a_n+1 \over a_n} = \lim_{n \to \infty } \frac{{(n+1)^ {(n+1)} \over (n+1)!} }{{n^n \over n!}} = \lim_{n \to \infty } \frac{{(n+1)^ {(n+1)} \over (n+1)} }{n^n} = \lim_{n \to \infty } \left( n+1 \over n\right)^n = \lim_{n \to \infty } \left(1+{1\over n}\right)^n = e$$ 377 | By the conclusion above, we have: 378 | $$ \lim_{n \to \infty }{n^n \over (n!)^{1 \over n}} = \lim_{n \to \infty } a_n ^ {1 \over n} =e$$ 379 | \end{proof} 380 | \section{Monotone and Cauchy Sequences} 381 | \begin{theorem}{} 382 | All bounded monotone sequences are convergent 383 | \label{bounded} 384 | \end{theorem} 385 | \begin{proof} 386 | I'll prove it for monotonically decreasing sequences\\ 387 | Let $(s_n)$ be a bounded decreasing sequence i.e. $\forall\ n \in\mathbb{N}\ [s_{n+1}0$ there exist a N such that $s_N < \lambda+\epsilon$. Also, the sequence is deceasing so $\lambda - \epsilon<\lambda \leq s_{n+1}\leq s_n\leq\lambda<\lambda + \epsilon\ $ 390 | $$ \Rightarrow\ - \epsilonN\}$$ 400 | and 401 | $$\liminf s_n = \lim_{N\to \infty} \inf \{s_n\ :\ n>N\}$$ 402 | 403 | \end{definition} 404 | \paragraph{} 405 | \begin{problem} 406 | Calculate the $\limsup a_n\ \text{and}\ \liminf a_n$ for $a_n = (-1)^n {(n+5) \over n}$ 407 | \end{problem} 408 | \begin{figure}[h!] 409 | \hfill \includegraphics[height=2in,width=3in]{lim_sup_prob.png} \hspace*{\fill} %%% \\hspace \includegraphics{imagefile} \hspace*{\fill} to align iage in center 410 | \caption{$(-1)^n {(n+5) \over n}$ v/s n} 411 | \end{figure} 412 | \begin{proof}[Solution] 413 | Since in the set of subsequences of $a_n$ the least element will be the $\liminf$ and the greatest term would be $\limsup$ so. Since the -ve values are less than the +ve ones and in case of $-1 - {5 \over n}$ the least value for n gives the least of all so,we can say the first negative term of each subsequence will be the element of subsequential infimimum.Therefore, $\lim (-1 - {5 \over n}) = -1$ is the $\liminf$. 414 | Similar argument for $\limsup$ as the greatest of all will be sup of each subsequence and the least value of n will give the largest of all. So, the sequence $ s_n = {(n+5) \over n}$ 415 | will be sequence of subsequential supremum.And $\lim {(n+5) \over n} = 1$. 416 | \end{proof} 417 | 418 | 419 | \begin{theorem}{} 420 | Let $(s_n)$ be a sequence $\mathbb{R}$ 421 | \begin{enumerate}[(i)] 422 | \item 423 | If $\lim s_n$ is defined then: 424 | $$\liminf s_n = \limsup s_n = \lim s_n$$ 425 | \item 426 | If $\liminf s_n = \limsup s_n$ then $\lim s_n$ is defined and $\liminf s_n = \limsup s_n = \lim s_n$ 427 | \end{enumerate} 428 | \end{theorem} 429 | \begin{definition}[\bf $\mathcal{CAUCHY\ SEQUENCE}$]{}\cite{wikicauchy} 430 | A sequence $(s_n)$ of real numbers is said to be Cauchy if: 431 | 432 | for each $\epsilon>0$ there exist a N such : 433 | 434 | \begin{equation*} 435 | \forall\ n,m>N\ \mathit{implies}\ |s_n - s_m|<\epsilon 436 | \end{equation*} 437 | \end{definition} 438 | \begin{figure}[hbt!] 439 | 440 | \subfloat[The plot of a Cauchy sequence $(x_n)$, shown in blue, as $(x_n)$ versus n] 441 | {{\includegraphics[width=2.5in]{cauchy_yes.png} }}% 442 | \qquad 443 | \subfloat[A sequence that is not Cauchy.But is bounded] 444 | {{\includegraphics[width=2.5in]{cauchy_no.png} }}% 445 | \caption{Illustartion for cauchy sequences}% 446 | \end{figure} 447 | \begin{lemma}{} 448 | Convergent sequence are Cauchy 449 | \end{lemma} 450 | \begin{lemma}{} 451 | Cauchy sequence are bounded 452 | \end{lemma} 453 | \begin{theorem}{} 454 | A sequence is a convergent sequence if and only if it is a Cauchy 455 | sequence. 456 | \end{theorem} 457 | \paragraph{} 458 | \begin{problem} 459 | Prove that $\abs{s_{n+1}-s_n} < 2^{-n}$ is cauchy. And hence convergent. 460 | \label{problem-2^n} 461 | \end{problem} 462 | \begin{proof} 463 | We've to prove that $\forall \epsilon>0$ there exist a N such that $\forall\ n,m>N$, $\abs{s_{m}-s_n} < \epsilon$. 464 | 465 | Take $m>n$ and let $m = n+k$. So, by triangular inequality 466 | $$ 467 | \abs{s_m - s_n} \leq \abs{s_m - s_{m-1}} + \abs{s_{m-1}+s_{m-2}}+ \ldots+ \abs{s_{n+1}+s_n} < {1 \over 2^{n}} + {1 \over 2^{n+1}} +\ldots+{1 \over 2^{m}} 468 | $$ 469 | Since, m = n+k and if we add more +ve terms then we will get a G.P. with ratio $\frac{1}{2}$ which look like this: 470 | $$ 471 | {1 \over 2^{n}} + {1 \over 2^{n+1}} +\ldots+{1 \over 2^{m}} = {1 \over 2^{n}} + {1 \over 2^{n+1}} +\ldots+{1 \over 2^{n+k}} = {1 \over 2^{n-1}}(1 - {1 \over 2^k}) 472 | $$ 473 | So,\todo[color = white,shadow]{We have not talked about Series / summation till now. Series, will be discussed in the next Section(3)} $$ 474 | \abs{s_m - s_n} \leq \abs{s_m - s_{m-1}} + \abs{s_{m-1}+s_{m-2}}+ \ldots+ \abs{s_{n+1}+s_n}<{1 \over 2^{n-1}}(1 - {1 \over 2^k})<{1 \over 2^{n-1}}$$ 475 | 476 | And applying Theorem \ref{def:basic-example} ,we can say ${1 \over 2^{n-1}}$ is convergent. Hence, the same N will work for the sequence $s_n$. 477 | \end{proof} 478 | \begin{problem} 479 | Let $s_n$ be an increasing sequence. Prove that $\sigma_n = {1 \over n}(s_1 + s_2 + \ldots+s_n)$ is an increasing sequence. 480 | \end{problem} 481 | \begin{proof} 482 | Since, since $\sigma_1 < \sigma_2$. Let us suppose $\sigma_n > \sigma_{n-1}$. Now, we have to prove that it is true for the term $\sigma_{n+1}$. 483 | 484 | Suppose, it's not true i.e. $\sigma_{n+1}<\sigma_n$ 485 | $$ {1 \over {n+1}}(s_1 + s_2 + \ldots + s_{n+1}) < {1 \over {n}}(s_1 + s_2 + \ldots + s_{n})$$ 486 | If we further simplify the inequality, we get $ns_{n+1}<(s_1 + s_2 + \ldots + s_{n})$ which is false. 487 | So, the assumption was false. Hence, $\sigma_{n+1} > \sigma_n$. 488 | \end{proof} 489 | \begin{problem} 490 | Define $x_1 = 2$ and: 491 | 492 | $$ x_{n+1} = {1 \over 2}\left(x_n + {2 \over x_n}\right) $$ 493 | 494 | Prove that the $\lim\limits_{n \to \infty} x_n = \sqrt{2}$. 495 | \end{problem} 496 | \begin{proof} 497 | Observer that $x_2 < x_1 = 2$. So, assume $x_{n+1}x_{n+1}$ (it would lead to a contradiction), it implies: 500 | $$ \Rightarrow {1 \over 2}\left(x_{n+1} + {2 \over x_{n+1}}\right) > x_{n+1}$$ 501 | $$ \Rightarrow (x_{n+1})^2<2$$ 502 | Substitute $x_{n+1} = {1 \over 2}\left(x_n + {2 \over x_n}\right)$ and solve. We get: 503 | $$ ({x_{n}}^2 - 2)^2 <0$$ 504 | which is a contradiction. So, the series is monotonically decreasing and bounded(because each element is greater than zero and less that two). Hence, convergent. 505 | To prove $\lim\limits_{n \to \infty} x_n = \sqrt{2}$, put $x_{n+1} = x_n = x$ in the definition.($\because \lim\limits_{n \to \infty} x_n = \lim\limits_{n \to \infty} x_{n+1}$ ) 506 | \end{proof} 507 | \paragraph{} 508 | \section{Subsequences} 509 | 510 | \paragraph{An unformal definition of subsequence:} 511 | 512 | 513 | A \textbf{Subsequence} is a sequence that can be derived from another sequence by deleting some or no elements 514 | without changing the order of the remaining elements 515 | 516 | \begin{theorem}{} 517 | If the sequence $(s_n)$ converges, then every subsequence converges to 518 | the same limit. 519 | \end{theorem} 520 | 521 | %%%%% Proof left %%%%%%% 522 | 523 | \begin{theorem}{} 524 | Every sequence $(s_n)$ has a monotonic subsequence. 525 | 526 | \end{theorem} 527 | %%%%%%%%% Proof Left %%%%%%%% 528 | 529 | \begin{theorem}[\bf Bolzano-Weierstrass Theorem]{}\cite{wikib} 530 | Every bounded sequence has a convergent subsequence. 