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diff --git a/Master/texmf-dist/doc/latex/rbt-mathnotes/examples/multivar.tex b/Master/texmf-dist/doc/latex/rbt-mathnotes/examples/multivar.tex new file mode 100644 index 00000000000..fc026754f68 --- /dev/null +++ b/Master/texmf-dist/doc/latex/rbt-mathnotes/examples/multivar.tex @@ -0,0 +1,507 @@ +%% multivar.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[knowledge]{rbt-mathnotes} +\title{Multivariable Calculus} +\mathnotes{ + instructor = Prof.~Corey Bregman , + name = Rebecca Turner , + email = rebeccaturner@brandeis.edu , + course = \textsc{math} 20a (multivariable calculus) , + institution = Brandeis University , + semester = Fall 2019 , +} + +\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 + +% \makeatletter +% \@ifpackageloaded{knowledge} +% { +\knowledge{notion, index={Derivative!Partial}} + | partial derivative +\AtBeginDocument{\index{Partial derivative|see{Derivative, partial}}} + +\knowledge{notion, index={Derivative!Directional}} + | directional derivative +\AtBeginDocument{\index{Directional derivative|see{Directional, partial}}} + +\knowledge{notion, index=Unit vector} + | unit vector + +\knowledge{notion, index=Limit} + | limit + +\knowledge{notion, index={Differentiable functions}} + | differentiable + +\knowledge{notion, index=Gradient} + | gradient + +\knowledge{notion, index={Neighborhood (topology)}} + | neighborhood + +\knowledge{notion, index=Local maximum} + | local maximum + | local maxima + +\knowledge{notion, index=Local minimum} + | local minimum + | local minima + +\knowledge{notion, index=Absolute maximum} + | absolute maximum + | absolute maxima + +\knowledge{notion, index=Absolute minimum} + | absolute minimum + | absolute minima + +\knowledge{notion, index=Local extremum} + | local extremum + | local extrema + +\knowledge{notion, index=Critical point} + | critical point + +\knowledge{notion, index=Saddle point} + | saddle point + +\knowledge{notion, index={Lagrange multiplier}} + | Lagrange multiplier + | Lagrange multipliers + | the method of Lagrange multipliers + +\knowledge{url={https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange}} + | Joseph-Louis Lagrange + | Lagrange + +\knowledge{url={https://en.wikipedia.org/wiki/Marquis_de_Condorcet}} + | Marquis de Condorcet + + \knowledge{url={https://en.wikipedia.org/wiki/Adrien-Marie_Legendre}} + | Adrien-Marie Legendre +% } +% {} +% \makeatother +\date{2019-10-23} +\begin{document} +\maketitle +\tableofcontents + +\chapter{Vectors} +I already know about vectors --- I've been taught them in about five +different courses so far. I'm skipping this. + +\chapter{Partial derivatives} + +If we have a function of multiple variables, say +\[f(a_1, a_2, a_3, \dots),\] +we might care about the change of $f$ with respect to only one variable. By +picking a fixed value for all but one of the variables, we can determine +this. + +Say that we want to find the "partial derivative" of $f$ with respect to +$a_2$; then, by constructing $g(a_2) = f(c_1, a_2, c_3, \dots)$, we've +created a function of \emph{one} variable, which we can differentiate as +usual. +\begin{notation} + We write the "partial derivative" of a function $f$ at a point $\vec p$ + with respect to a basis element $a$ of $\vec p$ as \fbox{$f_a(\vec p)$.} + + We may also use much more common notation + \[\pd[f]{a},\] + using the "partial derivative" symbol $\partial$, a stylized cursive + ``d''.\footnote{Introduced by "Marquis de Condorcet" in 1770, who used it + to represent a partial \emph{differential}, i.e.