From 230946b2aaec52b371ff6b6bd920cc3ebb4732c7 Mon Sep 17 00:00:00 2001 From: Benjamin Barenblat Date: Mon, 14 Mar 2011 23:46:17 -0400 Subject: Initial commit --- .gitignore | 4 + 18.022.sty | 105 ++++++++++++++ aidsheet.tex | 340 ++++++++++++++++++++++++++++++++++++++++++++++ matrix-multiplication.png | Bin 0 -> 13917 bytes 4 files changed, 449 insertions(+) create mode 100644 .gitignore create mode 100644 18.022.sty create mode 100644 aidsheet.tex create mode 100644 matrix-multiplication.png diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..ef5f7e5 --- /dev/null +++ b/.gitignore @@ -0,0 +1,4 @@ +*.aux +*.log +*.out +*.pdf diff --git a/18.022.sty b/18.022.sty new file mode 100644 index 0000000..a2fe1df --- /dev/null +++ b/18.022.sty @@ -0,0 +1,105 @@ +%% 18.022.sty - definitions for 18.022 +%% +%% Copyright (C) 2009, 2011 Benjamin Barenblat +%% http://benjamin.barenblat.name/ +%% +%% This document is licensed under the Creative Commons +%% Attribution-NonCommercial-ShareAlike 3.0 United States License. +%% For more information, see +%% http://creativecommons.org/licenses/by-nc-sa/3.0/us/. + +%% Identification +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{18.022}[2011/03/14 Definitions for 18.022] + +%% Preliminary declarations +\RequirePackage{amsmath} +\RequirePackage{amssymb} +\RequirePackage{amsthm} +\RequirePackage{mathrsfs} +\RequirePackage{amsbsy} + +%% Options + +%% Main package code +% Redefine proof environment to make it fit better +% \renewenvironment{proof}[1][\proofname]{\setlength{\parindent}{1em}\par +% \pushQED{\qed}% +% \normalfont \topsep6\p@\@plus6\p@\relax +% \trivlist +% \item[\hskip\labelsep +% \itshape +% #1\@addpunct{.}]\ignorespaces +% }{% +% \popQED\endtrivlist\@endpefalse +% } + +% Styling +\renewcommand{\vec}[1]{\mathbf{#1}} +\newcommand{\gvec}[1]{\boldsymbol{#1}} +\newcommand{\cvec}[1]{\begin{bmatrix}#1\end{bmatrix}} +\newcommand{\bra}{\begin{bmatrix}} +\newcommand{\ket}{\end{bmatrix}} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert} +\newcommand{\abs}[1]{\left\lvert#1\right\rvert} +\newcommand{\Epsilon}{\epsilon} +\renewcommand{\epsilon}{\varepsilon} +\renewcommand{\implies}{\Rightarrow} +\renewcommand{\le}{\leqslant} +\renewcommand{\ge}{\geqslant} +\renewcommand{\iff}{\Leftrightarrow} +\renewcommand{\leq}{\leqslant} +\renewcommand{\geq}{\geqslant} + +% Abbreviations +\renewcommand{\v}[1]{\vec{#1}} +\newcommand{\cv}[1]{\cvec{#1}} +\newcommand{\half}{\frac{1}{2}} +\newcommand{\cross}{\times} +\let\flux\dot +\renewcommand{\dot}{\cdot} +\newcommand{\R}{\mathbf{R}} +\newcommand{\0}{\vec{0}} +\newcommand{\of}{\circ} +\newcommand{\at}[1]{\Big\vert_{#1}} +\newcommand{\ds}{\displaystyle} +\newcommand{\grad}{\nabla} +\renewcommand{\div}{\nabla\dot} +\newcommand{\curl}{\nabla\cross} +\newcommand{\laplacian}{\nabla^2} + +% \dd is the basic partial derivative command, for forms like +% n +% d y +% --- , +% n +% dx +% the nth