531 | 532 | 533 | 534 | \end{theorem} 535 | \begin{figure}[!h] % the command in [] to keep it under the theorem or it will be placed above the page 536 | \begin{center} 537 | \includegraphics[width=3in]{bolanzo.png} 538 | \caption{Bolzano-Weierstrass Theorem} 539 | \end{center} 540 | \end{figure} 541 | \begin{proof} 542 | It's easy to prove that every bounded sequence has a convergent subsequence. Since, every sequence has a monotonic subsequence and since the sequence is bounded implies subsequence is bounded. And every bounded monotonic sequence is convergent. 543 | \end{proof} 544 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 545 | \subsection{Subsequential Limits} 546 | \begin{definition}{} 547 | Let $(s_n)$ be a sequence in $\mathbb{R}$. A subsequential limit is any real number 548 | or symbol $+\infty$ or $-\infty$ that is the limit of some subsequence of $(s_n)$. 549 | 550 | When a sequence has a limit s, then all subsequences have limit 551 | s, so \{s\} is the set of subsequential limits 552 | \end{definition} 553 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 554 | 555 | %%% Most part of subsequential limits was not covered in the class. sSo, I'll skip it 556 | 557 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 558 | \begin{theorem}{thm:problemlim} 559 | Let $(s_n)$ be any sequence. There exists a monotonic subsequence 560 | whose limit is $\limsup s_n$, and there exists a monotonic subsequence 561 | whose limit is $\liminf s_n$. 562 | \end{theorem} 563 | \paragraph{Recall } 564 | Let $(s_n)$ be any sequence of real numbers, and let S be the set of 565 | subsequential limits of $(s_n)$. Recall 566 | $$\liminf s_n = \lim_{N\to \infty} \inf \{s_n\ :\ n>N\} = \inf S$$ 567 | and 568 | $$\limsup s_n = \lim_{N\to \infty} \sup \{s_n\ :\ n>N\} = \sup S$$ 569 | \paragraph{} 570 | 571 | \chapter{Series} 572 | \section{Sum to infinity?} 573 | \begin{definition}[Summation Notation]{} 574 | $\sum_{n}^{m} a_k= a_n+a_{n+1}+\ldots + a_m$ 575 | 576 | 2.To assign meaning to $\sum_{n=m}^{\infty} a_n$, we consider the sequences $(s_n)_{n=m}^{\infty}$ 577 | of partial sums: 578 | $$s_n = a_m + a_{m+1} + \ldots + a_n = \sum_{k=m}^{n} a_k$$ 579 | The infinite series $\sum_{n=m}^{\infty} a_n$ an is said to converge provided the sequence 580 | $(s_n)$ of partial sums converges to a real number S, in which case we 581 | define $$\sum_{n=m}^{\infty} a_n = S$$ 582 | \end{definition} 583 | 584 | 585 | \begin{definition}[Cauchy Criterion for Series Convergence]{} 586 | We say a series $\sum a_n$ satisfies the Cauchy criterion if its sequence 587 | $(s_n)$ of partial sums is a Cauchy sequence i.e.: 588 | 589 | 590 | for each $\epsilon > 0$ there exists a number N such that: 591 | 592 | $$n \geq m>N\ \text{implies}\ \abs{s_n - s_{m-1} }<\epsilon$$ 593 | And, $s_n - s_{m-1} = \sum_{n}^{m} a_k$ 594 | 595 | 596 | A series converges iff it satisfies cauchy criterion 597 | \end{definition} 598 | \begin{corollary}{cor:lim0} 599 | If $\sum a_n$ converges then $\lim a_n = 0$ 600 | \end{corollary} 601 | %%%%%%%%%%%%%%%%%%%%% 602 | %% I'm leaving most of the part from the book 603 | 604 | %%%%%%%%%%%%%%%%%%%%% 605 | \section{Convergence Tests for Series} 606 | \paragraph{\S Comparison Test} 607 | Let $\sum a_n$ be a series where $a_n \geq 0$ for all n. 608 | \begin{enumerate} 609 | \item 610 | If $\sum a_n$ converges and $|b_n|\leq a_n $ for all n, then $\sum b_n$ converges. 611 | \item 612 | If $\sum a_n = + \infty$ and $b_n \geq a_n$ for all n, then $\sum b_n = + \infty$ 613 | \end{enumerate} 614 | \begin{problem} 615 | Show that the series $s_n = \sum {1 \over n^2}$ converges 616 | \label{1 by n^2} 617 | \end{problem} 618 | \begin{proof}[Solution] 619 | Observation:$$ 620 | \Rightarrow {1 \over n^2} < {1 \over n(n-1)} = {1 \over n-1} - {1 \over n} 621 | $$ 622 | $$ 623 | \Rightarrow \sum_{2}^{n} {1 \over n^2} < \sum_{2}^{n} \left({1 \over n-1} - {1 \over n}\right) 624 | $$ 625 | Or 626 | $$ 627 | \Rightarrow 1+\sum_{2}^{n} {1 \over n^2} < 1+\sum_{2}^{n} \left({1 \over n-1} - {1 \over n}\right) 628 | $$ 629 | $$ 630 | \Rightarrow \sum_{2}^{n} {1 \over n^2} <2 - \frac{1}{n} 631 | $$ 632 | As $n \to \infty$ we get: 633 | $$\sum_{2}^{\infty} {1 \over n^2} < 2 $$ 634 | Hence, it converges. 635 | \end{proof} 636 | 637 | \paragraph{\S\ Ratio Test:}\cite{wikiratio} 638 | A series $\sum a_n$ of nonzero terms.The usual form of the test makes use of the limit: 639 | $$L = \lim_{n \to \infty} \abs{a_{n+1 } \over a_n}$$ 640 | \begin{enumerate}[\bf (i)] 641 | \item 642 | if $L < 1$ then the series converges absolutely; 643 | \item 644 | if $L > 1$ then the series diverges; 645 | \item 646 | if $L = 1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case. 647 | \end{enumerate} 648 | It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let 649 | \begin{itemize} 650 | \item 651 | $ R = \limsup \abs{a_{n+1 } \over a_n}$ 652 | \item $ r = \liminf \abs{a_{n+1 } \over a_n}$ 653 | \end{itemize} 654 | Then the ratio test states that: 655 | \begin{itemize} 656 | \item if $R < 1$, the series converges absolutely; 657 | \item if $r > 1$, the series diverges; 658 | \item if $\abs{a_{n+1 } \over a_n} \geq 1$ for all large n (regardless of the value of r), the series also diverges; this is because $|a_{n}|$ is nonzero and increasing and hence an does not approach zero; 659 | \item the test is otherwise inconclusive 660 | \end{itemize} 661 | \paragraph{\S\ Root Test:}\cite{wikiroot} 662 | Let $\sum a_n$ be a series and let $ \alpha = \limsup |a_n|^{1/n}$. The series $\sum a_n$ 663 | 664 | \begin{enumerate}[\bf (i)] 665 | \item 666 | converges absolutely if $\alpha<1$. 667 | \item 668 | diverges if $\alpha>1$. 669 | \item 670 | Otherwise $\alpha = 1$ and the test gives no information 671 | \end{enumerate} 672 | Note that if: 673 | $$ \lim\limits_{n \to \infty} \sqrt[n]{|a_n|}$$ 674 | converges then it equals $\alpha$ and may be used in the root test instead. 675 | \begin{lemma}{} 676 | Let $|s_n|$ be a sequence of non-zero real numbers,\par 677 | Then, $$\liminf \abs{\frac{s_{n+1}}{s_n}} \leq \liminf\abs{|s_n|^{1 \over n}} \leq \limsup\abs{|s_n|^{1 \over n}} \leq \limsup \abs{\frac{s_{n+1}}{s_n}}$$ 678 | 679 | \end{lemma} 680 | \paragraph{} 681 | \begin{problem} 682 | Prove that $ \lim_{n \to \infty} {x^n \over n!} = 0$ 683 | \end{problem} 684 | 685 | \begin{proof}[\bf Solution] 686 | 687 | \begin{enumerate}[a.] 688 | \item 689 | Given: 690 | \begin{gather*} 691 | \Rightarrow a_n = {x^n \over n!} \\ 692 | \Rightarrow a_{n+1} = {x^{n+1} \over {n+1}!} \\ 693 | \Rightarrow {a_{n+1} \over a_n} = {x \over {n+1}} 694 | \end{gather*} 695 | As $ n \to \infty$ ratio ${a_{n+1} \over a_n} \to 0$. Thus it converges\textsuperscript{\href{https://math.stackexchange.com/q/712586}{1 }}. That means the series $\sum {x^n \over n!}$ converges therefore ${x^n \over n!}$ converges to zero. 696 | 697 | \item 698 | The series $e^x = \sum_{n=0}^{\infty} {x^n \over n!}$ converges. 699 | Hence ${x^n \over n!} \to 0$. 700 | \item 701 | \href{https://math.stackexchange.com/questions/712572/prove-that-xn-n-converges-to-0-for-all-x}{MathStackexchange} 702 | \end{enumerate} 703 | \end{proof} 704 | \begin{problem} 705 | Let $\limsup |a_n| > 0$. Then prove that $\limsup |a_n| ^ {1 \over n} \geq 1$. 