Multivariable Calculus

Let the function $f : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ satisfy the following two conditions:

1. $f(K)$ is compact whenever $K$ is a compact subset of $\mathbb {R}^{n}$.

2. If $\left\{ K_n\right\}$ is a decreasing sequence of compact subsets of $\mathbb {R}^{n}$, then

   $$f\left(\bigcap _1^{\infty } K_n\right) = \bigcap _1^{\infty } f(K_n).$$

Prove that $f$ is continuous.

Prove that a map $g : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is continuous only if its graph is closed in $\mathbb {R}^{n} \times \mathbb {R}^{n}$. Is the converse true?

See also Problem 1.2.14.

Let $U \subset \mathbb {R}^{n}$ be an open set. Suppose that the map $h : U \rightarrow \mathbb {R}^{n}$ is a homeomorphism from $U$ onto $\mathbb {R}^{n}$, which is uniformly continuous. Prove $U = \mathbb {R}^{n}$.

Let $f$ be a real valued function on $\mathbb {R}^{2}$ with the following properties:

Let $f$ be a continuous function from the ball $B_n = \left\{ x \in \mathbb {R}^{n} \mid \left\| x\right\| < 1\right\}$ into itself. (Here, $\left\| \cdot \right\|$ denotes the Euclidean norm.) Assume $\left\| f(x)\right\| < \left\| x\right\|$ for all nonzero $x \in B_n$. Let $x_0$ be a nonzero point of $B_n$, and define the sequence $(x_k)$ by setting $x_k = f(x_{k-1})$. Prove that $\lim x_k = 0$.

Let $N$ be a norm on the vector space $\mathbb {R}^{n}$; that is, $N : \mathbb {R}^{n} \rightarrow \mathbb {R}$ satisfies

for all $x, y \in \mathbb {R}^{n}$ and $\lambda \in \mathbb {R}$.

A map $f : \mathbb {R}^{m} \rightarrow \mathbb {R}^{n}$ is proper if it is continuous and $f^{-1}(B)$ is compact for each compact subset $B$ of $\mathbb {R}^{n}$; $f$ is closed if it is continuous and $f(A)$ is closed for each closed subset $A$ of $\mathbb {R}^{m}$.

Suppose that $F : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ is continuous and satisfies

for all $x, y \in \mathbb {R}^{n}$ and some $\lambda > 0$. Prove that $F$ is one-to-one, onto, and has a continuous inverse.

See also Problem 1.2.12.

Prove that $\dfrac {x^2+y^2}{4} \leq e^{x+y-2}$ for $x \geq 0$, $y \geq 0$.

Find the minimal value of the areas of hexagons circumscribing the unit circle in $\mathbb {R}^{2}$.

See also Problem 1.1.12.

Define $f : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$ by $f(x,0) = 0$ and

for $y \neq 0$.

Let $f : \mathbb {R}^{2} \rightarrow \mathbb {R}$ be defined by:

Determine all points at which $f$ is differentiable.

Let $F$, with components $F_1, \ldots , F_n$, be a differentiable map of $\mathbb {R}^{n}$ into $\mathbb {R}^{n}$ such that $F(0) = 0$. Assume that

Prove that there is a ball $B$ in $\mathbb {R}^{n}$ with center $0$ such that $F(B) \subset B$.

Let $p$ be a polynomial over $\mathbb {R}$ of positive degree. Define the function $f : \mathbb {R}^{2} \rightarrow \mathbb {R}^{2}$ by $f(x,y) = (p(x+y), p(x-y))$. Prove that the derivative $Df(x,y)$ is invertible for an open dense set of points $(x,y)$ in $\mathbb {R}^{2}$.

Find the most general continuously differentiable function $g : \mathbb {R} \rightarrow (0,\infty )$ such that the function $h(x,y) = g(x)g(y)$ on $\mathbb {R}^{2}$ is constant on each circle with center $(0,0)$.

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}$ be continuously differentiable. Assume the Jacobian matrix $(\partial f_i / \partial x_j)$ has rank $n$ everywhere. Suppose $f$ is proper; that is, $f^{-1}(K)$ is compact whenever $K$ is compact. Prove $f(\mathbb {R}^{n}) = \mathbb {R}^{n}$.