~the $dy$ or $dx$ in + $dy/dx$, and then adapted in 1786 by "Adrien-Marie Legendre" for use as the + partial derivative.} + + In the interest of completeness, I'll exhaustedly note that the book also + uses, on occasion, the notation $D_a f$. +\end{notation} + +We can also calculate ""higher partial derivatives"" --- similarly to the +higher ordinary derivatives. The notation is a fairly clear extension: +\[(f_x)_x = f_{xx} = \pd x \left( \pd[f]{x} \right) = \pd[^2 f]{x^2}.\] + +\begin{thm}[Clairaut's Theorem] + Suppose $f$ is defined on a neighborhood $N$ about a point $\vec p$. + If $f_{xy}$ and $f_{yx}$ are continuous in $N$, then $f_{xy}(\vec p) = + f_{yx}(\vec p)$. +\end{thm} + +\section{Gradients} +\begin{notation} + This ridiculous textbook denotes the "partial derivative" of a function + $f(x, y) = z$ with respect to $x$ as $f_x(x, y)$. +\end{notation} + +\begin{defn} + The ""directional derivative"" of a function $f$ at $(x_0, y_0)$ in the + direction of a "unit vector" $\vec u = \langle a, b \rangle$ is + \[D_{\vec u} f(x_0, y_0) = \lim_{h \to 0} \frac{f(x_0 + ha, y_0 + hb) - + f(x_0, y_0)}{h},\] + if the "limit" exists. +\end{defn} + +If $f : \Re^2 \mapsto \Re$ is a "differentiable" function, then $f$ has a +"directional derivative" in the direction of any "unit vector" $\vec u = +\langle a, b \rangle$ of +\begin{alignat*}{1} + D_{\vec u} f(x, y) &= f_x(x, y) a + f_y(x, y) b. +\intertext{Or, if $\vec u = \langle \cos \theta, \sin \theta \rangle$, then} + D_{\vec u} f(x, y) &= f_x(x, y) \cos \theta + f_y(x, y) \sin \theta. +\end{alignat*} + +Noticing that the "directional derivative" of a function can be written as +the dot product of two vectors, +\begin{alignat*}{1} + D_{\vec u} f(x, y) &= f_x(x, y) a + f_y(x, y) b \\ + &= \langle f_x(x, y), f_y(x, y) \rangle \cdot \langle a, b \rangle \\ + &= \langle f_x(x, y), f_y(x, y) \rangle \cdot \vec u, +\end{alignat*} +we call the first vector $\langle f_x(x, y), f_y(x, y) \rangle$ the +""gradient"" of $f$ and denote it as $\grad f$. + +\begin{defn} + The "gradient" of a function $f$ of two variables is defined as + \[\grad f(x, y) = \langle f_x(x, y), f_y(x, y) \rangle + = \frac{\partial f}{\partial x} \vec i + \frac{\partial f}{\partial y} + \vec j.\] +\end{defn} + +Therefore, we can rewrite the "directional derivative" of a function $f$ +as +\[D_{\vec u} f(x, y) = \grad f(x, y) \cdot \vec u.\] +It's intuitive, then, that the maximum value of the "directional derivative" +is $|\grad f(x, y)|$, when $\vec u$ is parallel to $\grad f(x, y)$. + +\section{Maximum and minimum values} +\begin{defn} + $f : A^k \mapsto B$ has a ""local maximum"" at $\vec a$ if for some + "neighborhood" $N \subset A$ about $\vec a$, for all $\vec x \in N$, + $f(\vec x) \le f(\vec a)$. + + Conversely, if $f(\vec x) \ge f(\vec a)$, then $f(\vec a)$ is a ""local + \emph{minimum}@local minimum"". + + If the statement also holds true for $N = A$, then $\vec a$ is an + ""absolute maximum"" (or ""absolute minimum""). +\end{defn} + +If $f$ has a "local maximum" or "minimum@local minimum" at $\vec a$ and the +partials of $f$ exist at $\vec a$, then $\partial f/\partial x (\vec a) = 0$ +and $f_y(a, b) = 0$; geometrically, the tangent plane to a maximum or +minimum must be horizontal. + +\begin{defn} + A point $\vec a$ is called a ""critical point"" of $f$ if $f_x(\vec a) = + 0$ or $f_x(\vec a)$ doesn't exist for all variables of $f$. +\end{defn} + +\begin{defn} + A ""saddle point"" of a function is a "critical point" which is not a + "local extremum" of the function. +\end{defn} + +If $(a, b)$ is a critical point of $f$, then let +\[D = D(a, b) = f_{xx} (a,b) \, f_{yy} (a,b) - (f_{xy}(a,b))^2.