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diff --git a/macros/latex/contrib/rbt-mathnotes/examples/cheat-sheet.tex b/macros/latex/contrib/rbt-mathnotes/examples/cheat-sheet.tex new file mode 100644 index 0000000000..11ff82a5a9 --- /dev/null +++ b/macros/latex/contrib/rbt-mathnotes/examples/cheat-sheet.tex @@ -0,0 +1,227 @@ +%% cheat-sheet.tex +%% Copyright 2021 Rebecca B. Turner. +% +% This work may be distributed and/or modified under the +% conditions of the LaTeX Project Public License, either version 1.3 +% of this license or (at your option) any later version. +% The latest version of this license is in +% http://www.latex-project.org/lppl.txt +% and version 1.3 or later is part of all distributions of LaTeX +% version 2005/12/01 or later. +% +% This work has the LPPL maintenance status `maintained'. +% +% The Current Maintainer of this work is Rebecca B. Turner. +% +% This work consists of the files: +% README.md +% rbt-mathnotes.tex +% rbt-mathnotes.sty +% rbt-mathnotes.cls +% rbt-mathnotes-util.sty +% rbt-mathnotes-messages.sty +% rbt-mathnotes-hw.cls +% rbt-mathnotes-formula-sheet.cls +% examples/cheat-sheet.tex +% examples/multivar.tex +% examples/topology-hw-1.tex +% and the derived files: +% rbt-mathnotes.pdf +% examples/cheat-sheet.pdf +% examples/multivar.pdf +% examples/topology-hw-1.pdf + +\documentclass{rbt-mathnotes-formula-sheet} +\usepackage{nicefrac} +\ExplSyntaxOn +\NewDocumentCommand \normalized { m } + { \frac { #1 } { \| #1 \| } } +\let \gr \grad +\def \ddx { \frac{d}{dx} } +% VL = vector literal +\NewDocumentCommand \vl { m } { \left\langle #1 \right\rangle } +\ExplSyntaxOff + +\title{Formula Sheet} +\author{Rebecca Turner} +\date{2019-11-12} + +% "The most common size for index cards in North America and UK is 3 by 5 +% inches (76.2 by 127.0 mm), hence the common name 3-by-5 card. Other sizes +% widely available include 4 by 6 inches (101.6 by 152.4 mm), 5 by 8 inches +% (127.0 by 203.2 mm) and ISO-size A7 (74 by 105 mm or 2.9 by 4.1 in)." +\mathnotes{ + height = 4in , + width = 6in , +} +\begin{document} +\maketitle +\begin{gather*} +% 12.2: Vectors +% 12.3: Dot product + \textstyle\vec a \cdot \vec b = \sum_i a_i b_i = |\vec a| |\vec b| \cos \theta. \\ +% 12.4: Cross product + \vec a \times \vec b + % = \left| \begin{array}{rrr} + % \hat{i} & \hat{j} & \hat{k} \\ + % a_1 & a_2 & a_3 \\ + % b_1 & b_2 & b_3 \\ + % \end{array} \right| \\ + = \langle a_2 b_3 - a_3 b_2, + \quad a_3b_1 - a_1b_3, \\ + a_1b_2 - a_2b_1 \rangle.\quad + |\vec a \times \vec b| = |\vec a| |\vec b| \sin \theta. +% 12.5: Equations of lines and planes. +\shortintertext{Param.\ eqns.\ of line through $\langle x_0,y_0,z_0 \rangle$ +par.\ to $\langle a,b,c \rangle$:} + x = x_0 + at, + \quad y = y_0 + bt, + \quad z = z_0 + ct. \\ +\text{Symm.\ eqns.: } + \frac{x-x_0}{a} + = \frac{y-y_0}{b} + = \frac{z-z_0}{c}. \\ +\shortintertext{Vec.\ eqn.