Suppose $f$ is a continuously differentiable map of $\mathbb {R}^{2}$ into $\mathbb {R}^{2}$. Assume that $f$ has only finitely many singular points, and that for each positive number $M$, the set $\left\{ z \in \mathbb {R}^{2} \mid \left|f(z)\right| \leq M\right\}$ is bounded. Prove that $f$ maps $\mathbb {R}^{2}$ onto $\mathbb {R}^{2}$.

Let $f$ be a real valued function on $\mathbb {R}^{n}$ of class $C^2$. A point $x \in \mathbb {R}^{n}$ is a critical point of $f$ if all the partial derivatives of $f$ vanish at $x$; a critical point is nondegenerate if the $n \times n$ matrix

is nonsingular.

Let $x$ be a nondegenerate critical point of $f$. Prove that there is an open neighborhood of $x$ which contains no other critical points (i.e., the nondegenerate critical points are isolated).

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ be a function whose partial derivatives of order $\leq 2$ are everywhere defined and continuous.

Prove that a real valued $C^3$ function $f$ on $\mathbb {R}^{2}$ whose Laplacian,

is everywhere positive cannot have a local maximum.

Let the function $u$ on $\mathbb {R}^{2}$ be harmonic, not identically $0$, and homogeneous of degree $d$, where $d > 0$.

To say that $u$ is homogeneous of degree $d$ means that $u(tx,ty) = t^d u(x,y)$ for $t > 0$.

Prove that $d$ is an integer.

Let $f : \mathbb {R}^{3} \rightarrow \mathbb {R}^{2}$ and assume that $0$ is a regular value of $f$ (i.e., the differential of $f$ has rank $2$ at each point of $f^{-1}(0)$). Prove that $\mathbb {R}^{3} \setminus f^{-1}(0)$ is arcwise connected.

Let the transformation $T$ from the subset $U = \left\{ (u,v) \mid u > v\right\}$ of $\mathbb {R}^{2}$ into $\mathbb {R}^{2}$ be defined by $T(u,v) = (u+v, u^2+v^2)$.

Let $f$ be a $C^1$ function from the interval $(-1,1)$ into $\mathbb {R}^{2}$ such that $f(0) = 0$ and $f'(0) \neq 0$. Prove that there is a number $\varepsilon$ in $(0,1)$ such that $\left\| f(t)\right\|$ is an increasing function of $t$ on $(0,\varepsilon )$.

For a real $2 \times 2$ matrix

let $\left\| X\right\| = x^2+y^2+z^2+t^2$, and define a metric by $d(X,Y) = \left\| X-Y\right\|$. Let $\Sigma = \left\{ X \mid \det (X) = 0\right\}$. Let

Find the minimum distance from $A$ to $\Sigma$ and exhibit an $S \in \Sigma$ that achieves this minimum.

Let $S \subset \mathbb {R}^{3}$ denote the ellipsoidal surface defined by

Let $T \subset \mathbb {R}^{3}$ be the surface defined by

Prove that there exist points $p \in S$, $q \in T$, such that the line $\overline{pq}$ is perpendicular to $S$ at $p$ and to $T$ at $q$.

Let $P_2$ denote the set of real polynomials of degree $\leq 2$. Define the map $J : P_2 \rightarrow \mathbb {R}$ by

Let $Q = \left\{ f \in P_2 \mid f(1) = 1\right\}$. Show that $J$ attains a minimum value on $Q$ and determine where the minimum occurs.

Let $X$ be the space of orthogonal real $n \times n$ matrices. Let $v_0 \in \mathbb {R}^{n}$. Locate and describe the elements of $X$, where the map

takes its maximum and minimum values.

Let $W \subset \mathbb {R}^{n}$ be an open connected set and $f$ a real valued function on $W$ such that all partial derivatives of $f$ are $0$. Prove that $f$ is constant.

In $\mathbb {R}^{2}$, consider the region $\mathcal{A}$ defined by $x^2+y^2 > 1$. Find differentiable real valued functions $f$ and $g$ on $\mathcal{A}$ such that $\partial f/\partial x = \partial g/\partial y$ but there is no real valued function $h$ on $\mathcal{A}$ such that $f = \partial h/\partial y$ and $g = \partial h/\partial x$.

Suppose that $u(x,t)$ is a continuous function of the real variables $x$ and $t$ with continuous second partial derivatives. Suppose that $u$ and its first partial derivatives are periodic in $x$ with period $1$, and that

Prove that

is a constant independent of $t$.