\] +If $D < 0$, then $(a, b)$ is a "saddle point" of $f$. + +\section{Lagrange multipliers} +Often we want to find the "local extrema" of a function subject to +constraints, i.e.~maximizing the volume of an object while keeping its +surface area constant. The method of "Lagrange multipliers"\footnote{After +"Joseph-Louis Lagrange" (1736--1813), ``an Italian Enlightenment Era +mathematician and astronomer [who] made significant contributions to the +fields of analysis, number theory, and both classical and celestial +mechanics.''} is a strategy for doing this. + +To find extrema of $f(\vec p)$ constrained with $g(\vec p) = k$, +we look for extrema of $f$ that are restricted to lie on the level curve +$g(\vec p) = k$; it happens that the largest $c$ such that $f(\vec p) = c$ +intersects with $g(\vec p) = k$ when the two level curves are tangent with +each other, i.e.~they have identical normals. In other words, for some +scalar $\lambda$, $\grad f(\vec p) = \lambda \grad g(\vec p)$. + +More formally, suppose $f$ has an extrema at $\vec p_0$. Then, let the level +surface generated by the constraint $g(\vec p) = k$ be called $S$, where +$\vec p_0 \in S$. Then, let $C$ be the set of points given by $\vec r(t)$ such +that $C \subset S$ and $\vec p_0 \in C$. Further, let $t_0$ be a point such +that $\vec r(t_0) = \vec p_0$. + +Then, $f \after \vec r$ gives the values of $f$ on the curve $C$. $f$ has an +extrema at $\vec p$, so $f \after \vec r$ must also, and $(f \after \vec +r)'(t_0) = 0$. If $f$ is "differentiable", we can use the chain rule to +write +\begin{alignat*}{1} + 0 &= (f \after \vec r)'(t_0) \\ + &= \grad f(\vec p_0) \cdot \vec r'(t_0). +\end{alignat*} +Therefore, the gradient of $f$ is orthogonal to the tangent of every such +curve $C$. We also know that $\grad g(\vec p_0)$ is orthogonal to $\vec +r'(t_0)$, so the gradients of $f$ and $g$ at $\vec p_0$ must be parallel. +Therefore, if $\grad g(\vec p_0) \ne 0$, there exists some $\lambda$ such +that +\begin{equation} + \grad f(\vec p_0) = \lambda \grad g(\vec p_0), +\end{equation} +where the constant $\lambda$ is called a "Lagrange multiplier". + +Then, the ""method of Lagrange multipliers"" gives us a process to +find the maximum and minimum values of a function $f(\vec p)$ subject to the +constraint $g(\vec p) = k$, where $\vec p \in \Re^n$. To use the method of +Lagrange multipliers, we assume that the extreme values exist and that +$\grad g \ne 0$ on the level surface $g(\vec p) = k$. +\begin{enumerate} + \item Find all values of $\vec p$ and $\lambda$ such that + \begin{alignat*}{1} + \grad f(\vec p) &= \lambda \grad g(\vec p) \\ + \text{and}\qquad g(\vec p) &= k. \\ + \end{alignat*} + + \item Next, evaluate $f$ at all of the points found in the first step. The + largest of these values is the maximum value of $f$, and the smallest of + them is the minimum value. +\end{enumerate} + +\chapter{Multiple integrals} +Single integrals are good for functions of one variable. To integrate +functions of multiple variables, we use multiple integrals. Straightforward +enough. + +Multiple integrals allow us to calculate things like surface areas and +volumes of geometric objects. + +In general, for some double integral +\[\underbrace{\int_a^b \overbrace{\int_c^d f(x,y)\,dx}^{\mathclap{\text{We +treat $y$ as constant while evaluating this.}}}\,dy,}_{\mathclap{\text{We've +eliminated $x$ from the equation before evaluating this.