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\begin{document}

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

\begin{enumerate}
\item Calculate the differential of the identity map $F\left(  x\right)  =x$
where $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}.$

\item Suppose $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and $g:\mathbb{R}%
^{m}\rightarrow\mathbb{R}^{n}$ are differentiable. Prove the chain rule for
the composition $g\circ f$.

\item Demonstrate the chain rule to differentiate $g\circ f$ for the following
pairs of functions:

\begin{enumerate}
\item $f\left(  x,y\right)  =x^{2}+y^{2},$ $g\left(  z\right)  =z^{5}$

\item $f\left(  x,y\right)  =\left(  x^{2}+y^{2},2xy\right)  ,$ $g\left(
z,w\right)  =\left(  we^{z},ze^{-w}\right)  $

\item $f\left(  r,\theta\right)  =\left(  r\cos\theta,r\sin\theta\right)  ,$
$g\left(  x,y\right)  =\left(  \sqrt{x^{2}+y^{2}},\arctan\left(  \frac{y}%
{x}\right)  \right)  $

\item $f\left(  \vec{x}\right)  =Ax,$ $g\left(  \vec{y}\right)  =By$ for
matrices $A$ and $B.$
\end{enumerate}

\item Recall that $GL\left(  n\right)  $ are the $n$ by $n$ invertible
matrices with real coefficients. Evaluate the differentials of the following
matrix functions:

(a) $inv:GL(n)\rightarrow GL(n)$ given by $inv(M)=M^{-1}$.

(b) The determinant function which maps $GL(n)$ to $\mathbb{R}$.

\item (a) Give an example of a function of two variables that is discontinuous
at the origin, but whose partial derivatives at the origin exist.

(b) Give an example of a function of two variables all of whose directional
derivatives exist at the origin, but the function itself is not differentiable
at the origin.

\item An open subset $O\subset\mathbb{R}^{n}$ has the property that for any
pair of points $x,y\in O$, there is a differentiable function $\gamma
:[0,1]\rightarrow O$ such that $\gamma(0)=x$ and $\gamma(1)=y$.
$f:O\rightarrow\mathbb{R}^{m}$ is differentiable and $Df=0$ on $O$. Show that
$f$ is a constant on $O$. (The Hypothesis can be slightly weakened. It is
sufficient to assume that $O$ is connected and open.)

\item Show that every point $p$ on the sphere $x^{2}+y^{2}+z^{2}=1$ has a (3
dimensional) neighborhood $U$ such that there is a smooth, one to one mapping
of an open neighborhood $V$ of the origin in $\mathbb{R}^{3}$ such that the
plane $z=0$ maps to the surface of the sphere.

\item Let $f$ and $\frac{\partial f}{\partial y}$ be continuous on
$[0,1]\times\lbrack0,1]$ and assume that $p,q:[0,1]\rightarrow\left[
0,1\right]  $ are differentiable. Define%

\[
F(y)=\int_{p(y)}^{q(y)}\,\ f(x,y)dx,
\]
\quad for $y\in\lbrack0,1]$. Use the chain rule to find $F^{\prime}(y)$. Hint:
Consider%
\[
G(x_{1},x_{2},x_{3})=\int_{x_{1}}^{x_{2}}\,f(t,x_{3})dt.
\]


\item Show that in a neighborhood of the origin, the systems of equations%
\begin{align*}
3x+y-z+u^{2}  &  =0\\
x-y+2z+u  &  =0\\
2x+2y-3z+2u  &  =0
\end{align*}
can be solved for $x,y,u$ in terms of $z$; for $x,z,u$ in terms of $y$; for
$y,z,u$ in terms of $x$. What can you say (using the implicit function
theorem) about solving them for $x,y,z$ in terms of $u$?

