From 51147664dd828770d04a5aae8f973146a8918fee Mon Sep 17 00:00:00 2001 From: Robert Date: Sun, 28 Mar 2021 22:31:30 +0200 Subject: [PATCH] added multivar calc --- chapters/sections/partial_total_diff.tex | 361 ++++++++++++++++++++++- script.pdf | Bin 780104 -> 817511 bytes script.tex | 3 + 3 files changed, 363 insertions(+), 1 deletion(-) diff --git a/chapters/sections/partial_total_diff.tex b/chapters/sections/partial_total_diff.tex index e91ae6d..dfbcc69 100644 --- a/chapters/sections/partial_total_diff.tex +++ b/chapters/sections/partial_total_diff.tex @@ -5,6 +5,365 @@ \section{Partial and Total Differentiability} \begin{defi} - + Let $U \subset \realn^n$ be open, $x \in (x_1, \cdots, x_n) \in U$ and define the function $f: U \rightarrow \realn^m$. + The mapping $f$ is said to be partially differentiable in $x$ in terms of $x_i$ if + \[ + t \longmapsto f(x_1, \cdots, x_{i-1}, t, x_{i+1}, \cdots, x_n) + \] + is differentiable in $x_i$, i.e. + \[ + \partial_i f(x) = \limes{h}{0} \frac{f(x_1, \cdots, x_{i-1}, x_i + h, x_{i+1}, \cdots, x_n) - f(x_1, \cdots, x_n)}{h} + \] + exists. $\partial_i f(x)$ is said to be the partial derivative of $f$ in $x$ in terms of $x_i$. Another notation is + \[ + \pdv{f}{x_i} + \] + This mapping is said to be partially differentiable in $x$ if it is partially differentiable in terms of $x_i ~~\forall i \in \set{1, \cdots, n}$. \end{defi} + +\begin{eg} + Consider + \begin{align*} + f: \realn^2 &\longrightarrow \realn \\ + (x, y) &\longmapsto \begin{cases} + 1, & x = 0 \vee y = 0 \\ + 0, & \text{else} + \end{cases} + \end{align*} + $f$ is partially differentiable in $(0, 0)$, but not continuous. +\end{eg} + +\begin{thm} + Let $U \subset \realn$ be open, $x \in U$ and $f: U \rightarrow \field$. + \begin{gather*} + f \text{ is differentiable in } x \\ + \iff \\ + \exists a \in \field, \phi: U \rightarrow \field: ~~f(y) = f(x) + a(y - x) + \phi(y) ~~\forall y \in U + \end{gather*} + and + \[ + \limes{y}{x} \frac{\phi(x)}{\abs{y - x}} = 0 + \] +\end{thm} +\begin{proof} + We will first prove the "$\impliedby$" direction. So let $a, \phi$ be as demanded in the theorem. Then + \begin{equation} + \frac{f(y) - f(x)}{y - x} = a + \frac{\phi(y)}{\abs{y - x}} \cdot \frac{\abs{y - x}}{y - x} \conv{y \rightarrow x} a + \end{equation} + which means $f$ is differentiable in $x$ and $f'(x) = a$. Now let $f$ be differentiable, and set + \begin{equation} + \phi(y) = f(y) - f(x) - f'(x)(y - x) + \end{equation} + Which is equivalent to the equation in the theorem, with $a = f'(x)$. Then + \begin{equation} + \limes{y}{x} \frac{\phi(x)}{\abs{y - x}} = \left( \frac{f(y) - f(x)}{y - x} - f'(x) \right) \cdot \frac{y - x}{\abs{y - x}} = 0 + \end{equation} +\end{proof} + +\begin{defi} + Let $U \subset \realn^n$, $x \in U$ and $f: U \rightarrow \realn^m$. + $f$ is said to be (totally) differentiable in $x$ if a matrix $A \in \realn^{m \times