\ of plane through $\vec r$ with $\vec n$ normal:} + \vec n \cdot (\vec r - \vec r_0) = 0, + \quad \vec n \cdot \vec r = \vec n \cdot \vec r_0. \\ +% 13.1: Vector functions +% 13.2: Derivatives/integrals of vector functions +% 13.3: Arc length and curvature +\shortintertext{Length along a vec.\ fn.\ $\vec r(t)$:} + \textstyle\int_a^b \left|\vec r'(t)\right|\,dt = \int_a^b \sqrt{\sum_i + r_i'(t)^2}\,dt, \\ +\shortintertext{Unit tang.\ $\vec T(t) = \vec r'(t)/\left|\vec +r'(t)\right|$, so curvature of $\vec r(t)$ w/r/t the arc len.\ fn. $s$:} + \kappa = \left|\frac{d\vec T}{ds}\right| + = \frac{\left| \vec T'(t) \right|}{\left| \vec r'(t) \right|} + = \frac{\left| \vec r'(t) \times \vec r''(t) \right|}{\left| \vec r'(t) + \right|^3}. \\ +\text{Unit normal:}\quad + \vec N(t) = \vec T'(t)\,/\,\left| \vec T'(t) \right| \\ +% 14.1: Functions of several variables +% 14.2: Limits and continuity +% 14.3: Partial derivatives +\text{Clairaut's thm.:}\quad + f_{xy}(a,b) = f_{yx}(a,b) \\ +% 14.4: Tangent planes & linear approximations +\shortintertext{Tan.\ plane to $z = f(x,y)$ at $\langle x_0, y_0, +z_0\rangle$:} + z - z_0 = f_x(x_0, y_0) (x-x_0) \\ + + f_y(x_0, y_0) (y-y_0). \\ +% Partial derivatives of f for each variable exist near a point and are +% continuous => f is differentiable at the po\int. +% 14.6: Directional derivatives and the gradient vector +\text{Grad.:}\quad + \grad f(x,y) = \pd[f]x \hat{i} + \pd[f]y \hat{j}. \\ +\shortintertext{Dir.\ deriv.\ towards $\vec u$ at $\langle x_0, y_0 \rangle$:} + D_{\langle a,b\rangle} f(x_0, y_0) = f_x(x,y) a + f_y(x,y) b \\ + = \grad f(x,y) \cdot \vec u. \\ +\shortintertext{Max of $D_{\vec u} f(\vec x) = \left|\grad f(\vec +x)\right|$. Tan.\ plane of $f$ at $\vec p$:} + 0 = + f_x(\vec p)(x-\vec p_x) + + f_y(\vec p)(y-\vec p_y) \\ + + f_z(\vec p)(z-\vec p_z). +% 14.7: Maximum and minimum values +\shortintertext{If $f$ has loc.\ extrem.\ at $\vec p$, then $f_x(\vec p) = +0$ (\& $f_y$, etc). If so, let} + D = \left| \begin{array}{ll} + f_{xx} & f_{xy} \\ + f_{yx} & f_{yy} + \end{array}\right| + = f_{xx} f_{yy} - (f_{xy})^2. +\shortintertext{% + $D = 0$: no information. + $D < 0$: saddle pt. + $D > 0$: $f_{xx}(\vec p) > 0 \implies$ loc.\ min; + $f_{xx}(\vec p) < 0 \implies$ loc.\ max. + ($D$ is the \textbf{Hessian mat.}) +\endgraf + Set of possible abs. min and max vals of $f$ in reg.\ $D$: $f$ at critical + pts.\ and extreme vals.\ on the boundary of $D$. +% 14.8: Lagrange multipliers +\endgraf + Lagrange mults.: extreme vals of $f(\vec p)$ when $g(\vec p) = k$. + Find all $\vec x, \lambda$ s.t. +} + \grad f(\vec x) = \lambda \grad g(\vec x),\quad g(\vec x) = k. +\shortintertext{i.e.\ $f_x = \lambda g_x$, etc.