derivative of y with respect to x. +\newcommand{\dd}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} +\newcommand{\ddx}[1]{\dd{#1}{x}} +\newcommand{\ddy}[1]{\dd{#1}{y}} +\newcommand{\ddz}[1]{\dd{#1}{z}} + +% For mixed partials, we have \ddd. +\newcommand{\ddd}[3][]{\frac{\partial^{#1} #2}{\partial #3}} +\newcommand{\dddx}[1]{\dds{#1}{x}} +\newcommand{\dddy}[1]{\dds{#1}{y}} +\newcommand{\dddz}[1]{\dds{#1}{z}} + +% For mixed second partials, we have \dds. +\newcommand{\dds}[2]{\ddd[2]{#1}{#2}} + +\newcommand{\xo}{x_0} +\newcommand{\yo}{y_0} +\newcommand{\zo}{z_0} + +% New stuff +\DeclareMathOperator{\graph}{Graph} +\DeclareMathOperator{\proj}{Proj} +\DeclareMathOperator{\sgn}{sgn} + +% Define 18.022 stuff for pset.cls +\def\@course{18.022} +\def\@fullcourse{Multivariate and Vector Calculus} +\def\@department{Department of Mathematics} +\def\@school{Massachusetts Institute of Technology} diff --git a/aidsheet.tex b/aidsheet.tex new file mode 100644 index 0000000..067de15 --- /dev/null +++ b/aidsheet.tex @@ -0,0 +1,340 @@ +%% 18.022 cheat sheet +%% +%% Copyright (C) 2009, 2010, 2011 Benjamin Barenblat +%% http://benjamin.barenblat.name/ +%% +%% This document is licensed under the Creative Commons +%% Attribution-NonCommercial-ShareAlike 3.0 United States License. +%% For more information, see +%% http://creativecommons.org/licenses/by-nc-sa/3.0/us/. +%% +%% This document is designed to be typeset with pdfLaTeX. +\documentclass[10pt,landscape]{article} +\usepackage{eco} +\usepackage{geometry} +\usepackage{multicol} +\usepackage{mathtools} +\usepackage{color} +\usepackage[colorlinks]{hyperref} +\definecolor{darkred}{rgb}{0.5,0,0} +\hypersetup{colorlinks,linkcolor=black,urlcolor=darkred} +\usepackage{graphicx} +\usepackage{wrapfig} +\usepackage{18.022} +\usepackage{esint} +\usepackage{titlesec} + +% Only show equation numbers of referenced equations. +\mathtoolsset{showonlyrefs=true} + +% Set margin. +\geometry{margin=6mm} + +% Turn off header and footer. +\pagestyle{empty} + +% Don't print subsection numbers. +\setcounter{secnumdepth}{0} + +%% Squash paragraph headings. +\titlespacing*{\paragraph}{0pt}{.94em}{*1} + +\setlength{\parindent}{0pt} +\setlength{\parskip}{0pt} + +\makeatletter +\newcommand{\@recitation}{} +\newcommand{\recitation}[1]{\renewcommand{\@recitation}{#1}} +\renewcommand{\maketitle}{% + \begin{center} + \LARGE{\textbf{\@title}} + \end{center} +} +\newcommand{\makeend}{% + \vfill + \rule{0.3\linewidth}{0.25pt} + \scriptsize + \begin{tabular}{@{}l} + Copyright \copyright\ \@date\ \@author. \href{http://creativecommons.org/licenses/by-nc-sa/3.0/us/}{\textsc{cc by$\cdot$nc$\cdot$sa}}. No warranty.