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\begin{document}

\title{Multivariate Calculus}
\author{David Glickenstein\thanks{based on work by Shankar Venkataramani, Ibrahim
Fatkullin, Tom Kennedy, and others}}
\maketitle

\section{Introduction}

We will cover the following:

\begin{enumerate}
\item The differential, its relationship to directional derivatives and the
Jacobian matrix, and chain rule.

\item Inverse and implicit functions.

\item Multivariate integration and integration over surfaces - mention of
divergence theorem, etc.
\end{enumerate}

\section{Differentiation}

\begin{definition}
Let $O\subset\mathbb{R}^{n}$ be open and $f:O\rightarrow\mathbb{R}^{m}$. Let
$c\in O$. The function $f$ is said to be differentiable at $c$ if there is a
linear function, called the differential, or total derivative, $T_{c}%
^{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$, such that
\end{definition}

%

\[
\lim_{v\ \rightarrow0}\frac{||f(c+v)-f(c)-T_{c}^{f}(v)||}{||v||}=0.
\]


We say $f$ is differentiable on $O$ if it is differentiable at every point in
$O$.

If the total derivative exists, then for all directions the directional
derivative exists%

\[
D_{v}f(c)=\lim_{\epsilon\rightarrow0}\frac{f(c+\epsilon v)-f(c)}{\epsilon}%
\]


and equals $T_{c}^{f}(v)$.

Let $e_{1},e_{2},\cdots,e_{n}$ be the standard basis for $\mathbb{R}^{n}$.
Then the partial derivatives are the directional derivative in the directions
of the standard basis:%

\[
\frac{\partial f}{\partial x_{k}}=D_{e_{k}}f.
\]


Note that both sides of this equation are vectors in $\mathbb{R}^{m}$. The
matrix representation of $T$ in this basis is called the Jacobian matrix%

\[
df(c)=%
\begin{array}
[c]{cccc}%
\frac{\partial f_{1}}{\partial x_{1}}(c) & \frac{\partial f_{1}}{\partial
x_{2}}(c) & \cdots & \frac{\partial f_{1}}{\partial x_{n}}(c)\\
\vdots &  & \ddots & \vdots\\
\frac{\partial f_{m}}{\partial x_{1}}(c) & \frac{\partial f_{m}}{\partial
x_{2}}(c) & \cdots & \frac{\partial f_{m}}{\partial x_{n}}(c)
\end{array}
\]


\begin{theorem}
[(Chain rule)]Suppose that $f:O\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$
and $g:U\subset\mathbb{R}^{k}\rightarrow O$ are both differentiable. Then
$h=f\circ g$ is differentable and
\[
T_{p}^{h}=T_{g(p)}^{f}\circ T_{p}^{g}.
\]
In matrix form $dh(p)=df(g(p))\,dg(p).$
\end{theorem}

We now consider higher order derivatives.

\begin{theorem}
Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$. Then the following conditions
are sufficient for the equality of the mixed partial derivatives
\[
\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}(c)=\frac{\partial^{2}%
f}{\partial x_{j}\partial x_{i}}(c)
\]

\end{theorem}

\begin{enumerate}
\item Both $\partial f/\partial x_{i}$ and $\partial f/\partial x_{j}$ exist
in an $n$-ball $B(c,\delta)$ and are differentiable at $c$.

\item Both $\partial f/\partial x_{i}$ and $\partial f/\partial x_{j}$ exist
in an $n$-ball $B(c,\delta)$ and $\partial^{2}f/\partial x_{i}\partial x_{j}$
and $\partial^{2}f/\partial x_{j}\partial x_{i}$ are both continuous at $c$.
\end{enumerate}

Call $\alpha=(\alpha_{1},\cdots,\alpha_{n})$ a multi-index if each of its
entries are non-negative integers. Write $|\alpha|=\alpha_{1}+\cdots
+\alpha_{n}$. This allows for the notational abbreviations%

\[
x^{\alpha}=x^{\alpha_{1}}\cdots x^{\alpha_{n}},
\]
\quad%

\[
D^{\alpha}=\frac{\partial}{\partial x^{\alpha_{1}}}\cdots\frac{\partial
}{\partial x^{\alpha_{n}}}%
\]


and provides for a compact notation for Taylor's formula for functions $f$
from $\mathbb{R}^{n}$ to $\mathbb{R}$. Write
\[
f^{(k)}(x;t)=\sum_{\alpha:\ |\alpha|=k}\,D_{\alpha}f(x)t^{\alpha}%
\]
and assume that $f$ and all of its partial derivatives of order up to $m-1$
are differentiable at each point of an open set $S\subset\mathbb{R}^{n}$.
Choose $x$ and $a$ so that $L(a,x)\subset S$. Then for some $c\in L(a,x)$,
\[
f(x)=f(a)+\sum_{k=1}^{m-1}\,\frac{1}{k!}f^{(k)}(x;t)+\frac{1}{m!}f^{(m)}(c;t)
\]