Let $f(x,t)$ be a $C^1$ function such that $\partial f/\partial x = \partial f/\partial t$. Suppose that $f(x,0) > 0$ for all $x$. Prove that $f(x,t) > 0$ for all $x$ and $t$.

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ have continuous partial derivatives and satisfy

for all $x = (x_1, \ldots , x_n)$, $j = 1, \ldots , n$. Prove that

(where $\left\| u\right\| = \sqrt{u_1^2 + \cdots + u_n^2}$).

Let $f : \mathbb {R}^{n} \setminus \left\{ 0\right\} \rightarrow \mathbb {R}$ be a function which is continuously differentiable and whose partial derivatives are uniformly bounded:

for all $(x_1, \ldots , x_n) \neq (0, \ldots , 0)$. Show that if $n \geq 2$, then $f$ can be extended to a continuous function defined on all of $\mathbb {R}^{n}$. Show that this is false if $n = 1$ by giving a counterexample.

Let $f : \mathbb {R}^{n} \setminus \left\{ 0\right\} \rightarrow \mathbb {R}$ be differentiable. Suppose

exists for each $j = 1, \ldots , n$.

Let $f : \mathbb {R}^{2} \rightarrow \mathbb {R}$ have directional derivatives in all directions at the origin. Is $f$ differentiable at the origin? Prove or give a counterexample.

Let $f : \mathbb {R}^{n} \rightarrow \mathbb {R}$ have the following properties: $f$ is differentiable on $\mathbb {R}^{n} \setminus \left\{ 0\right\}$, $f$ is continuous at $0$, and

for $i = 1, \ldots , n$. Prove that $f$ is differentiable at $0$.

Let $U \subset \mathbb {R}^{n}$ be a convex open set and $f : U \rightarrow \mathbb {R}^{n}$ a differentiable function whose partial derivatives are uniformly bounded but not necessarily continuous. Prove that $f$ has a unique continuous extension to the closure of $U$.

Let $f : \mathbb {R}^{3} \rightarrow \mathbb {R}$ be such that

Suppose $f$ has continuous partial derivatives of orders $\leq 2$. Is there a $p \in \mathbb {R}^{3}$ with $\left\| p\right\| \leq 1$ such that

Let $f$ be a differentiable function from $\mathbb {R}^{n}$ to $\mathbb {R}^{n}$. Assume that there is a differentiable function $g$ from $\mathbb {R}^{n}$ to $\mathbb {R}$ having no critical points such that $g \circ f$ vanishes identically. Prove that the Jacobian determinant of $f$ vanishes identically.

Let $f, g : \mathbb {R} \rightarrow \mathbb {R}$ be smooth functions with $f(0) = 0$ and $f'(0) \neq 0$. Consider the equation $f(x) = tg(x)$, $t \in \mathbb {R}$.

Consider the system of equations

Describe the two regions in $(a,b)$-space for which the function

restricted to the circle $x^2+y^2=1$, has exactly two, and exactly four critical points, respectively.

Let $u$ and $v$ be two real valued $C^1$ functions on $\mathbb {R}^{2}$ such that the gradient $\nabla u$ is never $0$, and such that, at each point, $\nabla v$ and $\nabla u$ are linearly dependent vectors. Given $p_0 = (x_0,y_0) \in \mathbb {R}^{2}$, show that there is a $C^1$ function $F$ of one variable such that $v(x,y) = F(u(x,y))$ in some neighborhood of $p_0$.

Let $f$ be a continuously differentiable function from $\mathbb {R}^{2}$ into $\mathbb {R}$. Prove that there is a continuous one-to-one function $g$ from $[0,1]$ into $\mathbb {R}^{2}$ such that the composite function $f \circ g$ is constant.

Let $f : \mathbb {R} \rightarrow \mathbb {R}$ be $C^1$ and let

If $f'(x_0) \neq 0$, show that this transformation is locally invertible near $(x_0,y_0)$ and the inverse has the form

Let $M_{n \times n}$ denote the vector space of real $n \times n$ matrices. Define a map $f : M_{n \times n} \rightarrow M_{n \times n}$ by $f(X) = X^2$. Find the derivative of $f$.