\\ + \url{http://benjamin.barenblat.name/} + \end{tabular} +} +\makeatother + +\title{18.022 Cheat Sheet} +\author{Benjamin Barenblat} +\date{2009--2011} + +\begin{document} +\raggedright +\begin{multicols*}{3} +\maketitle + +\section{Vectors} +\paragraph{Dot product} +Let $\vec v, \vec w, \vec u \in \R^n$. +Then, $\vec v \dot \vec w = \langle \vec v, \vec w \rangle = \langle\vec v \vert \vec w\rangle = \bra v_1w_1 & \cdots & v_nw_n \ket$. + +\paragraph{Cauchy-Schwarz inequality} +Let $\vec v, \vec w \in \R^n$. +Then, $(\vec v \dot \vec w)^2 \le (\vec v \dot \vec v)(\vec w \dot \vec w) \iff \abs{\vec v \dot \vec w} \le \norm{\vec v}\norm{\vec w}$. + +\paragraph{Triangle inequality} +Let $\vec v, \vec w \in \R^n$. +Then, $\norm{\vec v + \vec w} \le \norm{\vec x} + \norm{\vec y}$ and $\norm{\vec v - \vec w} \ge \big\lvert\norm{\vec v} - \norm{\vec w}\big\rvert$. + +\paragraph{Projection} +For $\vec a, \vec b \in \R^n$, the projection of $\vec b$ onto $\vec a$ +\begin{equation} + \label{eq:projection} + \proj_{\vec a}\vec b = \frac{\vec a \dot \vec b}{\norm{\vec a}}\vec{\hat a} = \frac{\vec a \dot \vec b}{\norm{\vec a}^2}\vec a = \frac{\vec a \dot \vec b}{\vec a \dot \vec a}\vec a +\end{equation} +and $\norm{\proj_{\vec a}\vec b} = \norm{\vec b\cos\theta}$. + +\paragraph{Distance from a point to a line} +Given a point $\vec p$ and a line $\vec l(t) = \vec vt + \vec q$, the shortest vector from $\vec p$ to $\vec l$ is +\begin{equation} + \vec q - \vec p - \proj_{\vec v}(\vec q - \vec p) = \vec q - \vec p - \frac{(\vec q - \vec p) \dot \vec v}{\norm{\vec q - \vec p}^2}\vec v. +\end{equation} + +\paragraph{Cross product} +Let $\vec v, \vec w \in \R^3$. +Then, +\begin{equation} + \label{eq:crossproduct} + \vec v \cross \vec w = \bra v_1 \\ v_2 \\ v_3 \ket \cross \bra w_1 \\ w_2 \\ w_3 \ket = \bra v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \ket +\end{equation} +and +$\norm{\vec v \cross \vec w} = \norm{\vec v}\norm{\vec w}\abs{\sin\theta}$. + +\paragraph{Planes} +Let $\vec x = \bra x_1 & \cdots & x_n \ket \in \R^n$. +For a point $\vec p$ and normal vector $\vec n$, $(\vec x - \vec p) \dot \vec n = 0$. +\begin{list}{\textbullet}{\setlength{\itemsep}{0pt}} +\item Point and two vectors: $\vec n = \vec u \cross \vec v$. +\item Three points: $\vec n = (\vec q - \vec p) \cross (\vec r - \vec p) = 0$. +\item Function and point: $\vec n = \grad f(\vec p)$. +\end{list} + +\paragraph{Triple scalar product / determinant} +For $\vec u, \vec v, \vec w \in \R^3$, +\begin{equation} + \vec u \dot \vec v \cross \vec w = \det \bra \vec u & \vec v & \vec w \ket +\end{equation} +is the volume of the parallelepiped spanned by $\vec u$, $\vec v$, and $\vec w$. +$(\vec u, \vec v, \vec w)$ is right-handed iff $\det \bra \vec u & \vec v & \vec w \ket > 0$. + +\paragraph{Matrix Multiplication} +Dimensionally, the number of columns in the first matrix must match the number of rows in the second. +\begin{wrapfigure}[5]{r}{1.4in} + \includegraphics[width=1.4in]{matrix-multiplication} + \scriptsize Copyright \textcopyright\ 2009 \href{http://en.wikipedia.org/wiki/User:Fangfufu}{Fangfufu}.