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zch969ABZ|W7IRc=wXXji^Lr1B$tUt$66vDH?ZL@W3hLu(v|ro7S>TPf$1w8u?d1JF i;T*^&BGVN)B~Ql;1wJ?q%ug$ikXN7)6~!27)BG1w$+b8D diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex index 0e5b451..4f4e1d1 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/00_continuity.tex @@ -66,3 +66,4 @@ Furthermore: $\{ x\in \R^3 : ||x - x_0|| = r \}$ is closed\\ \shorttheorem Let $(X \neq \emptyset) \subseteq \R^n$ compact and $f: X \rightarrow \R$ continuous. Then $f$ bounded, has $\max$ and $\min$, i.e. $\exists x_+, x_- \in X$ s.t. $\displaystyle f(x_+) = \sup_{x \in X} f(x)$ and $\displaystyle f(x_-) = \inf_{x \in X} f(x)$ % ──────────────────────────────────────────────────────────────────── +\rmvspace diff --git a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex index ad72069..50b9462 100644 --- a/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex +++ b/semester3/analysis-ii/cheat-sheet-jh/parts/vectors/differentiation/02_differential.tex @@ -1,14 +1,57 @@ -\newsectionNoPB +\newsection \subsection{The differential} \setLabelNumber{all}{2} \compactdef{Differentiable function} We have function $f: X \rightarrow \R^m$, linear map $u : \R^n \rightarrow \R^m$ and $x_0 \in X$. $f$ is differentiable at $x_0$ with differential $u$ if $\displaystyle \lim_{\elementstack{x \rightarrow x_0}{x \neq x_0}} \frac{1}{||x - x_0||} (f(x) - f(x_0) - u(x - x_0) = 0$ where the limit is in $\R^m$. We denote $\dx f(x_0) = u$. -If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$ - +If $f$ is differentiable at every $x_0 \in X$, then $f$ is differentiable on $X$. +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} +\shortproposition +\rmvspace Let $f: X \rightarrow \R^m$ be differentiable on $X$ +\begin{itemize}[noitemsep] + \item $f$ is continuous on $X$ + \item $f$ admits partial derivatives on $X$ with respect to each variable + \item Assume $m = 1$, let $x_0 \in X$ and let $u(x_1, \ldots, x_n) = a_1 x_1 + \ldots + a_n x_n$ be diff. of $f$ at $x_0$. + Then $\partial_{x_i} f(x_0) = a_i$ for $1 \leq i \leq n$ +\end{itemize} +\rmvspace % ──────────────────────────────────────────────────────────────────── \stepLabelNumber{all} \shortproposition Let $f, g : X \rightarrow \R^m$ with $X \subseteq \R^n$ open +\rmvspace \begin{itemize}[noitemsep] - \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$ + \item The function $f + g$ is differentiable with differential $\dx (f + g) = \dx f + \dx g$. If $m = 1$, then $fg$ is differentiable + \item If $m = 1$ and if $g(x) \neq 0 \forall x \in X$, then $f \div g$ is differentiable \end{itemize} +\rmvspace +% ──────────────────────────────────────────────────────────────────── +\shortproposition If $f$ as above has all partial derivatives on $X$ and if they are all continuous on $X$, then $f$ is differentiable on $X$. +The differential is the Jacobi Matrix of $f$ at $x_0$. +This implies that most elementary functions are differentiable.\\ +% ──────────────────────────────────────────────────────────────────── +\compactproposition{Chain Rule} For $X \subseteq \R^n$ and $Y \subseteq \R^m$ both open and $f: X \rightarrow Y$ and $g : Y \rightarrow \R^p$ are both differentiable. +Then $g \circ f$ is differentiable on $X$ and for any $x \in X$, its differential is given by +$\dx (g \circ f)(x_0) = \dx g(f(x_0)) \circ \dx f(x_0)$. +The Jacobi matrix is $J_{g \circ f}(x_0) = J_g(f(x_0)) J_f(x_0)$ (RHS is a matrix product)\\ +% ──────────────────────────────────────────────────────────────────── +\setLabelNumber{all}{11} +\compactdef{Tangent space} The graph of the affine linear approximation $g(x) = f(x_0) + u(x - x_0)$, or the set +\vspace{-0.75pc} +\begin{align*} + \{ (x, y) \in \R^n \times \R^m : y = f(x_0) + u(x - x_0) \} +\end{align*} + +\drmvspace\rmvspace +% ──────────────────────────────────────────────────────────────────── +\stepLabelNumber{all} +\compactdef{Directional derivative} $f$ has a directional derivative $w \in \R^m$ in the direction of $v \in \R^n$, +if the function $g$ defined on the set $I = \{ t \in \R : x_0 + tv \in X \}$ by $g(t) = f(x_0 + tv)$ has a derivative at $t = 0$ and is equal to $w$ +% ──────────────────────────────────────────────────────────────────── +\shortremark Because $X$ is open, the set $I$ contains an open interval $]-\delta, \delta[$ for some $\delta > 0$. +% ──────────────────────────────────────────────────────────────────── +\shortproposition Let $f$ as previously be differentiable. Then for any $x \in X$ and non-zero $v \in \R^n$, +$f$ has a directional derivative at $x_0$ in the direction of $v$, given by$\dx f(x_0)(v)$ +% ──────────────────────────────────────────────────────────────────── +\shortremark The values of the above directional derivative are linear with respect to the vector $v$. +Suppose we know the dir. der. $w_1$ and $w_2$ in directions $v_1$ and $v_2$, then the directional derivative in direction $v_1 + v_2$ is $w_1 + w_2$