n}$ and a mapping $\phi: U \rightarrow \realn^m$ exist, such that + \[ + f(y) = f(x) + A(y - x) + \phi(x) ~~\forall y \in U + \] + and + \[ + \limes{y}{x} \frac{\phi(y)}{\norm{y - x}} = 0 + \] + $f$ is said to be (totally) differentiable if it is (totally) differentiable in every point $x \in U$. +\end{defi} + +\begin{thm} + Let $U \subset \realn^n$ be open, $x \in U$ and $f: U \rightarrow \realn^m$ with + \[ + f = (f_1, \cdots, f_m), ~~f_1, \cdots, f_m: U \longrightarrow \realn + \] + If $f$ is totally differentiable in $x$, then it is partially differentiable as well, and the matrix $A$ is given by + \[ + a_{ji} = \partial_i f_j(x) + \] +\end{thm} +\begin{proof} + Let $A, \phi$ be as demanded above. Let $e_1, \cdots, e_n$ be the canonical basis for $\realn^n$. + We insert $y = x + he_i$ and receive + \begin{equation} + f(x + he_i) = f(x) + h \cdot Ae_i + \phi(x + he_i) + \end{equation} + By rearranging this yields + \begin{equation} + \frac{f(x + he_i) - f(x)}{h} = Ae_i + \frac{\phi(x + he_i)}{|h|} \cdot \frac{|h|}{h} \conv{h \rightarrow 0} Ae_i + \end{equation} + Thus, $f$ is partially differentiable in $x$ in terms of $x_i$ with $\partial_i f(x) = Ae_i$. +\end{proof} + +\begin{defi} + The matrix $(\partial_i f_j(x))_{ij}$ is called the Jacobian matrix of $f$ in $x$. + We write $Df(x)$. If $f$ is totally differentiable, then $Df(x)$ is said to be the (total) derivative of $f$ in $x$. + + For $m = 1$ (so $f: \realn^n \rightarrow \realn$), the Jacobian matrix has one column, and we call it gradient + \[ + Df(x) =: \grad f(x) + \] + Note: I will adhere to the physical notation of the gradient, using the Nabla operator $\nabla$. +\end{defi} + +\begin{eg} + Let $A \in \realn^{m \times n}$ and define + \begin{align*} + f_A: \realn^n &\longrightarrow \realn^m \\ + x &\longmapsto Ax + \end{align*} + Then we have + \[ + f_A(y) = Ay = Ax + A(y - x) = f_A(x) - f_A(y - x) + \] + Thus, $f_A$ is differentiable $(\phi = 0)$ and the derivative is + \[ + Df_A(x) = A ~~\forall x \in \realn^n + \] + For another example, let + \begin{align*} + f: (0, \infty) \times (0, 2\pi) &\longrightarrow \realn^2 \\ + (r, \phi) &\longmapsto (r \cos \phi, r \sin \phi) + \end{align*} + Then $f$ is partially differentiable. + \[ + Df(r, \phi) = \begin{pmatrix} + \cos\phi & -r\sin\phi \\ + \sin\phi & r\cos\phi + \end{pmatrix} + \] + So $f$ is also totally differentiable (We'll get back to this later). +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Let $U \subset \realn^n$ be open and $f: U \rightarrow \realn^m$ differentiable, + then the derivative $Df$ is a function $U \rightarrow \realn^{m \times n}$ + + \item Total differentiability is also called local linear approximation. Linearity is the property + \[ + A(x + \lambda y) = Ax + \lambda Ay ~~\forall x, y \in \realn^n ~\lambda \in \realn + \] + + \item For arbitrary vector spaces $V, W$, a mapping $V \rightarrow W$ is