\\\textsc{\href{http://creativecommons.org/licenses/by-sa/3.0/}{cc by$\cdot$sa}}. +\end{wrapfigure} +\begin{multline} + \bra a & b & c \\ d & e & f \\ g & h & i \ket \bra \alpha & \delta \\ \beta & \epsilon \\ \gamma & \zeta \ket = \\ \bra a\alpha + b\beta + c\gamma & a\delta + b\epsilon + c\zeta \\ d\alpha + e\beta + f\gamma & d\delta + e\epsilon + f\zeta \\ g\alpha + h\beta + i\gamma & g\delta + h\epsilon + i\zeta\ket +\end{multline} +$a_{1,2}$ refers to the element in the $1$st row, $2$nd column. +Generally, for two matrices $A$ and $B$, +\begin{equation} + (AB)_{i,j} = A_{i,1} + B_{1,j} + A_{i,2}B{2,j} + \cdots + A_{i,n}B_{n,j}. +\end{equation} + +\section{Differential calculus} +\paragraph{Gradient} +For a scalar field $f : \R^n \to \R$, the gradient of $f$ +\begin{equation} + \grad f = \bra D_1f \\ D_2f \\ D_3f \\ \vdots \\ D_nf \ket, +\end{equation} +a new vector field which consistently points in the direction of $f$'s greatest increase with magnitude equal to the rate of that increase. +By corollary, $\grad f$ is always perpendicular to $f$'s level curves. + +\paragraph{Divergence} +For a vector field $\vec F : \R^n \to \R^n$, the divergence of $\vec F$ +\begin{equation} + \div\vec F = D_1F_1 + D_2F_2 + D_3F_3 + \cdots + D_nF_n, +\end{equation} +a new scalar field. +Positive values of $\div\vec F$ indicate field sources, while negative values indicate field sinks. + +\paragraph{Curl} +For a vector field $\vec F : \R^3 \to \R^3$, the curl of $\vec F$ +\begin{equation} + \curl\vec F = \bra D_2F_3 - D_3F_2 \\ D_3F_1 - D_1F_3 \\ D_1F_2 - D_2F_1 \ket, +\end{equation} +a new vector field measuring the rate of rotation at each point. + +\paragraph{Laplacian} +For a scalar field $f : \R^n \to \R$, the Laplacian of $f$ +\begin{equation} + \nabla^2 f = \div\grad f = D_1^2f + D_2^2f + D_3^2f + \cdots + D_n^2f. +\end{equation} + +\paragraph{Conservative vector fields} +Let $\vec F : U \subseteq \R^n \to \R^n$ ($U$ open) be a vector field +of class $C^1$. If there exists a class $C^2$ scalar field $f : U \to +\R$ such that $\vec F = \grad f$ on $U$, then $\vec F$ is conservative +on $U$. If $\vec F$ is conservative on $U$, then $\vec F$ is also +curl-free on $U$. (The converses are true if $U$ is simply connected.) + +\paragraph{Chain rule} +If $\vec f : \R^n \to \R^m$ is differentiable at $\vec x$ and $\vec g +: \R^m \to \R^p$ is differentiable at $\vec f(\vec x)$, then $\vec g +\circ \vec f$ is differentiable at $\vec x$ and +\begin{equation} + \label{eq:chainrule} + D(\vec g \circ \vec f)_{\vec x} = \big(D\vec g_{\vec f(\vec x)}\big)(D\vec f_{\vec x}). +\end{equation} + +\paragraph{Implicit function theorem} +Let $\vec F : \R^{n+m} \to \R^m$ (i.e., $m$ functions in $n + m$ unknowns) be of class $C^1$ and let $\vec F(\vec x_0) = \0$ for some $\vec x_0 \in \R^{n+m}$. +Write $\vec x = (\vec a, \vec b)$, where $\vec a \in \R^n$ and $\vec b \in \R^m$; write $\vec x_0 = (\vec a_0, \vec b_0)$, where $\vec a_0 \in \R^n$ and $\vec b_0 \in \R^m$. +Note