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198 lines
8.2 KiB
TeX
198 lines
8.2 KiB
TeX
\subsection{Partial Derivatives}
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\begin{subbox}{Partial Derivative}
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\smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R,\quad 1 \leq i \leq n,\quad x_0 \in X$}
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$$\dfd{i}(x_{0}) := g'(x_{0,i})$$
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\smalltext{for $g: \{ t \in \R \sep (x_{0, 1}, \ldots,\ t\ ,\ldots, x_{0, n}) \in X \} \to \R^n$}
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$$ g(t) := \underbrace{f(x_{0,i}, \ldots, x_{0,t-1},\ t\ , x_{0, t+1},\ldots,x_{0, n})}_{ \text{ Freeze all }x_{0, k} \text{ except one } x_{0, i} \to t}$$
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\end{subbox}
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\notation $\dfd{i}(x_0) = \sdfd{i}(x_0) =\ssdfd{i}(x_0)$
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\lemma \textbf{Properties of Partial Derivatives}\\
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\smalltext{Assuming $\sdfd{i} \text{ and } \partial_{x_i} g \text{ exist }$:}
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$
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\begin{array}{ll}
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(i) & \partial x_i (f+g) = \partial x_i f + \partial x_i g \\
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(ii) & \partial x_i (fg) = \partial x_i (f)g + \partial x_i (g)f\quad \text{ if } m=1\\
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(iii) & \partial x_i \Bigr(\displaystyle\frac{f}{g}\Bigl) = \displaystyle\frac{\partial x_i(f)g - \partial x_i(g)f}{g^2}\quad \text{ if } g(x) \neq 0\ \forall x \in X\\
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\end{array}
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$\\
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\subtext{$X \subset \R^n \text{ open},\quad f.g: X \to \R^n,\quad 1 \leq i \leq n$}
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\begin{subbox}{The Jacobian}
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\smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R^n \text{ with partial derivatives existing}$}
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$$
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\textbf{J}_f(x) := \begin{bmatrix}
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\partial x_1 f_1(x) & \partial x_2 f_1(x) & \cdots & \partial x_n f_1(x) \\
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\partial x_1 f_2(x) & \partial x_2 f_2(x) & \ddots & \vdots \\
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\vdots & \vdots & \ddots & \vdots \\
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\partial x_1 f_n(x) & \partial x_2 f_n(x) & \cdots & \partial x_n f_m(x)
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\end{bmatrix}
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$$
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\end{subbox}
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\subtext{Think of $f$ as a vector of $f_i$, then $\textbf{J}_f$ is that vector stretched for all $x_j$}
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\definition \textbf{Gradient} $\nabla f(x_0) := \begin{bmatrix}
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\partial x_1 f(x_0) \\
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\vdots \\
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\partial x_n f(x_0)
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\end{bmatrix} = \textbf{J}_f(x)^\top$\\
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R$, i.e. \textit{must} map to $1$ dimension}
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\remark $\nabla f$ points in the direction of greatest increase.
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\subtext{This generalizes that in $\R$, $\text{sgn}(f)$ shows if $f$ increases/decreases}
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\definition \textbf{Divergence} $\text{div}(f)(x_0) := \text{Tr}\bigr(\textbf{J}_f(x_0)\bigl)$\\
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^n,\quad \textbf{J}_f \text{ exists}$}
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\subsection{The Differential}
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\smalltext{
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Partial derivatives don't provide a good approx. of $f$, unlike in the $1$-dimensional case. The \textit{differential} is a linear map which replicates this purpose in $\R^n$.
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}
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\begin{subbox}{Differentiability in $\R^n$ \& the Differential}
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\smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R^n,\quad u: \R^n \to \R^m \text{ linear map}$}
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$$
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df(x_0) := u
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$$
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If $f$ is differentiable at $x_0 \in X$ with $u$ s.t.
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$$
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\underset{x \neq x_0 \to x_0}{\lim} \frac{1}{\big\| x - x_0 \big\|}\Biggl( f(x) - f(x_0) - u(x - x_0) \Biggr) = 0
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$$
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\end{subbox}
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\subtext{Similarly, $f$ is differentiable if this holds for all $x \in X$}
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\lemma \textbf{Properties of Differentiable Functions}
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$
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\begin{array}{ll}
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(i) & \text{Continuous on } X \\
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(ii) & \forall i \leq m, j \leq n:\quad \partial_{x_j}f_i \text{ exists} \\
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(iii) & m=1:\quad \partial_{x_i} f(x_0) = a_i \\
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& \text{for:}\quad u(x_1,\ldots,x_n) = a_1x_1 + \cdots + a_nx_n
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\end{array}
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$
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m \text{ differentiable on } X$}
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\lemma \textbf{Preservation of Differentiability}
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$
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\begin{array}{ll}
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(i) & f + g \text{ is differentiable: } d(f+g)=df+dg \\
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(ii) & fg \text{ is differentiable, if } m=1 \\
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(iii) & \displaystyle\frac{f}{g}\ \text{ is differentiable, if } m=1,\ g(x) \neq 0\ \forall x \in X
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\end{array}
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$
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\subtext{$X \subset \R^n \text{ open},\quad f,g: X \to \R^m \text{ differentiable on }X$}
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\lemma \textbf{Cont. Partial Derivatives imply Differentiability}
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if all $\partial_{x_j} f_i$ exist and are continuous:
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$$
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f \text{ differentiable on } X,\quad df(x_0) = \textbf{J}_f(x_0)
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$$
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m$}