}}}\] +we do the opposite of partial differentiation and treat all variables other +than the one we're integrating for as constant, repeatedly, until we've +integrated with respect to all variables; each step in this process is +called, predictably, ""partial integration"". + +\section{Double integrals} +For an axis-aligned rectangle $R$ on the $xy$-plane from $(x_0, y_0)$ to $(x_1, +y_1)$, the area of a function $f(x, y)$ under $R$ is given by the double +integral +\begin{alignat*}{1} + \iint_{Y} f(x,y)\,dA &= \underbrace{\int_{x_0}^{x_1} \int_{y_0}^{y_1} + f(x,y)\,dy\,dx}_{\mathclap{\text{This is the iterated form of the + integral.}}} \\ + &= \int_{y_0}^{y_1} \int_{x_0}^{x_1} f(x,y)\,dx\,dy, \\ +\end{alignat*} +where we use $\iint_R$ to mean ``integrating over the area of $R$'' and +``$dA$'' to mean ``with respect to area.'' + +The right-hand side of the equation above is called the ""iterated form"", +or an \reintro*"iterated integral". + +We can also iterate over funkier regions if we're willing to play with the +limits of integration a bit. The easiest regions to integrate over are the +ones that are easily expressible as the region bounded above and below by +functions of one variable, e.g. ``the region under the line $y = 2x$ and +above the line $y = x^2$'' (note that this is bounded on the left at $x = 0$ +and on the right at $x = 2$). + +% Area[ImplicitRegion[x^2 < y \[And] y < 2 x, {x, y} +The area of that region is expressed by the integral +\begin{alignat*}{1} + A &= \int_0^2 \int_{x^2}^{2x}\,dy\,dx \\ + &= \int_0^2 {\bigg[ x \bigg]}_{x^2}^{2x}\,dx \\ + &= \int_0^2 {\bigg( 2x-x^2 \bigg)}\,dx \\ + &= {\left[ x^2 - \frac{x^3}{3} \right]}_0^2 \\ + &= 4 - \frac{8}{3} = \frac{4}{3}. +\end{alignat*} + +A more complicated region might be ``the region under the paraboloid $z = +x^2 + y^2$ and above the region in the $xy$-plane bounded by $y = \sqrt{x}$ +and $y=1-\cos x$.'' + +We can build larger regions out of pieces, by summing smaller integrals. + +\section{Polar coordinates} +Use the conversions +\begin{alignat*}{1} + r &= \sqrt{x^2+y^2} \\ + x &= r\cos\theta \\ + y &= r\sin\theta \\ +\intertext{for the coordinates and then we have that if $R$ is a ``polar +rectangle'' (arc-shaped region bounded by angles and radii) from $r=a$ to +$r=b$ and $\theta = \alpha$ to $\theta = \beta$, we have} + \iint_R f(x,y)\,dA &= \int_\alpha^\beta \int_a^b f(r\cos\theta, + r\sin\theta)r\,dr\,d\theta, \\ +\intertext{which makes our lives easier for circly areas and volumes. Don't +forget to multiply by $r$. +\endgraf +For squiggly and varying radii, we can use functions $h_1(\theta)$ and +$h_2(\theta)$ instead of constants $a$ and $b$:} + \iint_D f(x,y)\,dA &= \int_\alpha^\beta \int_{h_1(\theta)}^{h_2(\theta)} f(r\cos\theta, + r\sin\theta)r\,dr\,d\theta. \\ +\end{alignat*} + +\section{Cylindrical coordinates} +Just add $z$. + +\section{Spherical coordinates} +I can never remember how these work. If we have a point $P$, and we drop it +down to the $xy$-plane, the angle between the positive $x$-axis and the +segment from the origin to $P$ is $\theta$. + +Next, the angle between the positive $z$-axis and the segment from the origin +to $P$ is $\phi$. + +Finally, the length of the segment from the origin to $P$ is $\rho$. + +The conversions +\begin{alignat*}{1} + x &= \rho \sin \phi \cos \theta \\ + y &= \rho \sin \phi \sin \theta \\ + z &= \rho \cos \phi \\ +\intertext{give us the integral-conversion for the spherical wedge bounded +by $a \le \rho \le b, \alpha \le \theta \le \beta, c \le \phi \le d$ as} + \iiint_E f(x,y,z)\,dV &= \int_c^d \int_\alpha^\beta \int_a^b + f(\rho \sin\phi \cos\theta,\, + \rho \sin\phi \sin\theta,\, + \rho \cos \theta) + [\rho^2 \sin\phi] + \,d\rho\,d\theta\,d\phi. \\ +\end{alignat*} +Very gross! + +\section{Surface area} + +For $f(x,y)$ with $f_x$, $f_y$ continuous, the surface area of $f$ within +a region $D$ is +\[A = \iint_D \left(\sqrt{f_x(x,y)^2 + f_y(x,y)^2 + 1}\right) \,dA.