706 | \end{problem} 707 | \begin{proof} 708 | Assume that $\limsup |a_n| > 0$ but $\limsup |a_n| ^ {1 \over n} < 1$. We also conclude that $\limsup |a_n| ^ {1 \over n} < 1 $ implies $\sum a_n $ converges absolutely (by Root test).So, $\lim |a_n|=\limsup |a_n| = 0$. Contradiction!. So, $\limsup |a_n| ^ {1 \over n} \geq 1$. 709 | \end{proof} 710 | \begin{theorem}{} 711 | Let $\sum |a_n|$ be a convergent series \& let $(b_n)$ be a bounded sequence. 712 | Then, $\sum a_n b_n$ is also convergent. 713 | \end{theorem} 714 | \begin{proof} 715 | By triangular inequality we can show that: 716 | $$ \left|\sum_{k = m}^{n} a_k b_k \right|\leq \sum_{k = m}^{n}\abs{a_k b_k}$$ 717 | 718 | Given, $|b_n| \leq M$ (it's bounded), implies: 719 | $$ \Rightarrow \abs{a_k}\abs{b_k} \leq \abs{a_k }M$$ 720 | $$ \Rightarrow \sum \abs{a_k}\abs{b_k} \leq \sum \abs{a_k }M$$ 721 | Since, $\sum a_n$ converges; therefore by cauchy criterion $\exists\ N \in \mathbb{N}$ such that $\forall\ n\geq m>N$: 722 | 723 | $$\Rightarrow \sum_{k = m}^{n} |a_k| < {\epsilon \over M}$$ 724 | $$\Rightarrow \sum_{k = m}^{n} M|a_k| < \epsilon$$ 725 | Or 726 | $$\Rightarrow\left|\sum_{k = m}^{n} a_k b_k \right|\leq \sum \abs{a_k}\abs{b_k} \leq \sum_{k = m}^{n} M|a_k| < \epsilon$$ 727 | Hence, $\sum a_n b_n$ is convergent by comparison test. 728 | \end{proof} 729 | \begin{corollary}{} 730 | Let $ a_n \geq 0$ \& $\sum a_n$ converges. 731 | Then 732 | $ \sum (a_k) ^ p$ converges $\forall\ p>1$ 733 | \end{corollary} 734 | \begin{proof}The above expression can be rewritten as: 735 | $$ \sum_{k=m}^{n} \abs{a_k}^p = \sum_{k=m}^{n} \abs{a_k}\abs{a_k}^{p-1}$$ 736 | 737 | Since, $\sum a_n$ converges, therefore $a_n \to 0$. So, sequence $a_n$ is convergent, hence bounded.So, $a_k ^{p-1}$ is bounded. 738 | 739 | $\therefore$ By previous theorem, we can say 740 | $ \sum (a_k) ^ p$ converges. 741 | \end{proof} 742 | \section{Alternating Series Test} 743 | \begin{theorem}{} 744 | If $a_1 \geq a_2 \geq \ldots \geq a_n \geq \ldots \geq 0$ and $\lim a_n = 0$, then the alternating series $\sum (- 1)^{n+1}a_n$ converges. Moreover, the partial sums $s_n =\sum_{k = 1}^{n} ( - 1)^{k+1}a_k$ satisfy $|s - s_n|\leq a_n$ for all n. 745 | \end{theorem} 746 | \begin{proof} 747 | We need to show that the sequence $(s_n)$ converges. Note that the 748 | subsequence $(s_{2n})$ is increasing because $s_{2n+2} - s_{2n} = - a_{2n+2} + 749 | a_{2n+1}\geq 0$. Similarly, the subsequence $(s_{2n - 1})$ is decreasing since 750 | $s_{2n+1} - s_{2n - 1} = a_{2n+1} - a_{2n}\leq 0$. We claim: 751 | 752 | $$ s_{2m} \leq s_{2n+1}\quad \text{for all }\quad m,n \in \mathbb{N} $$ 753 | First note that $s_{2n} \leq s_{2n+1}$ for all n, because $s_{2n+1} - s_{2n} = a_{2n+1} \geq 0$. 754 | If $m\leq n$, then the above equation holds because $s_{2m} \leq s_{2n} \leq s_{2n+1}$. If $m \geq n$, 755 | then equation holds because $s_{2n+1}\geq s_{2m+1 }\geq s_{2m}$. We 756 | see that $(s_{2n}) $ is an increasing subsequence of $(s_n)$ bounded above 757 | by each odd partial sum, and $(s_{2n+1})$ is a decreasing subsequence 758 | of $(s_n)$ bounded below by each even partial sum. By Theorem (\ref{bounded}), 759 | these subsequences converge, say to s and t. Now 760 | $$ t - s = \lim_{n \to \infty} s_{2n+1} - \lim_{n \to \infty}s_{2n} = \lim\limits_{n \to \infty}(s_{2n+1} - s_{2n}) = \lim\limits_{n \to \infty}a_{2n+1} = 0$$ 761 | . so s = t, follows that $\lim_{n}s_n = s$. 762 | 763 | To check the last claim, note that $s_{2k} \leq s \leq s_{2k+1}$, so both 764 | $s_{2k+1} - s$ and $s - s_{2k}$ are clearly bounded by $s_{2k+1} - s_{2k} = a_{2k+1} \leq a_{2k}$. 765 | So, whether n is even or odd, we have $|s - s_n| \leq a_n$. 766 | \end{proof} 767 | \section{Integral Test} 768 | 769 | \begin{theorem}{} 770 | Consider an integer N and a non-negative function $f$ defined on the unbounded interval $ [N,\infty) $, on which it is monotone decreasing. Then the infinite series: 771 | $$ \sum_{n}^{\infty} f(n)$$ 772 | converges to a real number if and only if the improper integral 773 | 774 | $$ \int_{N}^{\infty} f(x)dx$$ 775 | is finite. In other words, if the integral diverges, then the series diverges as well. 776 | \end{theorem} 777 | 778 | 779 | \paragraph{} 780 | \begin{problem} 781 | Let $(a_n)_{n \in \mathbb{N}}$ be a sequence such that $\liminf |a_n| = 0$. Prove there is 782 | a subsequence $ (a_{n_k})_{k \in \mathbb{N}} $ such that $\sum_{k = 1}^{\infty}(a_{n_k})$ converges. 783 | \end{problem} 784 | \begin{proof} 785 | We first set $n_0=1$ and $c_1=1$. By the property of $\liminf$, there exists $n_1>n_0=1$ such that $|a_{n_1}| \mathbb{N}\}$ so that $1\leq inf \{a_n : n > \mathbb{N}\}$, in 788 | which case $1 \leq \lim_{N\to\infty} \inf \{a_n : n > \mathbb{N}\}$, so we would get $1 \leq0$. A contradiction. 789 | 790 | Then we set $c_2=1/4$. Again, there exists $n_2>n_1$ such that $|a_{n_2}|n_{k-1}$ such that $|a_{n_k}|0$ $\exists N \in \mathbb{N}$ such that $\forall n> m+1>N$: (we'll be using the fact that for $n>m$ we have $a_n 1$ 814 | \end{problem} 815 | \begin{proof} 816 | \begin{enumerate} 817 | \item 818 | In particular, for $p\leq 1$, we can write $\sum{1 \over n^p} = + \infty$. 819 | For $ p = 2$ we have already proved it in Problem \ref{1 by n^2}. 820 | 821 | For $p>2$, we can prove it by the comparison test as : 822 | 823 | $$ \left\{{1 \over n^p } < {1 \over n^2}\right\}, \quad \forall \quad p>0$$ 824 | Since, ${1 \over n^2}$ converges. Then ${1 \over n^p}$ converges by comparison test. 825 | \begin{center} 826 | Or 827 | \end{center} 828 | The above proposition can be used to prove the result for $p>2$ 829 | 830 | \item 831 | \begin{figure}[h!] % the command in [] to keep it under the theorem or it will be placed above the page 832 | \begin{center} 833 | \includegraphics[width=\textwidth]{geogebra.png} 834 | \caption{Geometric intution for $\sum{1 \over n^p }$} 835 | \end{center} 836 | \end{figure} 837 | For $ 1 < p<2$ 838 | 839 | We have, 840 | $$ \underbrace{\sum_{k=1}^{n} {1 \over k^p}}_{\substack{ \text{sum of rectangles going} \\ \text{from 1 $\to$ n}}} \leq \underbrace{1 + \int_{1}^{n} {1 \over x^p}dx}_\text{area under the curve} $$ 841 | $$ \Rightarrow \sum_{k=1}^{n} {1 \over k^p} \leq 1 + \left[ x^{1-p} \over {1-p} \right]_{1}^{n} $$ 842 | $$ \Rightarrow \sum_{k=1}^{n} {1 \over k^p} \leq 1 + \left[ {n^{1-p} \over {1-p} } - {1 \over {1-p}}\right] $$ 843 | $$ \Rightarrow \sum_{k=1}^{n} {1 \over k^p} \leq \left[ {p - n^{1-p} \over p-1} \right] $$ 844 | 845 | As $ n\to \infty$ we get( $\because$ $1-p<0$), 846 | $$ \Rightarrow \sum_{k=1}^{n} {1 \over k^p} \leq \left[ {p \over p-1} \right] $$ 847 | 848 | \end{enumerate} 849 | 850 | \end{proof} 851 | 852 | \chapter{Continuity} 853 | 854 | \section{Continuity and Functions} 855 | \paragraph{} 856 | \begin{definition}{} 857 | A function $f$ whose domain is defined over $\mathbb{R}$ is said to be continuous at a point 858 | $x_0\in dom(f)$ iff: 859 | 860 | for each $\epsilon>0$ there exist a $\delta>0$ such that: 861 | $$x\in dom(f)\ \textit{and}\ \abs{x-x_0}<\delta\ \textit{imply}\ \abs{f(x)-f(x_0)}<\epsilon$$. 