that $D\vec F = \bra D_{\vec a}\vec F & D_{\vec b}\vec F \ket$. +If $D_{\vec b}\vec F(\vec b_0)$ is invertible (i.e., $\det D_{\vec b}\vec F(\vec b_0) \ne 0$), then there exists a neighborhood $U$ of $\vec a_0$ in $\R^n$ and a neighborhood $V$ of $\vec b_0$ in $\R^m$ and a function $\vec f : U \to V$ such that $F\big(\vec a_0, f(\vec a_0)\big) = 0$. +$\vec f$ expresses $\vec b$ in terms of $\vec a$ in the neighborhood of $(\vec a_0, \vec b_0)$, and +\begin{equation} + \label{eq:implicitdiff} + D\vec f_{\vec a_0} = -\big(D_{\vec b}\vec F(\vec b_0)\big)^{-1}\big(D_{\vec a}\vec F(\vec a_0)\big). +\end{equation} + +\paragraph{Taylor's theorem} +The $k$th-order Taylor polynomial of a class $C^k$ function $f : \R^2 \to \R$ at $\vec x \in \R^2$ near a point $\vec a \in \R^2$ +\begin{equation} + T^kf_{\vec a}(\vec x - \vec a) = \sum_{\substack{n,m\\n+m \le k}} \frac{D_1^nD_2^m f(a_1,a_2)}{n!m!}(x - a_1)^n(y - a_2)^m. +\end{equation} + +\paragraph{Extrema} +Consider a function $f : \R^n \to \R$ of class $C^2$. +$f$ has critical points where $Df = \0$; if $f$ has local extrema, they will occur at critical points. +At each critical point, +\begin{itemize} +\item If $D^2f$'s minors are all positive, $D^2f$ is positive definite and the point is a local minimum. +\item If $-D^2f$'s minors are all positive, $D^2f$ is negative definite and the point is a local maximum. +\item If neither of these are true, but $D^2f$ is invertible, $D^2f$ is indefinite and the point is a saddle point. +\item If $D^2f$ is not invertible, then the point is degenerate. +\end{itemize} +If a function $f : K \to \R$ is of class $C^1$ on $K$ and continuous on the interior of $\partial K$ and $K$ is closed and bounded, then $f$ has at least one minimum and at least one maximum on $K$. +The extrema will occur inside $K$ at critical points or somewhere on $\partial K$. + +\paragraph{Constrained optimization (one constraint)} +Consider a differentiable function $f : \R^n \to \R$ on a set $\mathscr{C} \subseteq \R$ defined as the level set of some function -- i.e., for some $C^1$ function $g$, $\mathscr{C} = \{ \vec x \in \R^n : g(\vec x) = 0 \}$. +Define $L(\vec x, \lambda) = f(\vec x) - \lambda g(\vec x)$. +If, at a point $\vec x_0$, $\grad L(\vec x_0, \lambda_0) = 0$ for some constant $\lambda_0$ and $\grad g(\vec x_0) \ne 0$, then $f\vert_\mathscr{C}$ has a critical point at $\vec x_0$. + +\paragraph{Constrained optimization ($k$ constraints)} +Consider a differentiable function $f : \R^n \to \R$ on a set $\mathscr{C} \subseteq \R$ defined as the level set of some family of functions -- i.e., for some $C^1$ function $\vec g = \bra g_1 & g_2 & \cdots & g_n \ket$, $\mathscr{C} = \{ \vec x \in \R^n : \vec g(\vec x) = \0 \}$. +Define $L(\vec x, \boldsymbol\lambda) = f(\vec x) - \boldsymbol\lambda \dot \vec g(\vec x)$. +If, at a point $\vec x_0$, $\grad L(\vec x_0, \boldsymbol\lambda_0) = 0$ for some constant $\boldsymbol\lambda_0$ and $\bra \grad g_1(\vec x_0) & \cdots & \grad g_k(\vec x_0) \ket$ has a $k\times k$ submatrix that is invertible, then $f\vert_\mathscr{C}$ has a critical point at $\vec x_0$. + +\section{Integral calculus} +\paragraph{Double integrals} +Let $f : \R^2 \to \R$ be Riemann integrable on some nice domain $D \subseteq \R^2$. +Define $a, b, c, d \in \R$ such that $[a,b] \times [c,d]$ is $D$'s bounding box. +Now define $f$'s extension +\begin{equation} + f^{\text{ext}}(x,y,z) = \begin{cases} + f(x,y) & \text{if $(x,y) \in D$},\\ + 0 & \text{otherwise}. + \end{cases} +\end{equation} +Then, +\begin{equation} + \iint_D f(x,y) dxdy = \int_a^b \int_c^d f^{\text{ext}}(x,y) dydx. +\end{equation} +In practice, +\begin{equation} + \iint_D f(x,y) dxdy = \int_a^b \int_{g(x)}^{h(x)} f^{\text{ext}}(x,y) dydx +\end{equation}j +for some functions $g$ and $h$. + +\paragraph{Triple integrals} +Let $f : \R^3 \to \R$ be Riemann integrable on some nice domain $D \subseteq \R^3$. +Define $a, b, c, d, e, f \in \R$ such that $[a,b] \times [c,d] \times [e,f]$ is $D$'s bounding box. +Now define $f$'s extension +\begin{equation} + f^{\text{ext}}(x,y,z) = \begin{cases} + f(x,y,z) & \text{if $(x,y,z) \in D$},\\ + 0 & \text{otherwise}. + \end{cases} +\end{equation} +Then, +\begin{equation} + \iiint_D f(x,y,z) dxdydz = \int_a^b \int_c^d \int_e^f f^{\text{ext}}(x,y,z) dzdydx. +\end{equation} +In practice, +\begin{multline} + \iiint_D f(x,y,z) dxdydz\\ = \int_a^b \int_{g(x)}^{h(x)} \int_{p(x,y)}^{q(x,y)} f^{\text{ext}}(x,y,z) dzdydx +\end{multline} +for some functions $g$, $h$, $p$, and $q$. + +\paragraph{Change of variables} +Let $f : D \subset \R^n \to \R$ be Riemann integrable on some nasty domain $D$ and let $\boldsymbol\Phi : D^* \subset \R^n \to D$ be such that $\boldsymbol\Phi$ is of class $C^1$, $\boldsymbol\Phi$ is one-to-one, $\boldsymbol\Phi$ is invertible on its domain (i.e., $\det D\boldsymbol\Phi_{\vec u} \ne 0$ for all $\vec u \in D^*$), and $\boldsymbol\Phi(D^*) = D$. Then, +\begin{equation} + \int_D f(\vec x) d\vec x = \int_{D^*} f\big(\boldsymbol\Phi(\vec u)\big)\abs{\det D\boldsymbol\Phi_{\vec u}} d\vec u. +\end{equation} +For well-known coordinate systems, $dxdy = rdrd\theta$; $dxdydz = \rho^2\sin\phi d\rho d\phi d\theta = rdrd\theta dz$. + +\paragraph{Scalar line integrals} +Let $\vec x(t) : \R \to \R^n$ parametrize a curve $C$ in $\R^n$ with endpoints $\vec x(a)$ and $\vec x(b)$; let $f(\vec x) : \R^n \to \R$ be a function defined on $C$. +Then, +\begin{equation} + \int_C fds = \int_a^b f\big(\vec x(t)\big) \norm{\vec{\flux x}(t)} dt. +\end{equation} + +\paragraph{Vector line integrals} +Let $\vec x(t) : \R \to \R^n$ parametrize a curve $C$ in $\R^n$ with endpoints $\vec x(a)$ and $\vec x(b)$; let $\vec F(\vec x) : \R^n \to \R^m$ be a vector field defined on $C$. +Then, +\begin{equation} + \int_C \vec F \dot d\vec s = \int_a^b \vec F\big(\vec x(t)\big) \dot \vec{\flux