\item In this problem we will calculate the exact value of the Gaussian
integral. Consider the integral
\[
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-\left(  x^{2}+y^{2}\right)
}dydx.
\]
Change variables to polar coordinates to compute this double integral, and
then use the properties of integrals to infer that
\[
\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}.
\]


\item Let $f(x,y)=(x^{2}-y^{2})/(x^{2}+y^{2})^{2}$. Show that $\int_{0}%
^{1}\left(  \int_{0}^{1}f(x,y)dx\right)  dy=-\pi/4,$but $\int_{0}^{1}\left(
\int_{0}^{1}f(x,y)dy\right)  dx=\pi/4.$ Why does this not have contract change
of order of integration?

\item Calculate the volume and surface area of a sphere $\mathbb{S}^{2}%
\subset\mathbb{R}^{3}$ using integration. Can you generalize to any dimension?
What happens as the dimension gets large?

\item Let $P$ be a convex polyhedron in $\mathbb{R}^{3}$ with surface
$\partial P.$ Let $n_{1},\ldots,n_{f}$ denote the normal vectors to the facets
of $\partial P$ such that the length corresponds to the area of the facet. Use
the divergence theorem to show that the sum $\sum_{i=1}^{f}n_{i}=0$ by using
the divergence theorem on the integral
\[
\sum_{i=1}^{f}\int_{F_{i}}X\cdot n_{i}~dA
\]
for constant vectors $X,$ where $F_{1},\ldots,F_{f}$ denote the facets of $P.$
The converse of this theorem is called Minkowski's Theorem, that given such
normals there is always a convex polyhedron corresponding to those normals.

\item Let $F$ denote the vector field $(x^{2}-z^{2},2xy,z)$ on $\mathbb{R}%
^{3}$. Compute in two different ways the surface integral%

\[
\int\int_{T}\,\ F\cdot n\,\ dA.
\]


where $T$ denotes the surface of the tetrahedron bounded by $x\geq
0,y\geq0,z\geq0,x+y+z\leq1$ and $n$ denotes the outward normal to $T$. Use the
two answers to verify the divergence theorem.

\item Compute in two different ways the line integral%
\[
\oint~ydx-xdy+z^{2}dz
\]
where $C$ is the intersection of the paraboloid $z=x^{2}+4y^{2}$ with the
cylinder $x^{2}+y^{2}=9$, traversed counter-clockwise when viewed from the
point $(0,0,100)$. Use the two answers to verify Stokes' theorem.

\item A smooth function $F$ on the (open) unit disc in $\mathbb{R}^{2}$
satisfies $F(0,0)=(0,0)$ and $\Vert F(x)\Vert\geq3\Vert x\Vert$.

(a) Show that $F$ is one to one in a neighborhood of the origin.

(b) Let $F=(u,v)$ in components. Show that, for all $a<1$, the integral
$\frac{1}{2\pi}\oint_{\Vert r\Vert=a}\frac{u\nabla v-v\nabla u}{u^{2}+v^{2}%
}\cdot dr$ is an integer and this value does not depend on $a$.

(c) Can you determine the value of the integral in terms of the Jacobian
$A=DF(0,0)$ of the mapping $F$ at the origin?

(d) Can you construct a smooth function $F$ on the (open) unit disc in
$\mathbb{R}^{2}$ satisfying $F(0,0)=(0,0)$, $F(x)\neq(0,0)$ for $x\neq(0,0)$,
such that the value of the integral is 2?

\item $M:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is the mapping%

\[
M(x,y)=(2x-\sin(xy),\frac{1}{3}y+x^{2}-y^{2}).
\]


(a) Show that $M$ is invertible in a neighborhood of the origin.

(b) The unstable manifold of the origin is the set $\Gamma_{u}=\{(x,y)|M^{-n}%
(x,y)\rightarrow(0,0)$ as $n\rightarrow\infty\}$. Show that $\Gamma_{u}$ is
invariant under $M$.

(c) Show that, there is a unique function $f$ such that $f(x)\rightarrow0$
when $x\rightarrow0$, and the unstable manifold is given by the graph of the
function $f$ for sufficiently small $x$, i.e.$\Gamma_{u}\cap B_{\varepsilon
}(0,0)=\{(x,f(x)):|x|<\varepsilon\}\cap B_{\varepsilon}(0,0)\ $
\end{enumerate}


\end{document}