862 | 863 | 864 | 865 | \begin{center} 866 | \textbf{Or} 867 | \end{center} 868 | 869 | Let $f$ be a real-valued function whose domain is a subset of $\mathbb{R}$. The 870 | function $f$ is continuous at $(x_0)\in dom(f)$ if, for every sequence $(x_n)\in dom(f)$ converging to $x_0$, we have $lim_{n} f(x_n) = f(x_0)$ i.e.: 871 | 872 | if for every sequence in domain if $x_n\to x_0$ we have $f(x_n)\to f(x_0)$ 873 | 874 | \end{definition} 875 | 876 | \begin{theorem}{} 877 | Let $f$ be a real-valued function with $dom(f) \subseteq \mathbb{R}$. If $f$ is continuous 878 | at $x_0$ in $dom(f)$, then $|f|$ and $kf$, $k \in \mathbb{R}$, are continuous at $x_0$ 879 | 880 | \end{theorem} 881 | \subsection{Properties of Continuous Functions} 882 | 883 | \paragraph{\S\ Bounded Function} 884 | A real-valued function $f$ is said to be bounded if $\{f(x) \colon x \in dom(f)\}$ 885 | is a bounded set, i.e., if there exists a real number M such that 886 | $|f(x)| \leq M$ for all $x \in dom(f)$. 887 | 888 | \begin{theorem}{} 889 | Let $f$ be a continuous real-valued function on a closed interval $[a, b]$. 890 | Then $f$ is a bounded function. Moreover, $f$ assumes its maximum 891 | and minimum values on $[a, b]$; that is, there exist $ x_0, y_0\ \in \ [a, b]$ such 892 | that $f(x_0) \leq f(x) \leq f(y_0)$ for all $x \in [a, b]$. 893 | \end{theorem} 894 | 895 | \begin{theorem}[\bf Intermediate value theorem]{} 896 | If $f$ is a continuous real-valued function on an interval I, then $f$ has 897 | the intermediate value property on I: Whenever $a, b \in I$, $a 0 $ that works, any smaller value will also work. By choosing $ \delta < 1$ we would have: 935 | 936 | $$ |x - a| < 1\ \text{ when ever } \ |x - a| < \delta$$ 937 | 938 | $$\Rightarrow ||x| - |a|| \leq |x - a| < 1$$ 939 | Or 940 | $$\Rightarrow |x| < 1 + |a|$$ 941 | $$\Rightarrow |x + a| \leq |x| + |a| < |2a| + 1$$ 942 | $$ \Rightarrow [1 + 3 |x+a |\ ] < [1 + 3 (1+2|a |)\ ]$$ 943 | 944 | Since, we need both the above assumptions i.e. $ | x- a| < \delta$ \& $| x- a| < 1$ to be satisfied.So, take : 945 | 946 | $$ \delta = \min \left( 1, { \epsilon \over 1 + 3(1 + 2 | a| )} \right) $$ 947 | And, whole discussion above proves that it would work. 948 | 949 | \end{proof} 950 | \begin{problem} 951 | Is $ \lim {{4x+1} \over 3x-4} $ is continuous for $x \not = 4/3$? 952 | \end{problem} 953 | 954 | %\begin{proof} 955 | %We will break the interval in two sets 956 | %$ |x| > 4/3$ and $ |x|<4/3$. 957 | 958 | %\begin{enumerate}[\bf 1.] 959 | %\item 960 | %Case 1: $|x|>4/3$ . $ \abs{{{4x+1} \over 3x-4} - {{4a+1} \over 3a-4}} = \frac{|x-a|19}{|3a - 4||3x-4|} $ 961 | %$$ \Rightarrow |x-a| < \frac{\epsilon |3a - 4||3x-4|}{19}$$ 962 | 963 | %We need to find a lower bound for $|x|$, so that we can find $\delta$ such that $|x-a| < \delta \leq \frac{\epsilon |3a - 4||3x-4|}{19}$. To do so, we have assumed $|x|>4/3$ implies $3|x| - 4>0$.So, using the same argument in above pooof, we can say $ ||x| - |a|| \leq |x-a| < \delta = 1$ 964 | 965 | 966 | 967 | 968 | %\item 969 | %\end{enumerate} 970 | 971 | 972 | %\end{proof} 973 | \begin{problem} 974 | Let f and g be continuous functions on $[a, b]$ such that $f(a) \geq 975 | g(a)$ and $f(b) \leq g(b)$. Prove $f(x_0) = g(x_0)$ for at least one $x_0$ in 976 | $[a, b]$ 977 | \end{problem} 978 | \begin{proof} 979 | Define $h(x) = f(x)-g(x)$. It's given that $f(a) - g(a)\geq 0$ and $f(b) - g(b)\leq 0$. So, by intermediate value theorem there exists a $x_0 \in [a,b]$. Such that $h(x_0) = 0$. 980 | 981 | \end{proof} 982 | \begin{problem} 983 | Suppose f is continuous on $[0, 2]$ and $f(0) = f(2)$. Prove there exist 984 | $x, y \in [0, 2]$ such that $|y - x| = 1$ and $f(x) = f(y)$. 985 | \end{problem} 986 | \begin{proof} 987 | Define $g(x) = f(x+1) - f(x)$ . Let $|x-y| = 1$ implies either $x = y+1$ or $y = x+1$. 988 | Take anyone of them, it won't matter as we can exchange y with x for another case. 989 | 990 | Also, $ g(0) = f(1)-f(0)$ and $g(1) = f(2)-f(1)$. Adding both the equations $g(0)+g(1) = 0$, which implies that one of them is negative of other i.e. if any one of the value is + ve the other is - ve. 991 | So, there exists a value of $x \in [0,1]$ such that $g(x) = 0$ or $f(x+1) = f(x) \equiv f(y) = f(x)$. 992 | \end{proof} 993 | \section{Uniform Continuity} 994 | 995 | \begin{definition}[Uniform Continuity]{} 996 | Let $f$ be a real-valued function defined on a set $S \subseteq \mathbb{R}$. Then f is 997 | uniformly continuous on S if 998 | for each $\epsilon> 0$ there exists $\delta > 0$ such that 999 | $$ x,y \in S \ \text{and} \ \abs{x - y} < \delta \ \text{imply} \ \abs{f(x) - f(y)} < \epsilon $$ 1000 | We will say f is uniformly continuous if f is uniformly continuous 1001 | on dom(f). 1002 | 1003 | \end{definition} 1004 | \begin{figure}[h!] 1005 | \centering 1006 | \subfloat[For uniformly continuous functions, there is for each $\epsilon >0$ a $\delta >0$ such that when we draw a rectangle around each point of the graph with width $2\delta$ and height $2\epsilon$, the graph lies completely inside the rectangle.] 1007 | {{\includegraphics[width= 2.5in]{uniform_yes.png} }}% 1008 | \qquad 1009 | \subfloat[For functions that are not uniformly continuous, there is an $\epsilon>0$ such that regardless of the $\delta>0$ here are always points on the graph, when we draw a $2\epsilon-2\delta$ rectangle around it, there are values directly above or below the rectangle.] 1010 | {{\includegraphics[width= 2.5in]{uniform_no.png} }}% 1011 | \caption{Uniform Continuity}% 1012 | \end{figure} 1013 | \paragraph{If your function happens to satisfy $ 0<|f'(x)|0$ such that for all $x' \in [a,b]$ with $|x' - x| <\delta$ it follows that $|f(x) - f(x')|<\epsilon$. As , there is however a rational $x' \in [a,b]$ with $|x - x'|<\delta$ . But now $|f(x) - \underbrace{f(x')}_{0}| < |f(x)| < |f(x)|/2 $ . Contradiction! 1067 | 1068 | \begin{center} 1069 | \textbf{Or} \end{center} 1070 | \end{enumerate} 1071 | Let $x\in[a,b]$, and let ${q_n}$ be a sequence of rational numbers, such that $q_n\to x$. By continuity of f, we have: 1072 | 1073 | $$ 1074 | f(x) = f(\lim_{n \to \infty} q_n) = \lim_{n \to \infty} f(q_n) = 0 1075 | $$ 1076 | 1077 | \item 1078 | Using the above result we can define a function $f(x) - g(x)$, and $f(x) = g(x)$ or $f(x) - g(x) = 0$ for all rationals. 1079 | 1080 | \end{proof} 1081 | 1082 | \begin{problem} Let 1083 | $$ 1084 | f(x) = \begin{cases} 1085 | 0;\ \text{for x irratonal} \\ 1086 | {1 \over q};\ \text{for}\ x = {p \over q } \\ 1087 | 1,\ \text{for}\ x=0 1088 | \end{cases} 1089 | $$ 1090 | Show f is 1091 | continuous at each point of $\mathbb{R \sim Q}$ and discontinuous at each point 1092 | of $\mathbb{Q}$. 1093 | \end{problem} 1094 | 1095 | \begin{proof} 1096 | It's easy to proof for x $\in \mathbb{Q}$. Take a sequence $(x_n)$ of irrationals such that $x_n \to p/q$. But, $f(x_n) = 0 \not = f(p/q) = 1/q$. Also, observe that the function is periodic with period 1, i.e. $f(x) = f(x+1)$. 1097 | 1098 | For x in set of irrational numbers. For a sequence of irrationals it's obvious that the difference b/w the function values will be 0 always. 1099 | For a sequence of rationals we can think of any sequence of rationals coverging to x, then the denominator value will approach infinity(it's not a formal proof). An example is shown below shown in the footnote below. 1100 | \end{proof} 1101 | 1102 | \begin{problem} Let 1103 | $$ 1104 | f(x) = \begin{cases} 1105 | 0;\ \text{for x irratonal} \\ 1106 | 1;\ \text{for}\ x\ {rational } \\ 1107 | \end{cases} 1108 | $$ 1109 | Show f is 1110 | discontinuous at each point of $\mathbb{R}$. 