\] + +% Need notes on: +% - Change of variables (in multiple integrals) + +\chapter{Vector calculus} +A vector field is a mapping $\R^k \mapsto \R^n$; for each point in +$k$-dimensional Euclidean space, we associate an $n$-dimensional vector. +These vectors can represent velocity, distance, or anything else, and come +up in all sorts of applied fields. + +We'll be mostly concerned with vector fields $\R^2\mapsto\R^2$ and +$\R^3\mapsto\R^3$. + +If we have a plane curve given by the vector equation +\begin{alignat*}{1} + \vec r(t) &= \left< x(t), y(t) \right> \qquad a \le t \le b, \\ +\intertext{then the line integral of $f$ along $\vec r(t)$ from $a$ to $b$ +is} + & \int_a^b f(x(t), y(t)) \sqrt{{\left(\dd[x]{t}\right)}^2 + + {\left(\dd[y]{t}\right)}^2}\,dt, +\end{alignat*} +i.e.\ the length of the curve multiplied, at each point, by the value of the +vector field $f$ at that point. + +% Need notes on: +% 16.1, 16.2, 16.3, 16.4, 16.5, +% Need to update cheat sheet. + +\appendix +\chapter{Common formulas for derivatives and integrals} +\backmatter +\section{Derivatives} + +\begin{alignat*}{2} + \ddx&\;& (f + g) &= f' + g' \\ + \ddx&& x^n &= nx^{n - 1} \\ + \ddx&& (fg) &= fg' + f'g \\ + \ddx&& \frac{h}{l} &= \frac{l h' - h l'}{l^2} \\ + \ddx&& f(g(x)) &= f'(g(x)) g'(x) \quad\text{(Chain rule.)} \\ + \ddx&& b^x &= b^x \ln b \\ + \ddx&& f^{-1}(x) &= \frac{1}{f'(f^{-1}(x)} \\ + \ddx&& c &= 0 \\ + \ddx&& c\,f &= c\,f' \\ + \ddx&& e^x &= e^x \\ + \ddx&& e^{f(x)} &= f'(x) e^{f(x)} \quad\text{(By the chain rule.)} \\ + \ddx&& \ln x &= \frac{1}{x} \\ + \ddx&& \log_b x &= \frac{1}{x \ln b} \\ + \ddx&& [\vec u \cdot \vec v] &= \vec u' \cdot \vec v + \vec u \cdot \vec v' \\ + \ddx&& [\vec u \times \vec v] &= \vec u' \times \vec v + \vec u \times \vec v' \\ +\end{alignat*} + +\subsection{Trigenometric} +\begin{alignat*}{2} + \ddx&\;& \sin x &= \cos x \\ + \ddx&& \cos x &= -\sin x \\ + \ddx&& \tan x &= \sec^2 x \\ + \ddx&& \cot x &= -\csc^2 x \\ + \ddx&& \sec x &= \sec x \tan x \\ + \ddx&& \csc x &= -\csc x \cot x \\ + % inverse + \ddx&& \sin^{-1} x &= \frac{ 1}{\sqrt{1 - x^2}} \\ + \ddx&& \cos^{-1} x &= \frac{-1}{\sqrt{1 - x^2}} \\ + \ddx&& \tan^{-1} x &= \frac{ 1}{1 + x^2} \\ + \ddx&& \cot^{-1} x &= \frac{-1}{1 + x^2} \\ + \ddx&& \sec^{-1} x &= \frac{ 1}{|x| \sqrt{x^2 - 1}} \\ + \ddx&& \csc^{-1} x &= \frac{-1}{|x| \sqrt{x^2 - 1}} \\ +\end{alignat*} + +\section{Integrals} +See also: +\emph{\href{https://www.whitman.edu/mathematics/calculus/calculus_08_Techniques_of_Integration.pdf}{Techniques +of Integration}}. +\begin{alignat*}{1} + \int x^n\,dx &= \frac{x^{n + 1}}{n + 1} + C \quad \text{when } n \ne -1 \\ + \int x^{-1}\,dx &= \ln|x| + C \\ + \int e^x\,dx &= e^x + C \\ + \dd{t} \int_{a(t)}^{b(t)} g(s)\,ds &= b'(t) g(b(t)) - a'(t) g(a(t)) + \quad\text{(Leibniz' rule.)} \\ + \int uv'\,dx &= uv - \int u'v\,dx \\ +\end{alignat*} + +\subsection{Trigenometric} +\begin{alignat*}{1} + \int \sin x\,dx &= -\cos x + C \\ + \int \cos x\,dx &= \sin x + C \\ + \int \sec^2 x\,dx &= \tan x + C \\ + \int \sec x \tan x\,dx &= \sec x + C \\ + \int \frac{1}{1 + x^2}\,dx &= \tan^{-1} x + C \\ + \int \frac{1}{\sqrt{1 + x^2}}\,dx &= \sin^{-1} x + C \\ +\end{alignat*} + +\printindex +\end{document} |