} +% 15.1: Double integrals over rectangles +% 15.2: Iterated integrals +% 15.3: Double integrals over general regions + \iint f(r\cos\theta, r\sin\theta)r\,dr\,d\theta. \\ + A = \iint_D \left(\sqrt{f_x(x,y)^2 + f_y(x,y)^2 + 1}\right) \,dA. \\ +\shortintertext{Line int.s} + \int_C f(x,y)\,ds = \\ + \int_a^b f(x(t), y(t))\sqrt{\left(\pd[x]t\right)^2 + \left(\pd[y]t\right)^2}\,dt \\ +\shortintertext{If $C$ is a smooth curve given by $\vec r(t)$ from $a \le t +\le b$,} + \int_C \grad f \cdot d\vec r = f(\vec r(b)) - f(\vec r(a)) \\ +\text{Spherical coords:}\quad + x = \rho \sin \phi \cos \theta \\ + y = \rho \sin \phi \sin \theta, z = \rho \cos \phi \\ + \curl \vec F = \\ \left< \pd[R]y - \pd[Q]z, \pd[P]z - \pd[R]x, \pd[Q]x - + \pd[P]y\right>. \\ + \vec F = \langle P,Q,R \rangle,\quad + \curl \vec F = \grad \times \vec F \\ + \vec F \text{ ``conservative''} \implies \exists f, \vec F = \grad f. \\ + \dive \vec F = \grad \cdot \vec F = \pd[P]x + \pd[Q]y + \pd[R]z. \\ + \curl(\grad f) = \vec 0,\quad \dive \curl \vec F = 0 \\ +\shortintertext{If $C$ is a positively-oriented (ccw) closed curve, $D$ +is bounded by $C$, and $\vec n$ represents the normal,} + % \int_C P\,dx + Q\,dy = \iint_D\left( \pd[Q]{x} - \pd[P]{y} \right). \\ + \oint_C \vec F \cdot \vec n\,ds = \iint_D \dive \vec F(x,y)\,dA. +\end{gather*} + +\pagebreak +\raggedright Common derivs: +$f(g(x)) \to g'(x) f'(g(x))$, +$b^x \to b^x \ln b$, +$f^{-1}(x) \to 1/f'(f^{-1}(x))$, +$\ln x \to 1/x$, +$\sin x \to \cos x$, $\cos x \to -\sin x$, +$\tan x \to \sec^2 x$, +$\sin^{-1} x \to 1/\sqrt{1-x^2}$, +$\cos^{-1} x \to -(\sin^{-1}x)'$ (etc.), +$\tan^{-1} x \to 1/(1+x^2)$, +$\sec^{-1} x \to 1/(|x|\sqrt{x^2-1})$. + +Common ints (don't forget $+C$): +\begin{gather*} + x^n \to \frac{x^{n + 1}}{n + 1} + C \quad \text{when } n \ne -1 \\ + 1/x \to \ln |x| \\ + \tan x \to -\ln(\cos x) \\ + \int uv'\,dx = uv - \int u'v\,dx \quad\text{(Int.\ by parts)} \\ + \int u\,dv = uv-\int v\,du \\ + \int_{g(a)}^{g(b)} f(u)\,du = \int_a^b f(g(x))g'(x)\,dx + \quad\text{$u$-substitution.} +\intertext{E.x.\ in $\int 2x \cos x^2\,dx$, let $u=x^2$, find $du/dx=2x +\implies du = 2x\,dx$, subs.\ $\int \cos u\,du = \sin u + C = \sin x^2 + +C$.} + \iint_R f(x,y)\,dA = \int_\alpha^\beta \int_a^b f(r\cos\theta, + r\sin\theta)r\,dr\,d\theta +\end{gather*} +\begin{itemize} + \item Integrand contains $a^2-x^2$, let $x = a\sin\theta$ and use $1 - + \sin^2 \theta = \cos^2 \theta$. + \item $a^2 + x^2$, let $x = a\tan\theta$, use $1 + \tan^2 \theta = \sec^2 + \theta$. + \item $x^2 - a^2$, let $x = a\sec\theta$, use $\sec^2\theta - 1 = \tan^2 + \theta$. +\end{itemize} + +\begin{gather*} + \lim_{x \to 0} \sin x/x = 1 \\ + \lim_{x \to 0} (1-\cos x)/x = 0 \\ + \lim_{x \to \infty} x \sin(1/x) = 1 \\ + \lim_{x \to 0} (1+x)^{1/x} = e \\ + \lim_{x \to 0} (e^{ax}-1)/(bx) = a/b \\ + \lim_{x \to 0^+} x^x = 1 \\ + \lim_{x \to 0^+} x^{-n} = \infty \\ + \text{For $0/0$ or $\pm\infty/\infty$,}\quad + \lim_{x \to c} f(x)/g(x) = \lim_{x \to c} f'(x)/g'(x) \\ + \text{For $g(x)$ cont.\ at $L$,} + \lim_{x \to c} f(x) = L \implies \lim_{x \to c} g(L) +\end{gather*} + +\end{document} |