Multivariable Calculus Foundation

Directional Derivative

Let $U\subseteq {\mathbb{R}}^m$ be an open set, and let $f:U\to {\mathbb{R}}^n$ be a function, the directional derivative for $v\in {\mathbb{R}}^m$ is defined as $D_{v}f(x)={\frac{d}{dt}|}_{t=0}f(x+t\cdot v)={\lim }_{t\to 0}\frac{f(x+tv)-f(x)}{t}={\lim }_{s\to 0}\frac{L(sv)+o(\Vert sv\Vert )}{s}=L(v)$ In particular the partial derivatives exist and we have $\frac{\partial f}{\partial x^i}=Df{|}_x(e_i)$

Differentiable Function

A function $f$ from an open subset $U$ of ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ is differentiable at a point $x\in U$ if it is well approximated by a linear function $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ near $x$, in the sense that $f(y)=f(x)+L(y-x)+o(\Vert y-x{\Vert }_{{\mathbb{R}}^m})\text{as }y\to x,$ or equivalently if ${\lim }_{y\to x}\frac{\Vert f(y)-f(x)-L(y-x){\Vert }_{{\mathbb{R}}^n}}{\Vert y-x{\Vert }_{{\mathbb{R}}^m}}=0.$

Writing $f$ and $v\in {\mathbb{R}}^m$ in components as

$f=\left[\begin{array}{c}{f}^1\\ ⋮\\ f^n\end{array}\right],v=\left[\begin{array}{c}{v}^1\\ ⋮\\ v^m\end{array}\right]$

we have $Df{|}_x(v)=\left[\begin{array}{ccc}\frac{\partial f^1}{\partial x^1}& \dots & \frac{\partial f^1}{\partial x^m}\\ ⋮& \ddots & ⋮\\ \frac{\partial f^n}{\partial x^1}& \dots & \frac{\partial f^n}{\partial x^m}\end{array}\right]\left[\begin{array}{c}{v}^1\\ ⋮\\ v^m\end{array}\right].$ The matrix above is called Jacobi Matrix

Chain Rule

Let $U\subset {\mathbb{R}}^k$ and $V\subset {\mathbb{R}}^m$ be open sets, and $f:U\to {\mathbb{R}}^m$ and $g:V\to {\mathbb{R}}^n$ maps, and $x\in U$ with $f(x)\in V$. If $f$ is differentiable at $x$ and $g$ is differentiable at $f(x)$, then $g\circ f$ is differentiable at $x$ and we have $D(g\circ f){|}_x=Dg{|}_{f(x)}\circ Df{|}_x.$

Higher derivatives and Taylor approximations

We can similarly define higher derivatives inductively as best polynomial approximations: Recall that a map $L:\underset{k\text{ times}}{\underbrace{V\times \cdots \times V}}\to W$ is called $k$-multilinear if it is separately linear in each argument. Such a map is called symmetric if it is independent of the order of the arguments: $L(v_1,v_2,\dots ,v_k)=L(v_{\sigma (1)},\dots ,v_{\sigma (k)})$ where $\sigma$ is any permutation of $\{1,\dots ,k\}$. The $k$th derivative is a symmetric $k$-multilinear map from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ for each $k$: If we have already defined such maps $Df{|}_x,D^{2}f{|}_x,\dots D^{k-1}f{|}_x$, then we say $f$ is $k$ times differentiable at $x$ if there is a symmetric $k$-multilinear map $L:{\mathbb{R}}^m\times \cdots \times {\mathbb{R}}^m\to {\mathbb{R}}^n$ such that $\begin{array}{rl}f(x+v)& =f(x)+Df{|}_x(v)+\frac{1}{2}{D}^{2}f{|}_x(v,v)+\cdots +\frac{1}{(k-1)!}{D}^{k-1}f{|}_x(v,\dots ,v)\\ & +\frac{1}{k!}L(v,\dots ,v)+o(\Vert v{\Vert }^k)\text{as }v\to 0.\end{array}$ In this case the map $L$ is unique, and we denote it by $D^{k}f{|}_x$