\section{Inverse and Implicit Function Theorems}

\begin{theorem}
[Inverse function theorem]Let $f:S\rightarrow\mathbb{R}^{n}$ be continuously
differentiable on an open set $S\subset\mathbb{R}^{n}$. If the differential
$Df\left(  a\right)  $ is nondegeneratefor some point $a\in S$,
\end{theorem}

then there exists two open sets $X\subset S$ and $Y\subset f(S)$ such that
$a\in X$ and $f(a)\in Y$ and $Y=f(X)$ and a unique function $g$ defined on $Y$
such that $g\left(  f\left(  x\right)  \right)  =x$ for every $x\in X$ and $g$
is continuously differentiable on $Y$, with $dg\left(  f\left(  x\right)
\right)  =df\left(  x\right)  ^{-1}.$

\begin{theorem}
[Implicit function theorem]Let $S\subset\mathbb{R}^{n}\times\mathbb{R}^{k}$
and suppose that $f:S\rightarrow\mathbb{R}^{n}$ is continuously
differentiable. Assume that $f(x_{0},y_{0})=0$ and that the $n\times n$ determinant
\end{theorem}

$[\partial f_{j}/\partial x_{i}(x_{0},y_{0})]$ is nondegenerate. Then there
exists a $k$-dimensional set $Y_{0}$ containing $y_{0}$ and a unique vector
valued function $g:Y_{0}\rightarrow\mathbb{R}^{n}$ such that

1. $g$ is continuously differentiable

2. $g(y_{0})=x_{0}$

3. $f(g(y),y)=0$ for every $y\in Y_{0}$.

\bigskip

\section{Integration}

Finally, we recall integration,

Let $A=[a_{1},b_{1}]\times\cdots\times\lbrack a_{n},b_{n}]\subset
\mathbb{R}^{n}$ and let $P_{k}$ be a partition of $[a_{k},b_{k}]$ into $m_{k}$
intervals. Then%

\[
P=P_{1}\times\cdots\times P_{n}%
\]


divides $A$ into $m_{1}\times\cdots\times m_{n}$ intervals each of dimension
$n$. A partition $Q$ is called a refinement of $P$ if $P\subset Q$. For $I\in
P$ let $Vol(I)$ be the product of the lengths of the one dimensional intervals
that determine $I$. Let $f$ be a real valued function defined on $A$. For any
choice of sample points $t_{I}\in I$, define the Riemann sum%
\[
S(f,t_{I},P)=\sum_{I}f(t_{I})Vol(I)
\]


The function $f$ is Riemann integrable on $A$ if there exist a number, $r$,
having the following property:

For every $\epsilon>0$, there exists a partition $P_{\epsilon}$ such that for
any refinement $Q$ of $P_{\epsilon}$ and any choice of sample points $t_{I}\in
I$%

\[
|S(f,t_{I},P)-r|<\epsilon.
\]


We typically write the integral%
\[
\int_{A}f(x_{1},\ldots,x_{n})\ d(x_{1},\ldots,x_{n})=\int_{A}f(x)\ dx.
\]


Let $T$ be a one-to-one continuously differentiable mapping of an open set
$V\subset\mathbb{R}^{k}$ into $\mathbb{R}^{k}$ such that the determinant of
the Jacobian is nonzero for all $x\in V$. Let $f$ be a continuous function on
$\mathbb{R}^{k}$ whose support is compact and lies in $T(V)$. Then
\[
\int_{\mathbb{R}^{k}}\,\ f(y)\,\ dy=\int_{\mathbb{R}^{k}}\,\ f(T(x))\,\ |\det
J_{T}(x)|\,\ dx.
\]


\begin{theorem}
[Green's Theorem]Let $C$ be a simple closed curve in the $xy$-plane Let
$M(x,y)$ and $N(x,y)$ be continuously differentiable on an open set containing
$C$ and $R$, the region it encloses. Then
\[
\int_{C}(Mdx+Ndy)=\int_{R}\left(  \frac{\partial N}{\partial x}-\frac{\partial
M}{\partial y}\right)  dxdy.
\]

\end{theorem}

\begin{theorem}
[Divergence Theorem]Let $F$ be a continuously differentiable vector field on
an open set $V\subset\mathbb{R}^{3}$, and let $D\subset V$ be closed with
positively oriented boundary $\partial D$. Then
\[
\int_{D}(\nabla\cdot F)dV=\int_{\partial D}(F\cdot\mathbf{n})\,\ dA
\]
where $\mathbf{n}$ is a unit normal vector, pointing outwards.
\end{theorem}

\begin{theorem}
[classical Stokes' Theorem]Let $F$ be a continuously differentiable vector
field on an open set $V\subset\mathbb{R}^{3}$, and let $S\subset V$ be a
2-surface of class $C^{2}$. Then%
\[
\int_{S}(\nabla\times F)\cdot\mathbf{n}\,\ dV=\int_{\partial S}(F\cdot
\mathbf{t})\,\ ds
\]
where $\mathbf{t}$ is a oriented unit tangent vector.
\end{theorem}


\end{document}