said to be linear if + \[ + A(x + \lambda y) = Ax + \lambda Ay ~~\forall x, y \in \realn^n ~\lambda \in \realn + \] + So we can analogously define differentiability for mappings $f: V \rightarrow W$ between arbitrary normed vector spaces. + + \item $f$ is totally differentiable in $x$ if and only if the Jacobian matrix exists and + \[ + \limes{x}{y} \frac{f(y) - f(x) - Df(x)(y-x)}{\norm{y-x}} = 0 + \] + + \item Let $f = (f_1, \cdots, f_m)$ with $f_1, \cdots, f_m: U \rightarrow \realn$. + \[ + f \text{ totally differentiable} \iff f_i \text{ totally differentiable } ~\forall i \in \set{1, \cdots, n} + \] + The Jacobian matrix $Df_i(x)$ is the $i$-th row of $Df(x)$. + + \item Total differentiability implies continuity. + \item Partial and total differentiability are local properties. + \item The mapping $h \mapsto Df(x) \cdot h$ is linear. + \item The derivative $x \mapsto Df(x)$ is not linear in general. + \end{enumerate} +\end{rem} + +\begin{thm}[Chain rule] + Let $U \subset \realn^n$ be open, $V \subset \realn^m$ open, $x \in U$, $g: U \rightarrow V$ differentiable in $x$, and $f: V \rightarrow \realn^k$ differentiable in $g(x)$. + Then $f \circ g$ is differentiable and + \[ + D(f \circ g) = Df(g(x)) \cdot Dg(x) + \] +\end{thm} +\begin{proof} + Differentiability of $g$ in $x$ means + \begin{equation} + \exists \phi_g: U \longrightarrow \realn^m: ~~g(y) - g(x) = D_g(x)(y-x) + \phi_g(y) + \end{equation} + Differentiability of $f$ in $g(x)$ means + \begin{equation} + \exists \phi_f: V \rightarrow \realn^k:: ~\limes{z}{g(x)} \phi_f(z) \inv{\norm{z - g(x)}} = 0 + \end{equation} + and + \begin{equation} + f(z) = f(g(x)) + D_f(g(x))(z - g(x)) + \phi_f(z) + \end{equation} + Now set $z = g(y)$, then + \begin{equation} + \begin{split} + \underbrace{f(g(y))}_{(f \circ g)(y)} = \underbrace{f(g(x))}_{(f \circ g)(x)} &+ D_f(g(x)) \cdot D_g(x)(y-x) \\ + &+ (D_f(g(x)) \phi_g(y) + \phi_f(g(y))) + \end{split} + \end{equation} + And we finally need to show + \begin{equation} + \frac{D_f(g(x)) \phi_g(y) + \phi_f(g(y))}{\norm{y - x}} \conv{y \rightarrow x} 0 + \end{equation} + We know that + \begin{equation} + Df(g(x)) \frac{\phi_g(y)}{\norm{y - x}} \conv{} 0 + \end{equation} + because + \begin{equation} + z \longmapsto Df(g(x)) z \text{ linear and thus continuous} + \end{equation} + We define a new mapping + \begin{equation} + \begin{split} + \psi: U &\longrightarrow \realn \\ + z &\longmapsto \begin{cases} + \phi_f(z) - \inv{\norm{z - g(x)}}, & z \ne g(x) \\ + 0, & z = g(x) + \end{cases} + \end{split} + \end{equation} + $\psi$ is continuous in $g(x)$. Then $\forall y \in U$ we have + \begin{equation} + \frac{\phi_f(g(y))}{\norm{y - x}} = \underbrace{\psi(g(y))}_{\conv{y \rightarrow x} 0} \cdot \frac{\norm{g(y) - g(x)}}{\norm{y - x}} + \end{equation} + and + \begin{equation} + \begin{split} + \frac{\norm{g(y) - g(x)}}{\norm{y - x}} &= \norm{Dg(x) \frac{y - x}{\norm{y - x}} + \frac{\phi_g(y)}{\norm{y - x}}} \\ + &\le \underbrace{\norm{Dg(x) \frac{y - x}{\norm{y - x}}}}_{\le \norm{Dg(x)}} + \underbrace{\norm{\frac{\phi_g(y)}{\norm{y - x}}}}_{\conv{y \rightarrow x} 0} + \end{split} + \end{equation} + thus $\psi$ is bounded. + \begin{equation} + \implies \psi(g(y)) \cdot \frac{\norm{g(y) - g(x)}}{\norm{y - x}} \conv{} 0 + \end{equation} +\end{proof} + +\begin{thm} + Let $U \subset \realn^n$ and $f: U \longrightarrow \realn^m$. If $\forall x \in U$ the partial derivatives $\partial_i f(x)$ exist and are continuous $\forall i \in \set{1, \cdots, n}$. + then $f$ is totally differentiable. +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{defi} + Let $U \subset \realn^n$ be open. $f: U \rightarrow \realn^m$ is said to be continuously differentiable if all partial derivatives exist and are continuous. + The vector space of all such functions is denoted as $C^1(U, \realn^m)$, or in the special case $m=1$ as $C^1(U)$. +\end{defi} + +\begin{eg} + \begin{enumerate} + \item Coming back to a previous example, we consider + \[ + Df(r, \phi) = \begin{pmatrix} + \cos \phi & -r \sin \phi \\ + \sin \phi & \cos \phi + \end{pmatrix} + \] + Thus, $f$ is continuously differentiable, and therefore totally differentiable. + + \item Let $N \in \natn$ and $c_{\eta} \in \field$ for every multiindex $\eta \in \natn_0^n$ with $\abs{\eta} \le N$. + Then the polynomial + \begin{align*} + P: \realn^n &\longrightarrow \field \\ + x &\longmapsto \sum_{\substack{\eta \\ \abs{\eta} \le N}} c_{\eta} x^{\eta} + \end{align*} + is continuously differentiable, and therefore totally differentiable. + \begin{align*} + \partial_i x^{\eta} &= \partial_i \left( x_1^{\eta_1}, x_2^{\eta_2}, \cdots, x_n^{\eta_n} \right) \\ + &= \eta_i x_1^{\eta_1} \cdots x_{i-1}^{\eta_{i-1}} x_i^{\eta_{i-1}} x_{i+1}^{\eta_{i+1}} \cdots x_n^{\eta_n} + \end{align*} + This is another polynomial, and therefore continuous. + \end{enumerate} +\end{eg} + +We introduce the following new notation, for $x, y \in \realn^n$: +\begin{align*} + \oline &:= \set[t \in (0, 1)]{x + t(y - x)} \\ + \cline &:= \set[{t \in [0, 1]}]{x + t(y - x)} +\end{align*} +They denote the connecting line between $x$ and $y$. + +\begin{center} + \begin{tikzpicture} + \draw (0, 0) circle [radius=2pt] node[above left] {$x$}; + \draw (3, 1.5) circle [radius=2pt] node[above right] {$y$}; + + \draw (0, 0) -- node[below right] {$\oline$} (3, 1.5); + \end{tikzpicture} +\end{center} + +\begin{thm}[Intermediate value theorem for $\realn$-valued functions] + Let $U \subset \realn^n$ be oppen, $x, y \in U$ and $\cline \subset U$. + Now let $f: U \rightarrow \realn$ differentiable on $\oline$ and continuous in $x, y$. Then + \[ + \exists \xi \in \cline: ~~f(y) - f(x) = Df(\xi) (y-x) + \] +\end{thm} +\begin{proof} + Consider + \begin{equation} + \begin{split} + g: [0, 1] &\longrightarrow \realn \\ + t &\longmapsto