1111 | \end{problem} 1112 | \begin{proof} 1113 | We begin by considering a sequence of irrational numbers $x_n$ converging to a rational $x_0$. $x_n = x_0 + {\lambda \over n}$ where $\lambda \in \mathbb{R \sim Q}$.Since, every element of $x_n$ is irrational implies $f(x_n) = 0 \not= f(x_0) = 1$. So, there exist a sequence in the domain such that $x_n \to x_0$ but $ f(x_n) \not\to f(x_0)$ for rational $x_0$ 1114 | 1115 | Similarly, using the density property of rational numbers \footnote[1]{you need a sequence of rationals converging to the irrational x. In theory, we already know one: consider the decimal expansion of x. When x is irrational, the sequence is necessarily infinite, doesn't eventually repeat itself forever. Suppose 1116 | $$x=m+0.d_1 d_2\ldots$$ 1117 | where m is an integer. Let 1118 | $x_n=m+0.d_1\ldots d_n$ 1119 | In other words, $x_n$ is the decimal representation of x cut off at the n\textsubscript{th} digit after the decimal point $(x_0=m)$. Then every $x_n$ is rational, and: 1120 | 1121 | $$\lim_{n \to \infty}x_n=x$$ }there exist a sequence of rational numbers $(x_n)$ in $\mathbb{R}$ such that $x_n \to x_0$ for $x_0$ irrational.But $f(x_n) = 1 \not= f(x_0) = 0$. Hence , it is not continuous. 1122 | \end{proof} 1123 | \begin{problem} 1124 | Let $f$ be a continuous function on $[0,\infty)$. Prove that if $f$ is 1125 | uniformly continuous on $[k, \infty)$ for some $k$, then $f$ is uniformly 1126 | continuous on $[0,\infty)$. 1127 | \end{problem} 1128 | \begin{problem} 1129 | Let $f$ be a continuous function on $[a, b]$. Show that the function $f^{*}$ defined as 1130 | $f^{*}(x) = \sup \{f(y) : a \leq y \leq x\}$, for $x \in [a, b]$, is an increasing 1131 | continuous function on $[a, b]$. 1132 | \end{problem} 1133 | \begin{proof} 1134 | Take $x_2 > x_1$ . Let $S_1 = [a,x_1]\quad\&\ S_2 = [a,x_2]$. Observe $S_1 \subset S_2$.By definition $f^{*}(x_1) = \sup \{f(y) : a \leq y \leq x_1\} $ \& $f^{*}(x_2) = \sup \{f(y) : a \leq y \leq x_2\}$. So, $f(x_1) \geq f(x) \forall x \in S_1$ and $f(x_2) \geq f(x) \forall x \in S_2$. Since, $S_1 \subset S_2$ so, $x_1 \in S_2$ implies $f(x_2) \geq f(x_1)$. 1135 | \end{proof} 1136 | %%%%% Problems%%%%%%% 1137 | %%%%%%%%%% 1138 | %%%%%%%%%%%%%%%%%%% 1139 | 1140 | \section{Limits of Functions} 1141 | \paragraph*{In this section we'll formalize the notion of limit of a function and this will help us for a careful study of derivatives} 1142 | \begin{definition}{} 1143 | Let $S \subseteq \mathbb{R}\ \text{and }\ S \not = \varnothing$, let a be a real number or symbol $ - \infty$ or $\infty$ that is a limit of a sequence in S. And let L be a real number or symbol $ - \infty$ or $\infty$. We write $\lim_{x \to a} f(x) = L$ if: 1144 | 1145 | $$ \mathrm{f\ is\ a\ function\ defined\ on\ S}$$ 1146 | 1147 | and 1148 | 1149 | $$ \mathrm{for\ every\ sequence\ (x_n)\ in\ S\ converging\ to\ a,\ we\ have }\ \lim_{n \to \infty} f(x_n) = L$$ 1150 | 1151 | The expression ``$\lim_{x \to a^S} f(x)$" is read as ``limit, as x tends to a along S,of f(x)," 1152 | 1153 | 1154 | \end{definition} 1155 | \paragraph{Notations} 1156 | $$ S = (-\infty,b): 1157 | \lim_{x \to a} f(x) = \lim_{x \to a^{ - }} f(x)$$ 1158 | and 1159 | $$ S = (a,\infty): \lim_{x \to a} f(x) = \lim_{x \to a^{ + }} f(x)$$ 1160 | 1161 | \paragraph{Question:} Let a be a real number \& $S = (a,b) \subseteq \mathbb{R}$. Does $\lim_{x \to a^S} f(x)$ depends on S. If we take a different set $T = (a,b_1)\subseteq \mathbb{R}$ then what's the relation b/w 1162 | $ \lim_{x \to a^S} f(x) \& \lim_{x \to a^T} f(x)$ 1163 | \paragraph{Answer:} Since, as the $t_n \in T$ converge to a, then after some iterations the elements of sequence $t_n$ will be the elements of set S (Assuming $ S \subseteq T$).then: 1164 | 1165 | $$ \lim_{x \to a^S} f(x) = \lim_{x \to a^T} f(x)$$ 1166 | 1167 | \begin{theorem}{} 1168 | Let f be a function for which the limit $L = \lim_{x\to a^S} f(x)$ exists and 1169 | is finite. If g is a function defined on $\{f(x) : x \in S\} \cup \{L\}$ that is 1170 | continuous at L, then $\lim_{x\to a^S} g\circ f (x)$ exists and equals g(L). 1171 | \end{theorem} 1172 | \begin{theorem}{} 1173 | Let f be a function defined on a subset S of R, let a be a real number 1174 | that is the limit of some sequence in S, and let L be a real number. 1175 | then $\lim_{x \to a^S} f(x) = L$ if and only if 1176 | \begin{equation*} 1177 | \text{for each}\ \epsilon>0\ \text{there exists}\ \delta> 0\ \text{such that}\ 1178 | \end{equation*} 1179 | \begin{equation} 1180 | x\in S\ \text{and}\ |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon. 1181 | \end{equation} 1182 | \end{theorem} 1183 | \begin{proof} 1184 | Assuming the statement and proving (3) is trivial so I leave it. 1185 | 1186 | Let $\lim_{n \to \in \infty } f(x_n) = L$: 1187 | 1188 | 1189 | Now assume $\lim_{n \to \infty} x_n = a$, but (3) fails. So, there exist a $\epsilon>0$ such that for all $\delta >0$ $|f(x_n) - f(a) |\geq \epsilon$. Take $\delta = 1/n$.So, it implies for all $n\in \mathbb{N}$ there exist $x_n$ in S such that $|x_n - a| < 1/n$ while $|f(x_n) - f(x_0) |\geq \epsilon$. Hence aur assumption was wrong and there exist a sequence $(x_n)$ which converges to a , but $f(x_n)$ doesn't converge to $f(a)$ 1190 | \end{proof} 1191 | 1192 | \begin{corollary}{} 1193 | Let f be a function defined on $J \backslash \{a\}$ for some open interval $J$ 1194 | containing a, and let L be a real number. Then $\lim_{x \to a} f(x) = L$ iff: 1195 | 1196 | \begin{center} 1197 | for each $\epsilon > 0$ there exists $\delta> 0$ such that \\ 1198 | $|x - a| < \delta$ implies $|f(x) - L| < \epsilon$. 1199 | \end{center} 1200 | \end{corollary} 1201 | \chapter{Sequence and Series of Functions} 1202 | \section{Power Series} 1203 | \begin{definition}[Power Series]{} 1204 | $\sum_{0}^{\infty} a_n x^n$ is called a Power Series. 1205 | \end{definition} 1206 | \begin{definition}[Radius of Convergence]{def:radius-of-convergence} 1207 | Let $\beta = \limsup |a_n|^{1 \over n}$.Then \textbf{Radius of convergence} is defined as: 1208 | $$ R = {1 \over \beta}$$ 1209 | \end{definition} 1210 | %\begin{problem} 1211 | % $\sum_{0}^{\infty} a_n x^n$ converges for what value of x. 1212 | %\end{problem} 1213 | \begin{theorem}{} 1214 | \begin{enumerate} 1215 | \item 1216 | $\sum_{0}^{\infty} a_n x^n$ converges for $|x| < R$. 1217 | \item 1218 | $\sum_{0}^{\infty} a_n x^n$ diverges for $|x| >R$. 1219 | \end{enumerate} 1220 | \end{theorem} 1221 | \begin{proof} 1222 | Take $t \in \mathbb{R}$. Take $r_t = \limsup |a_n t^n|^{1 \over n}$.Then 1223 | 1224 | $$ 1225 | r_t = \limsup |a_n t^n|^{1 \over n} = |t| \limsup|a_n|^{1 \over n}= |t|\beta 1226 | $$ 1227 | 1228 | \paragraph{Case 1.} Let $0 R \Rightarrow \underbrace{r_t > 1}_{\text{diverges by root test }} 1234 | \end{cases} 1235 | \end{equation*} 1236 | \paragraph{Case 2.} Let $R = \infty$ 1237 | 1238 | Here $\beta = 0$ implies $r_t = 0 <1$.So, converges for all x by root test. 1239 | 1240 | \paragraph{Case 3.} Let $R<0$ 1241 | 1242 | $\Rightarrow \beta = \infty$ implies $r_t = \infty$ ; for all $|t| \not = 0$. 1243 | Therefore Diverges. 1244 | \end{proof} 1245 | \paragraph*{Example:} 1246 | $\sum x^n$ . Here $a_n =1$. 1247 | $$ 1248 | \therefore \lim \abs{a_{n+1} \over a_n} = 1 \quad \therefore \beta = 1, R = 1$$. 1249 | Therefore the series converges for $|x| < 1$. A strict inequality!. 