Let $M_{2 \times 2}$ be the four-dimensional vector space of all $2 \times 2$ real matrices and define $f : M_{2 \times 2} \rightarrow M_{2 \times 2}$ by $f(X) = X^2$.

Show that there is an $\varepsilon > 0$ such that if $A$ is any real $2 \times 2$ matrix satisfying $\left|a_{ij}\right| \leq \varepsilon$ for all entries $a_{ij}$ of $A$, then there is a real $2 \times 2$ matrix $X$ such that $X^2 + X^t = A$, where $X^t$ is the transpose of $X$. Is $X$ unique?

Let $M_{2 \times 2}$ be the space of $2 \times 2$ matrices over $\mathbb {R}$, identified in the usual way with $\mathbb {R}^{4}$. Let the function $F$ from $M_{2 \times 2}$ into $M_{2 \times 2}$ be defined by

Prove that the range of $F$ contains a neighborhood of the origin.

Let $M_{n \times n}$ denote the vector space of $n \times n$ real matrices. Prove that there are neighborhoods $U$ and $V$ in $M_{n \times n}$ of the identity matrix such that for every $A$ in $U$, there is a unique $X$ in $V$ such that $X^4 = A$.

Let $M_{n \times n}$ denote the vector space of $n \times n$ real matrices for $n \geq 2$. Let $\det : M_{n \times n} \rightarrow \mathbb {R}$ be the determinant map.

Let $F(t) = (f_{ij}(t))$ be an $n \times n$ matrix of continuously differentiable functions $f_{ij} : \mathbb {R} \rightarrow \mathbb {R}$, and let

Show that $u$ is differentiable and

Let $A = (a_{ij})$ be an $n \times n$ matrix whose entries $a_{ij}$ are real valued differentiable functions defined on $\mathbb {R}$. Assume that the determinant $\det (A)$ of $A$ is everywhere positive. Let $B = (b_{ij})$ be the inverse matrix of $A$. Prove the formula

What is the volume enclosed by the ellipsoid

Evaluate

where $\mathcal{A} = \left\{ (x,y) \in \mathbb {R}^{2} \mid x^2+y^2 \leq 1\right\}$.

Given the fact that $\displaystyle \int_{-\infty }^{\infty } e^{-x^2} \, dx = \sqrt{\pi }$, evaluate the integral

Evaluate

where

Let $\varphi (x,y)$ be a function with continuous second order partial derivatives such that

Let $S = \left\{ (x,y,z) \in \mathbb {R}^{3} \mid x^2+y^2+z^2=1\right\}$ denote the unit sphere in $\mathbb {R}^{3}$. Evaluate the surface integral over $S$:

Let $\overrightarrow {i}$, $\overrightarrow {j}$, and $\overrightarrow {k}$ be the usual unit vectors in $\mathbb {R}^{3}$. Let $\overrightarrow {F}$ denote the vector field

Let the vector field $F$ in $\mathbb {R}^{3}$ have the form

where $g$ is a real valued smooth function on $(0,\infty )$ and $\left\| \cdot \right\|$ denotes the Euclidean norm. ($F$ is undefined at $(0,0,0)$.) Prove that

for any smooth closed path $C$ in $\mathbb {R}^{3}$ that does not pass through the origin.

Let $\mathcal{B}$ denote the unit ball of $\mathbb {R}^{3}$, $\mathcal{B} = \left\{ r \in \mathbb {R}^{3} \mid \left\| r\right\| \leq 1\right\}$. Let $\overrightarrow {J} = (J_1, J_2, J_3)$ be a smooth vector field on $\mathbb {R}^{3}$ that vanishes outside of $\mathcal{B}$ and satisfies $\nabla \cdot \overrightarrow {J} = 0$.

Let $\mathcal{D}$ denote the open unit disc in $\mathbb {R}^{2}$. Let $u$ be an eigenfunction for the Laplacian in $\mathcal{D}$; that is, a real valued function of class $C^2$ defined in $\overline{\mathcal{D}}$, zero on the boundary of $\mathcal{D}$ but not identically zero, and satisfying the differential equation

where $\lambda$ is a constant. Prove that

and hence that $\lambda < 0$.

Let $\lambda , a \in \mathbb {R}$, with $a < 0$. Let $u(x,y)$ be an infinitely differentiable function defined on an open neighborhood of closed unit disc $\mathcal{D}$ such that