f(x + t(y - x)) + \end{split} + \end{equation} + Apply the one dimensional intermediate value theorem. Due to the chain rule, $g$ fulfils the prerequisites. + $\exists \theta \in (0, 1)$ such that + \begin{equation} + f(y) - f(x) = g(1) - g(0) = g(\theta) = Df(x + \theta(y - x))(y - x) + \end{equation} + For $\xi = x + \theta(y - x)$ follows the initial statement. +\end{proof} + +\begin{thm}[Intermediate value theorem] + Let $U \subset \realn^n$ be open, $\cline \subset U$ and $f: U \rightarrow \realn^m$ differentiable on $\oline$ and continuous in $x, y$. Then + \[ + \exists \xi \in \oline: ~~\norm{f(y) - f(x)} \le \norm{Df(\xi)(y - x)} + \] +\end{thm} +\begin{proof} + For $a \in \realn^m$, consider the (real) helper function + \begin{equation} + a^Tf(x) = \innerproduct{a}{f(x)} + \end{equation} + According to the previous theorem + \begin{equation} + \exists \xi \in \oball: ~~a^T f(y) - a^T f(x) = a^T D f(\xi)(y - x) + \end{equation} + In this implication the chain rule has been applied. We can rewrite this using the scalar product + \begin{equation} + \begin{split} + \norm{f(y) - f(x)}^2 &= \abs{\innerproduct{f(y) - f(x)}{Df(\xi)(y-x)}} \\ + &\le \norm{f(y) - f(x)}\norm{Df(\xi)(y - x)} + \end{split} + \end{equation} +\end{proof} + +\begin{cor} + Let $U \subset \realn^n$ be open and $f: U \rightarrow \realn^m$ a differentiable function. + \[ + Df = 0 \text{ on } U \implies \exists V \subset U: f \text{ constant on } V + \] +\end{cor} +\begin{proof} + Let $x \in U$, choose $\epsilon > 0$ such that $\oball(x) \subset U$. Then + \begin{equation} + \forall y \in \oball(x) ~\exists \xi \in \oline: ~~\norm{f(y) - f(x)} \le \norm{Df(\xi)(y - x)} = 0 + \end{equation} + This implies + \begin{equation} + \norm{f(y) - f(x)} = 0 \implies f(y) = f(x) ~~\forall y \in \oball(x) + \end{equation} +\end{proof} + +\begin{rem} + Functions with vanishing derivatives must be constant. Consider + \begin{align*} + f: (-2, -1) \cup (1, 2) &\longrightarrow \\ + x &\longmapsto \begin{cases} + -1, & x < 0 \\ + 1, & x > 0 + \end{cases} + \end{align*} + Local constancy implies constancy on connected sets. +\end{rem} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 7de147747f94d092d19e48980d5d09092b076aca..dd7840495c0a797d2bb976521f6a6ca507892a3b 100644 GIT binary patch delta 154167 zcmaHyRa9KvwylG^ySqCD6z=Zs?hqtEkU*fY;K8AANpN>}m*5V;-5r93Oa8svJ$s+r zcAu(M%`xVhHJ`rTM_)tZZ2MOn{=pEzkik&F(7`ZuuSY1-YnHS1-I~Q< z`K)Sp!a}XhM8>fvYnzwJ#nwKS@#}A1k%v3FQ>5Y4`f2RzYjN3L!;3F2$ppKR4>ewZ zPaf4xCE5!!sjk-Mh39*0*EVT<=aSD+cmR?V5FR7w;&`luQADvkgSld~|Ij;0#J52= zd$r8Dz`W*tD*RYtLyfgLvF_22K2DBI1bD6o#}a5>tzGj>0v4XpXZCDZp(X9HhwCdA zJbU-{mnMp5wII4(w0cEZ;XB6H6~Tg!Q|w@^ujq}5JR8o65)=#w z7e#?5H{2O*5ppbp!0@b=WI3w{f*+6&Ak3tkhbFH{HKE+R`5#n_260|dV#mvL>Dlz> z`nq}UqY2MTESn_1Vtlw`!8b8-=RTV3u{n|+O}f#@Lh+8K2Mc*@)IGUcf^_)bOj9*- zqr;EPaXO_Mg5e1u`$S4zc4EK@r+B 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