1250 | 1251 | \paragraph{Example:} 1252 | Consider the series 1253 | 1254 | $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(x-1)^n$$ 1255 | 1256 | The radius of convergence for the series $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(y)^n$ is $R = 1$, so it converges for $|y|<1$ or $ x \in (0,2)$ at $x = 0$ and $y = -1$, we have 1257 | 1258 | $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(-1)^n = - \sum {1 \over n}$$ 1259 | 1260 | therefore it diverges to $- \infty$. 1261 | at x = 2 1262 | 1263 | $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(1)^n = \sum \frac{(-1)^{n+1}}{n} = - \ln2$$ 1264 | 1265 | 1266 | Hence,it converges \& the 1267 | interval of convergence is $(0,2]$. 1268 | \begin{problem} 1269 | Consider a power series $\sum a_n x_n$ with radius of convergence R.Prove that if all the coefficients an are integers and if infinitely 1270 | many of them are nonzero, then $R \leq 1$. 1271 | \end{problem} 1272 | \begin{proof} 1273 | Given $|a_n| \geq 1$ ($\because a_n$ are integers). So, $s_n = \sup \{|a_k|: k \geq n\} \geq 1$. Further $|s_n|^{1 \over n} \geq 1$ implies $\lim |s_n|^{1 \over n} \geq 1$. Or $R \leq 1$ using \ref{def:radius-of-convergence}. 1274 | \end{proof} 1275 | \begin{problem} 1276 | Prove that if $\limsup |a_n| > 0$, then $\limsup |a_n|^{1\over n} \geq 1$. 1277 | \end{problem} 1278 | \begin{proof} 1279 | Let $\limsup |a_n| > 0$ (notice the strict inequality!). But for sake of contradiction assume $\limsup |a_n|^{1\over n} < 1$. Then $\sum a_n$ converges, which implies $\lim a_n = 0$ (See \ref{cor:lim0}). A contradiction! 1280 | \end{proof} 1281 | \section{Uniform Convergence} 1282 | 1283 | \begin{definition}[Pointwise Convergence]{} 1284 | Let $(f_n)$ be a sequence of real-valued functions defined on a set $S \subseteq \mathbb{R}$. The sequence $(f_n)$ converges pointwise [i.e., at each point] to a 1285 | function $f$ defined on S if 1286 | 1287 | $$ \lim\limits_{n\to \infty} f_n (x) = f(x) \quad \text{for all} \quad x\in S$$ 1288 | We often write $\lim f_n = f$ pointwise $[on\ S]$ or $f_n \to f$ pointwise [on S]. 1289 | 1290 | 1291 | Now observe $f_n\to f$ pointwise on S means exactly the following: 1292 | 1293 | for each $\epsilon>0$ and x in S there exists N such that 1294 | $$ |f_n(x) - f(x)| < \epsilon\quad \text{for}\quad n>N$$ 1295 | 1296 | Note the value of N depends on both $\epsilon$ and x in S. 1297 | \end{definition} 1298 | \paragraph{Example} 1299 | Let $f_n(x) = x^n$ for $x\in [0, 1]$. Then $f_n\to f$ pointwise on $[0, 1]$ where 1300 | $f(x) = 0$ for $x\in [0, 1)$ and $f(1) = 1$. Or we can write: 1301 | 1302 | $$ \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} x^n = \begin{cases} 1303 | 1\quad \text{if} \ x = 1\\ 1304 | 0 \quad \text{else} 1305 | \end{cases} $$ 1306 | Consider, 1307 | $$ f(x) = \begin{cases} 1308 | 1\quad \text{if} \ x = 1\\ 1309 | 0 \quad \text{else} 1310 | \end{cases} $$ 1311 | Then we can write $\lim_{n \to \infty} f_n (x) = f(x)$. 1312 | \paragraph{} 1313 | \begin{definition}[Uniform Convergence]{} 1314 | Let $(f_n)$ be a sequence of real-valued functions defined on a set 1315 | $S\subseteq \mathbb{R}$. The sequence $(f_n)$ converges uniformly on S to a function $f$ defined on S if 1316 | 1317 | for each $\epsilon>0$ there exists a number N such that 1318 | 1319 | $$ |f_n(x) - f(x)| < \epsilon\quad \text{for all}\quad x \in S \quad and \quad n>N$$. 1320 | We write $\lim f_n = f$ uniformly on S or $f_n \to f$ uiniformly on S. 1321 | \end{definition} 1322 | 1323 | 1324 | \paragraph{Example:}Define, 1325 | 1326 | $$f_n(x) = x^n\quad for\quad x\in [0,1]$$ 1327 | 1328 | $$ f(x) = \begin{cases} 1329 | 0, x\not = 0 \\ 1330 | 1, x = 0 1331 | \end{cases}$$ 1332 | 1333 | then $\epsilon = 1/2$,then we consider for all $x \in [0,1]$ and all $n >N$. 1334 | 1335 | $$ |f_n(x) - f(x)| < \epsilon = 1/2$$ 1336 | 1337 | $|x| \leq 1$ \& $n>N$ 1338 | 1339 | $$\Rightarrow |f_n - f| < 1/2$$ 1340 | at $x = 0$ it's not possible as $|0 - 1| < 1/2$ is not true. So, not uniformly converging. 1341 | 1342 | \paragraph{Example} 1343 | 1344 | 1345 | Let $f(x) = {1 \over n} sin(nx)$ $\forall x \in \mathbb{R}$ 1346 | 1347 | for any x $\lim_{n \to \infty} f_n(x) = 0$ pointwise on R. Define $f(x) = 0$ then $f_n \to f$. In fact $f_n \to f$ uniformly on R. Also, let $N = 1/\epsilon$ .Then for $n>N$ and all $x \in \mathbb{R}$ we have 1348 | 1349 | $$ |f_n(x) - f(x)| = |f_n(x) - 0| = \abs{{1 \over n}sin(nx)} \leq {1 \over n} <{1 \over N} = \epsilon$$ 1350 | Since, N is independent of x. 1351 | 1352 | \paragraph{} 1353 | 1354 | \begin{theorem}{} 1355 | Uniform limit of continuous function is continuous.More precisely, let $(f_n)$ be a sequence of functions on a set $S\subseteq \mathbb{R}$, suppose $f_n\to f$ uniformly on S, and suppose $S = dom(f)$. If each $f_n$ is 1356 | continuous at $x_0$ in S, then f is continuous at $x_0$. [So if each $f_n$ is continuous on S, then f is continuous on S.] 1357 | \end{theorem} 1358 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1359 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1360 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1361 | %%%%%%%% proof left %%%%%%%%%%%%%%\\ 1362 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1363 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1364 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1365 | \paragraph{Remark} Remember $f_n \to f$ on S uniformly iff, 1366 | 1367 | $$ \limsup \limits_{n \to \infty} \{|f_n(x) - f(x)|: x \in S\} = 0$$ 1368 | \paragraph{} 1369 | 1370 | We can also consider $\sum_{k = 0}^{\infty} g_k(x)$, where $g_k$ is a function of $S\subset\mathbb{R}$. 1371 | \begin{theorem}{} 1372 | If $g_k$ is continuous $\forall k$ and if $\sum g_k$ converges uniformly, then it converges to a continuous function on S. 1373 | \end{theorem} 1374 | 1375 | \begin{proof} 1376 | Let $f_n = \sum_{k = 0}^{n} g_k(x)$ is continuous on S and $f_n(x) = \sum_{k = 0}^{\infty} g_k(x)$ uniformly. 1377 | 1378 | $\therefore f(x) = \sum_{k = 0}^{\infty} g_k(x)$ is continuous. 1379 | \end{proof} 1380 | 1381 | \begin{corollary}{} 1382 | $f(x) = \sum a_n x^n $ is a continuous function if the convergence is uniform. 1383 | \end{corollary} 1384 | \begin{problem} 1385 | Prove that if $(f_n)$ is a sequence of uniformly continuous functions 1386 | on an interval $(a, b)$, and if $f_n\to f$ uniformly on $(a, b)$, then $f$ is 1387 | also uniformly continuous on $(a, b)$. 1388 | \end{problem} 1389 | \begin{proof} 1390 | We need to show that $\forall \delta>0\ \exists \epsilon>0$ such that $|x-y|<\delta$ implies $|f(x)-f(y)|<\epsilon.$ 1391 | 1392 | Using triangualar inequality we can show that: 1393 | $$|f(x)-f(y)| \leq |f(x)-f_n (x)|+|f_n(x)-f_n(y)|+|f_n (y)-f(y)|$$ 1394 | Using the $\epsilon/3$ argument we can show that, there exists a $N$ such that $n>N_1$ implies $ |f_n(x)-f_n(y)| <\epsilon/3$ (it's uniform cont.) and $|x-y| < \delta$.And we can apply definition of uniform convergence to show $|f_n (y)-f(y)| < \epsilon/3$ by choosing some $N_2$ for x and then some $N_2$ for y.So we have for $n> N = max(N_1,N_2,N_3)$. 1395 | $$|f(x)-f(y)| < 3\cdot {\epsilon \over 3}= \epsilon$$ 1396 | 1397 | \end{proof} 1398 | 1399 | 1400 | \chapter{Differentiation} 1401 | \section{Why differentiation? \\ Better ask Newton.