x}(t) dt. +\end{equation} + +\paragraph{Scalar surface integrals} +Let $\vec X(u,v) : \R^2 \to \R^n$ be a piecewise smooth parametrization of a surface $\mathscr{S}$ in $\R^n$ such that $\vec X(D) = \mathscr{S}$; let $f(\vec X) : \R^n \to \R$ be a function defined on $\mathscr{S}$. +Then, +\begin{equation} + \iint_\mathscr{S} fdS = \iint_D f\big(\vec X(u,v)\big) \norm{\frac{\partial\vec X}{\partial u} \cross \frac{\partial\vec X}{\partial v}} dudv. +\end{equation} + +\paragraph{Vector surface integrals} +Let $\vec X(u,v) : \R^2 \to \R^n$ be a piecewise smooth parametrization of a surface $\mathscr{S}$ in $\R^n$ such that $\vec X(D) = \mathscr{S}$; let $\vec F(\vec X) : \R^n \to \R^m$ be a vector field defined on $\mathscr{S}$. +Then, +\begin{equation} + \iint_\mathscr{S} \vec F \dot d\vec S = \iint_D \vec F\big(\vec X(u,v)\big) \dot \frac{\partial\vec X}{\partial u} \cross \frac{\partial\vec X}{\partial v} dudv. +\end{equation} +The first member of this definition, $\iint_\mathscr{S} \vec F \dot d\vec S$, is also called the flux of $\vec F$ through $\mathscr{S}$. + +\section{Fundamental theorems} +\paragraph{The first fundamental theorem of calculus} +If $f : \R^n \to \R$ is of class $C^1$ and $C$ is a smooth curve in $\R^n$ with endpoints $\vec x_0$ and $\vec x_1$, then +\begin{equation} + \int_C \grad f \dot d\vec s = f(\vec x_1) - f(\vec x_0). +\end{equation} + +\paragraph{Green's theorem} +Let $D$ be a closed set in $\R^2$ such that $\partial D$ is a collection of closed curves oriented such that $D$ is to the left. +If $\vec F: D \to \R^2$ is of class $C^1$, then +\begin{equation} + \oint_{\partial D} \vec F \dot d\vec s = \iint_D \curl \vec F \dot \vec k dxdy. +\end{equation} +This is a special case of Stokes' theorem. + +\paragraph{Gauss's theorem (the second fundamental theorem of calculus)} +Let $\Omega \in \R^3$ be a closed domain whose boundary is a piecewise smooth surface $\partial\Omega$. +Give $\partial\Omega$ outward-pointing orientation. +If $\vec F$ is a $C^1$ vector field in $\Omega$, then +\begin{equation} + \oiint_{\partial\Omega} \vec F \dot d\vec S = \iiint_\Omega \div\vec F dxdydz. +\end{equation} + +\paragraph{Stokes' theorem (the third fundamental theorem of calculus)} +Let $\mathscr{S}$ be a piecewise smooth surface in $\R^3$ with a given continuous normal vector field $\vec N$. +Let $\partial\mathscr{S}$ be a collection of piecewise smooth curves. +Orient the curves such that the outside of the surface is to the left. +Now let $\vec F : \mathscr{S} \to \R$ be a $C^1$ vector field. +Then, +\begin{equation} + \oint_{\partial\mathscr{S}} \vec F \dot d\vec s = \iint_\mathscr{S} \curl \vec F \dot d\vec S. +\end{equation} + +\makeend +\end{multicols*} +\end{document} diff --git a/matrix-multiplication.png b/matrix-multiplication.png new file mode 100644 index 0000000..79fe8d3 Binary files /dev/null and b/matrix-multiplication.png differ -- cgit v1.2.3