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\lemma \textbf{Chain Rule} $\quad g \circ f \text{ is differentiable on } X$
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\begin{align*}
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& d(g \circ f)(x_0) &= dg\bigl( f(x_0) \bigr) \circ df(x_0) \\
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& \textbf{J}_{g \circ f}(x_0) &= \textbf{J}_g\bigl( f(x_0) \bigr) \cdot \textbf{J}_f(x_0)
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\end{align*}
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\subtext{$X \subset \R^n \text{ open},\quad Y \subset \R^m \text{ open},\quad f: X \to Y, g: Y \to \R^p, f,g \text{ diff.-able}$}
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\definition \textbf{Tangent Space}
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$$
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T_f(x_0) := \Bigl\{ (x,y) \in \R^n \times \R^m \sep y = f(x_0) + u(x-x_0) \Bigr\}
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$$
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m \text{ diff.-able},\quad x_0 \in X,\quad u = df(x_0)$}
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\definition \textbf{Directional Derivative}
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$$
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D_v f(x_0) = \underset{t \neq 0 \to 0}{\lim} \frac{f(x_0 + tv) - f(x_0)}{t}
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$$
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m,\quad v \neq 0 \in \R^n,\quad x_0 \in X$}
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\lemma \textbf{Directional Derivatives for Diff.-able Functions}
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$$
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D_vf(x_0) = df(x_0)(v) = \textbf{J}_f(x_0) \cdot v
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$$
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\subtext{$X \subset \R^n \text{ open},\quad f: X \to \R^m \text{ diff.-able},\quad v \neq 0 \in \R^n,\quad x_0 \in X$}
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\remark $D_vf$ is linear w.r.t $v$, so: $D_{v_1 + v_2}f = D_{v_1}f + D_{v_2}f$
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\remark $D_vf(x_0) = \nabla f(x_0) \cdot v = \big\| \nabla f(x_0) \big\| \cos(\theta)$\\
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\subtext{In the case $f: X \to \R$, where $\theta$ is the angle between $v$ and $\nabla f(x_0)$}
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\newpage
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\subsection{Higher Derivatives}
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\definition \textbf{Differentiability Classes}
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\begin{align*}
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& f \in C^1(X;\R^m) &\iffdef& f \text{ diff.-able on } X, \text{ all } \partial_{x_j} f_i \text{ exist} \\
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& f \in C^k(X;\R^m) &\iffdef& f \text{ diff.-able on } X, \text{ all } \partial_{x_j} f_i \in C^{k-1} \\
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& f \in C^\infty(X;\R^m) &\iffdef& f \in C^k(X;\R^m)\ \forall k \geq 1
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\end{align*}
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\subtext{$X \subset \R^n \text{ open},\quad f:X\to\R^m$}
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\lemma Polynomials, Trig. functions and $\exp$ are in $C^\infty$
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\lemma \textbf{Operations preserve Differentiability Classes}
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$
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\begin{array}{lcll}
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(i) & f + g & \in C^k \\
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(ii) & fg & \in C^k & \text{ if } m=1 \\
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(iii) & \displaystyle\frac{f}{g} & \in C^k & \text{ if } m=1, g(x) \neq 0\ \forall x \in X
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\end{array}
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$\\
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\subtext{$f,g \in C^k$}
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\lemma \textbf{Composition preserves Differentiability Classes}
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$$
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g \circ f \in C^k
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$$
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\subtext{$f \in C^k,\quad f(X) \subset Y,\quad Y \subset \R^m \text{ open},\quad g: Y \to \R^p,\quad g \in C^k$}
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\begin{subbox}{Partial Derivatives commute in $C^k$}
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\smalltext{$k \geq 2,\quad X \subset \R^n \text{ open},\quad f: X \to \R^m,\quad f \in C^k$}
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$$
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\forall x,y:\quad \partial_{x,y}f = \partial_{y,x}f
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$$
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\smalltext{This generalizes for $\partial_{x_1,\ldots,x_n}f$.}
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\end{subbox}
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\remark Linearity of Partial Derivatives
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$$
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\partial_x^m(af_1 + bf_2) = a\partial_x^mf_1 + b\partial_x^mf_2
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$$
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\subtext{Assuming both $\partial_x f_{1,2}$ exist.}
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\definition \textbf{Laplace Operator}
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$$
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\Delta f := \text{div}\bigl( \nabla f(x) \bigr) = \sum_{i=0}^{n} \frac{\partial}{\partial x_i}\Bigl( \frac{\partial f}{\partial x_i} \Bigr) = \sum_{i=0}^{n} \frac{\partial^2f}{\partial x_i^2}
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$$
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\begin{subbox}{The Hessian}
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\smalltext{$X \subset \R^n \text{ open},\quad f: X \to \R,\quad f \in C^2,\quad x_0 \in X$}
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$$
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\textbf{H}_f(x) := \begin{bmatrix}
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\partial_{1,1}f(x_0) & \partial_{2,1}f(x_0) & \cdots & \partial_{n,1}f(x_0) \\
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\partial_{1,2}f(x_0) & \partial_{2,2}f(x_0) & \cdots & \partial_{n,2}f(x_0) \\
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\vdots & \vdots & \ddots & \vdots \\
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\partial_{1,n}f(x_0) & \partial_{2,n}f(x_0) & \cdots & \partial_{n,n}f(x_0)
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\end{bmatrix}
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$$
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Where $\bigl( \textbf{H}_f(x) \bigr)_{i,j} = \partial_{x_i,x_j}f(x)$
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\end{subbox}
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\subtext{Note that $f: X \to \R$, i.e. $\textbf{H}_f$ only exists for $1$-dimensionally valued $f$}
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\notation $\textbf{H}_f(x) = \text{Hess}_f(x) = \nabla^2f(x)$
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\remark $\textbf{H}_f(x_0)$ is symmetric: $\bigl( \textbf{H}_f(x_0) \bigr)_{i,j} = \bigl( \textbf{H}_f(x_0) \bigr)_{j, i}$
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\subsection{Change of Variable} |