} 1402 | \begin{definition}{} 1403 | $f$ is a real-valued function on an open interval $I$ Let $a \in I$. Then $f$ is differentiable at ``a" if the limit : 1404 | $$ \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$$ 1405 | exists \& is finite. We also sayy $f$ has a derivative at a. 1406 | 1407 | $$ f'(x) = \lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$$ 1408 | 1409 | \paragraph{Remark} 1410 | '$f'(x)$ itself is a function with domain $f \subseteq dom f$. 1411 | \end{definition} 1412 | \begin{theorem}{} 1413 | If $f$ be diff. at $x = a$. Then $f$ is continuous at a. 1414 | \end{theorem} 1415 | 1416 | \begin{proof} 1417 | $f(x) = (x - a) \frac{f(x)-f(a)}{x-a} + f(a)$ 1418 | \begin{align} 1419 | \therefore \lim_{x \to a}f(x) = \lim_{x \to a}(x - a) \lim_{x \to a}\frac{f(x)-f(a)}{x-a} + \lim_{x \to a}f(a) = 0\cdot f'(a) + f(a)\\ 1420 | \end{align} 1421 | \end{proof} 1422 | 1423 | \begin{theorem}[Properties]{} 1424 | If $f, g$ are diff. at $x = a$ and $c \in \mathbb{R}$ then 1425 | \begin{enumerate} 1426 | \item 1427 | $(cf)'(a) = cf'(a)$ 1428 | \item 1429 | $(f+g)'(a) = f'(a)+g'(a)$ 1430 | \item 1431 | $(fg)'(a) = f'(a)g(a)+f(a)g'(a)$ 1432 | \item 1433 | $({f \over g})'(a) = \frac{f'(a)g(a) - f(a)g'(a)}{g(a)^2}$ 1434 | \end{enumerate} 1435 | \end{theorem} 1436 | 1437 | 1438 | \begin{theorem}[\textbf{Chain Rule}]{} 1439 | $f $ be differentiable at $x = a$ and $g$ be differentiable at $f(a)$. Then $(g \circ f )(x)$ is diff at x = a. And $(g \circ f )'(x) = g'(f(x)) \cdot f'(x)$ 1440 | \end{theorem} 1441 | 1442 | \section{Mean Value Theorem} 1443 | 1444 | \begin{theorem}{} 1445 | If $f$ is defined on an open interval containing $x_0$, if $f$ assumes its 1446 | maximum or minimum at $x_0$, and if $f$ is differentiable at $x_0$, then 1447 | $f '(x_0) = 0$. 1448 | \end{theorem} 1449 | \begin{theorem}[Rolle’s Theorem]{thm:rolles theorem} 1450 | Let $f$ be a continuous function on $[a, b]$ that is differentiable on $(a, b)$ 1451 | and satisfies $f(a) = f(b)$. There exists [at least one] $x$ in $(a, b)$ such 1452 | that $f '(x) = 0$. 1453 | \end{theorem} 1454 | \begin{theorem}[Mean Value Theorem]{thm:mean value} 1455 | Let $f$ be a continuous function on $[a, b]$ that is differentiable on $(a, b)$. 1456 | Then there exists [at least one] x in $(a, b)$ such that: 1457 | $$ f'(x) = \frac{f(b) - f(a)}{b - a }$$ 1458 | \end{theorem} 1459 | \begin{corollary}{} 1460 | Let $f$ be a differentiable function on $(a, b)$ such that $f '(x) = 0$ for all 1461 | $x \in(a, b)$. Then $f$ is a constant function on $(a, b)$. 1462 | \end{corollary} 1463 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1464 | \begin{corollary}{} 1465 | Let $f$ and $g$ be differentiable functions on $(a, b)$ such that $f ' = g'$ on 1466 | $(a, b)$. Then there exists a constant c such that $f(x) = g(x) + c$ for 1467 | all $x \in(a, b)$. 1468 | \end{corollary} 1469 | 1470 | \begin{corollary}{} 1471 | Let $f$ be a differentiable function on interval $(a,b)$, then: 1472 | \begin{enumerate}[(i.)] 1473 | \item $f$ is strictly increasing if $f '(x) > 0$ for all $x \in(a, b)$; 1474 | \item $f$ is strictly decreasing if $f '(x) < 0$ for all $x \in(a, b)$; 1475 | \item $f$ is increasing if $f '(x) \geq 0$ for all $x \in(a, b)$; 1476 | \item $f$ is decreasing if $f '(x) \leq 0$ for all $x \in(a, b)$; 1477 | \end{enumerate} 1478 | \end{corollary} 1479 | 1480 | \begin{theorem}[Intermediate Value Theorem for Derivatives.]{} 1481 | Let $f$ be a differentiable function on $(a, b)$. If $a < x_1 < x_2 < b$, and 1482 | if c lies between $f'(x_1)$ and $f '(x_2)$, there exists [at least one] x in 1483 | $(x_1, x_2)$ such that $f '(x) = c$. 1484 | \end{theorem} 1485 | \begin{problem} 1486 | Let $f$ be differentiable on R with $a = \sup\{|f'(x)| : x \in \mathbb{R}\} < 1$. Select $x_0 \in \mathbb{R}$ and define $x_n =f(x_{n - 1})$ for $n \geq 1$. Thus $x_1 = f (x_0 ), x_2 = f (x_1 )$, etc. Prove $(x_n )$ is a convergent sequence. 1487 | \end{problem} 1488 | \begin{proof} 1489 | Using the mean value theorem we can see $f$ is a contraction, because 1490 | 1491 | $$|f(x) - f(y)| = |f'(c)||x-y| \leq a|x-y|$$ 1492 | With $a < 1$. 1493 | 1494 | 1495 | Now to see it is a convergent / cauchy sequence, select $x_0$ and note $x_n = f(x_{n-1})$, what can you say of $$|x_n - x_m|$$ for arbitrary $m, n$? 1496 | 1497 | HINT: Use triangle inequality and $|x_n - x_{n+1}| \leq a^n|x_0 - x_1| $ . For alternate proof refer to Problem \ref{problem-2^n} 1498 | 1499 | Suppose $m > n$ then: 1500 | 1501 | $$|x_n - x_m| \leq |x_n - x_{n+1}| + |x_{n+1} - x_{n+2}| + \dots + |x_{m-1} + x_{m}|$$ 1502 | 1503 | $$\leq \sum_{k = n}^m a^k |x_0 - x_1| \leq \sum_{k = n}^{\infty} a^k |x_0 - x_1| < +\infty $$ 1504 | Now, because this last series converges ($a < 1$), by the Cauchy criterion, given $\varepsilon >0$, there is some $N$ such that $$\sum_{k = N}^{\infty} a^k |x_0 - x_1| < \varepsilon $$ 1505 | If we pick $n,m \geq N$ we're done, now we know $x_n$ is a Cauchy sequence in $\mathbb{R}$, and thus has a limit $x$, now $$x = \lim x_n = \lim f(x_{n-1}) = f(\lim x_{n-1}) = f(x)$$ hence $x$ is a fixed point. To see its uniquenes suppose $x_1, x_2$ are two fixed points, then: 1506 | 1507 | $$|x_1 - x_2| = |f(x_1) - f(x_2)| \leq a|x_1 - x_2| $$ 1508 | 1509 | With $a < 1$ this is only true if $x_1 = x_2$. 1510 | \end{proof} 1511 | 1512 | \section{Taylor Theorem} 1513 | \begin{definition}[Taylors Series]{def:taylor} 1514 | Let $f$ be a function defined on some open interval containing c. If $f^k (c)$ 1515 | exists $\forall k$, then the series: 1516 | $$ \sum_{k=0}^{\infty} \frac{f^{(k)} (c)}{k!} (x -c)^k $$ 1517 | is called as \textbf{Taylor Series} of function $f(x)$ about c. For $n \geq 1$; remainder $R_n (x)$ is defined as: 1518 | $$R_n (x) = f(x) - \sum_{k=0}^{n-1} \frac{f^{(k)} (c)}{k!} (x -c)^k $$ 1519 | The remainder is important because, for any x; 1520 | $$ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)} (c)}{k!} (x -c)^k \quad \text{if and only if}\quad \lim\limits_{n \to \infty} R_n(x) = 0 $$ 1521 | \end{definition} 1522 | \begin{theorem}{thm:taylor} 1523 | Let $f$ be defined on $(a, b)$ where $a < c < b;$ here we allow $a = - \infty$ 1524 | or $b = \infty$. Suppose the $n\textsuperscript{th}$ derivative $f ^{n}$ exists on $(a, b)$. Then for 1525 | each $x \not = c$ in $(a, b)$ there is some $y$ between $c$ and $x$ such that 1526 | $$ R_n (x) = \frac{f^n (y)}{n!}(x-c)^n$$ 1527 | \end{theorem} 1528 | \begin{corollary}{} 1529 | Let $f$ be defined on interval $(a,b)$ and $a 0$ there exists a partition P of $[a, b]$ such that 1584 | $$U(f,P)-L(f,P) < \epsilon$$ 1585 | \end{theorem} 1586 | 1587 | \begin{definition}[Mesh]{} 1588 | The $mesh$ of a partition P is the maximum length of the subintervals 1589 | comprising P. Thus if: 1590 | $$P = \{a = t_0 < t_1 0$ there exists 1606 | $\delta > 0$ such that 1607 | $$|S-r| < \delta $$ 1608 | for every Rienmann sum of $f$ associated with the partition P having mesh$(P)< \delta$. The number $r$ is the Riemann integral of $f$ on $[a,b]$ and will be provisionally written as $\mathcal{R} \int_{a}^{b}f$. 1609 | \end{definition} 1610 | 1611 | \begin{theorem}{} 1612 | A bounded function $f$ on $[a, b]$ is integrable if and only if for each 1613 | $\epsilon > 0$ there exists a $\delta > 0$ such that 1614 | $mesh(P) < \delta$ implies 1615 | $$U(f, P) - L(f, P) < \epsilon$$ 1616 | 1617 | for all partitions P of $[a, b]$. 1618 | \end{theorem} 1619 | 1620 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1621 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1622 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\ 1623 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\ 1624 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\\ 1625 | 1626 | 1627 | \subsection{Fundamental Theorem of Calculus} 1628 | 1629 | \begin{theorem}{thm:fund-cal-1} 1630 | If $g$ is a continuous function on $[a, b]$ that is differentiable on $(a, b)$, 1631 | and if $g'$ is integrable on $[a, b]$, then 1632 | $$\int_{a}^{b} g' = g(b) - g(a)$$ 1633 | \end{theorem} 1634 | \begin{theorem}{thm:fund-cal-2} 1635 | Let $f$ be an integrable function on $[a, b]$. For $x$ in [a, b], let 1636 | $$F(x) = \int_{a}^{x}f(t)dt.$$ 1637 | Then $F$ is continuous on $[a, b]$. If f is continuous at $x_0$ in $(a, b)$, then 1638 | $F $ is differentiable at $x_0$ and 1639 | $$F'(x) = f(x)$$ 1640 | \end{theorem} 1641 | 1642 | 1643 | \subsection{Intermediate Value Theorem for Integrals.} 1644 | 1645 | \begin{theorem}[Intermediate Value Theorem for Integrals.]{} 1646 | If $f$ is a continuous function on $[a, b]$, then for at least one $x$ in $(a, b)$ 1647 | we have: 1648 | \[ f(x) = \int_{a}^{b} f \] 1649 | \end{theorem} 1650 | 1651 | 1652 | \begin{problem} 1653 | Let $f(x) = x$ for rational $x$ and $f(x) = 0$ for irrational $x$. 1654 | \begin{enumerate} 1655 | \item 1656 | Calculate the upper and lower Darboux integrals for $f$ on the 1657 | interval $[0, b]$. 1658 | \item 1659 | Is $f$ integrable on $[0, b]$? 1660 | \end{enumerate} 1661 | \end{problem} 1662 | \chapter{Differentiation and Integration of Power Series} 1663 | \begin{theorem}[Weierstrass M-test]{} 1664 | Let $(M_k)$ be a sequence of non-negative real numbers where $\sum M_k < \infty$. If $|g_k(x)| \leq M_k$ for all $x$ in a set $S$, then $\sum g_k$ converges 1665 | uniformly on S 1666 | \end{theorem} 1667 | \begin{proof} 1668 | Let $\epsilon>0$ , $\exists N $ such that $n \geq m > N \Rightarrow \sum_{k = m}^{\infty} M_k < \epsilon$. Then if $ n \geq m > N $ and $ x \in S $, 1669 | $$\abs{\sum_{k = m}^{n}g_k(x)} \leq\sum_{k = m}^{n}|g_k(x)| \leq \sum_{k = m}^{n}M_k < \epsilon $$ 1670 | $ \therefore $ by cauchy criterion on uniform cont. on S, 1671 | $ \sum g_k $ converges uniformly on S. 1672 | \end{proof} 1673 | \begin{problem} 1674 | Show that if the series $ \sum g_n $ converges uniformly on a set S, then 1675 | $$\lim\limits_{n \to \infty} \sup\{|g_n(x)|: x\in S\} = 0 $$ 1676 | \end{problem} 1677 | \begin{proof} 1678 | For $\epsilon > 0$ , $\exists N$ such that $n \geq m > N \Rightarrow \left| \sum_{m}^{n} g_k (x)\right| < \epsilon$.\\ 1679 | In particular $n>N$ 1680 | $$\Rightarrow |g_n| < \epsilon\ \forall \ x \in S $$ 1681 | $$\therefore \sup \{|g_n(x)| : x \in S \} \leq \epsilon$$ 1682 | $$\lim_{n \to \infty }\sup \{|g_n(x)|: x\in s\} = 0$$ 1683 | \end{proof} 1684 | 1685 | 1686 | \begin{theorem}{} 1687 | Let $ \sum a_n x^n $ be a power series with radius of convergence $R > 0$. If $0 < R_1 < R$, then the power series converges 1688 | uniformly on $[-R_1, R_1]$ to a continuous function. 1689 | \end{theorem} 1690 | 1691 | \begin{lemma} 1692 | If the power series $ \sum a_n x^n $ has radius of convergence R, then the 1693 | power series 1694 | $$\sum n a_n x^n \qquad \& \qquad \sum \dfrac{a_n}{n+1} x^{n+1}$$ 1695 | have the same radius of convergence R. 1696 | \end{lemma} 1697 | 1698 | \begin{theorem}{} 1699 | Suppose $f(x) = \sum a_n x^n $ has radius of convergence $R > 0$ Then 1700 | $$\int_{0}^{x}f(t) dt = \sum \frac{a_n}{n+1}x^{n+1}\ \ \forall x < R$$ 1701 | \end{theorem} 1702 | 1703 | \begin{theorem}{} 1704 | Let $f(x) = \sum a_n x^n $ have radius of convergence $R > 0$. Then $f$ is 1705 | differentiable on $(-R, R)$ and 1706 | $$f'(x) = \sum_{n=1}^{\infty} a a_n x^{n-1}\quad \forall |x| < R$$ 1707 | \end{theorem} 1708 | \begin{theorem}[Abel's Theorem]{} 1709 | Let $f(x) = \sum a_n x^n $ be a power series with $0< R<\infty$. If the series converges at $x = R$, then $f$ is continuous 1710 | at $x = R$. If the series converges at $x = -R$, then $f$ is continuous 1711 | at $x = -R$. 1712 | \end{theorem} 1713 | 1714 | 1715 | 1716 | \appendix 1717 | \addappheadtotoc 1718 | \chapter{Power series of some common functions} 1719 | 1720 | \begin{description} 1721 | \item [\textbf{Exponential Series}] 1722 | $$e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots.$$ 1723 | It converges for all $x \in \mathbb{R}$. 1724 | \item[Natural logarithm] 1725 | \[\begin{aligned} 1726 | \ln(1-x) &= - \sum^{\infty}_{n=1} \frac{x^n}n = -x - \frac{x^2}2 - \frac{x^3}3 - \cdots , \\ 1727 | \ln(1+x) &= \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n = x - \frac{x^2}2 + \frac{x^3}3 - \cdots . 1728 | \end{aligned}\] 1729 | They converge for $|x| < 1$ 1730 | \item[Geometric series] 1731 | \[\begin{aligned} 1732 | \frac{1}{1-x} &= \sum^\infty_{n=0} x^n \\ 1733 | \frac{1}{(1-x)^2} &= \sum^\infty_{n=1} nx^{n-1}\\ 1734 | \frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}. 1735 | \end{aligned}\] All are convergent for \(|x| < 1\). These are special cases of Binomial Series. 1736 | 1737 | \item[Binomial series] The binomial series is a power series 1738 | \[(1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n\] 1739 | whose coefficients are the generalized 1740 | \href{https://en.wikipedia.org/wiki/Binomial_coefficient}{binomial coefficients} 1741 | \[\binom{\alpha}{n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.\] 1742 | \item[Trignometric Functions] 1743 | The usual trigonometric functions and 1744 | their inverses have the following power series: 1745 | 1746 | \[\begin{aligned} 1747 | \sin x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} &&= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots && \text{for all } x\\[6pt] 1748 | \cos x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} &&= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots && \text{for all } x\\[6pt] 1749 | \tan x &= \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n \left(1-4^n\right)}{(2n)!} x^{2n-1} &&= x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt] 1750 | \sec x &= \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n} &&=1+\frac{x^2}{2}+\frac{5x^4}{24}+\cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt] 1751 | \arcsin x &= \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} &&=x+\frac{x^3}{6}+\frac{3x^5}{40}+\cdots && \text{for }|x| \le 1\\[6pt] 1752 | \arccos x &=\frac{\pi}{2}-\arcsin x\\&=\frac{\pi}{2}- \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}&&=\frac{\pi}{2}-x-\frac{x^3}{6}-\frac{3x^5}{40}-\cdots&& \text{for }|x| \le 1\\[6pt] 1753 | \arctan x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1} &&=x-\frac{x^3}{3} + \frac{x^5}{5}-\cdots && \text{for }|x| \le 1,\ x\neq\pm i 1754 | \end{aligned}\] 1755 | 1756 | All angles are expressed in radians.The numbers $B_k$ 1757 | appearing in the expansions of $\tan x$ are the 1758 | \href{https://en.wikipedia.org/wiki/Bernoulli_number}{Bernoulli numbers}. The $E_k$ in the expansion of $\sec x$ are 1759 | \href{https://en.wikipedia.org/wiki/Euler_number}{Euler numbers}. 1760 | \end{description} 1761 | 1762 | 1763 | 1764 | \bibliographystyle{unsrt} 1765 | \bibliography{ref} 1766 | 1767 | 1768 | \end{document} 1769 | --------------------------------------------------------------------------------