Vector Calculus

Jerrold E. Marsden, Anthony Tromba

Chapter 3

Higher-Order Derivatives; Maxima and Minima - all with Video Answers

Educators

Section 1

Iterated Partial Derivatives

Problem 1

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$f(x, y)=2 x y /\left(x^{2}+y^{2}\right)^{2},$ on the region where
$(x, y) \neq(0,0)$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$f(x, y, z)=e^{z}+(1 / x)+x e^{-y},$ on the region where $x \neq 0$

Lucas Finney

Numerade Educator

Problem 3

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$$f(x, y)=\cos \left(x y^{2}\right)$$

Lucas Finney

Numerade Educator

Problem 4

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$$f(x, y)=e^{-x y^{2}}+y^{3} x^{4}$$

Lucas Finney

Numerade Educator

Problem 5

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$$f(x, y)=1 /\left(\cos ^{2} x+e^{-y}\right)$$

Lucas Finney

Numerade Educator

Problem 6

Compute the second partial derivatives $a^{2} f / \partial x^{2}, a^{2} f / \partial x \partial y, \partial^{2} f / \partial y \partial x, \partial^{2} f / \partial y^{2}$ for each of the following functions. Verify Theorem I in each case.
$$f(x, y)=\log (x-y)$$

Lucas Finney

Numerade Educator

Problem 7

Find all second partial derivatives of the following functions at the point $\mathbf{x}_{0}$
(a) $f(x, y)=\sin (x y) ; x_{0}=(\pi, 1)$
(b) $f(x, y)=x y^{8}+x^{2}+y^{4} ; x_{0}=(2,-1)$
(c) $f(x, y, z)=e^{x y z} ; \mathbf{x}_{0}=(0,0,0)$

Lucas Finney

Numerade Educator

Problem 8

Find all second partial derivatives of $f(x, y)=\sec ^{3}(4 y-3 x)$.

Lucas Finney

Numerade Educator

Problem 9

Can there exist a $C^{2}$ function $f(x, y)$ with $f_{x}=2 x-5 y$ and $f_{y}=4 x+y ?$

Lucas Finney

Numerade Educator

Problem 10

The heat conduction equation is $u_{t}=k u_{x x} .$ Determine whether $u(x, t)=e^{-k i} \sin (x)$ is a solution.

Lucas Finney

Numerade Educator

Problem 11

Show that the following functions satisfy the one-dimensional wave equation
$\frac{\partial^{2} f}{\partial x^{2}}=\frac{1}{c^{2}} \frac{\partial^{2} f}{\partial t^{2}}$
(a) $f(x, t)=\sin (x-c t)$
(b) $f(x, t)=\sin (x) \sin (c t)$
(c) $f(x, t)=(x-c t)^{6}+(x+c t)^{6}$

Lucas Finney

Numerade Educator

Problem 12

(a) Show that $T(x, t)=e^{-k t} \cos x$ satisfies the onc-dimensional heat cquation $$ k \frac{\partial^{2} T}{\partial x^{2}}=\frac{\partial T}{\partial t} $$
(b) Show that $T(x, y, t)=e^{-k t}(\cos x+\cos y)$ satisfics the two-dimcnsional heat equation$$
k\left(\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}\right)=\frac{\partial T}{\partial t} $$
(c) Show that $T(x, y, z, t)=e^{-k t}(\cos x+$
$\cos y+\cos z)$ satisfics the throc-dimensional heat equation$$
k\left(\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}+\frac{\partial^{2} T}{\partial z^{2}}\right)=\frac{\partial T}{\partial t} $$

Nick Johnson

Numerade Educator

Problem 13

Find $a^{2} z / \partial x^{2}, \partial^{2} z / \partial x \partial y, \partial^{2} z / \partial y \partial x,$ and $\partial^{2} z / \partial y^{2}$ for
(a) $=-3 x^{2}+2 y^{2}$
(b) $z=\left(2 x^{2}+7 x^{2} y\right) / 3 x y,$ on the region where $x \neq 0$
and $y \neq 0$

John Nicolle

Numerade Educator

Problem 14

Find all the second partial derivatives of
(a) $z=\sin \left(x^{2}-3 x y\right)$
(b) $z=x^{2} y^{2} e^{2 x y}$

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 15

Find $f_{x y}, f_{y z}, f_{z x},$ and $f_{x y z}$ for $$ f(x, y, z)=x^{2} y+x y^{2}+y z^{2} $$

Lucas Finney

Numerade Educator

Problem 16

Let $z=x^{4} y^{3}-x^{8}+y^{4}$
(a) Compute $a^{3} z / \partial y \partial x \partial x, \partial^{3} z / \partial x \partial y \partial x,$ and
$\left.a^{3} z / \partial x \partial x \partial y \text { (also denoted } \partial^{3} z / \partial x^{2} \partial y\right)$
(b) Compute $a^{3}=/ 2 x$ ay $\partial y, \partial^{3} z / \partial y \partial x \partial y,$ and
$\left.a^{3} z / \partial y \text { ay } \partial x \text { (also denoted } \partial^{3} z / \partial y^{2} \partial x\right)$

Lucas Finney

Numerade Educator

Problem 17

Use Theorem 1 to show that if $f(x, y, z)$ is of class $C^{3}$ then $$ \frac{\partial^{3} f}{\partial x \partial y \partial z}=\frac{\partial^{3} f}{\partial y \partial z \partial x} $$

Lucas Finney

Numerade Educator

Problem 18

Verify that $$ \frac{\partial^{3} f}{\partial x \partial y \partial z}=\frac{\partial^{3} f}{\partial z \partial y \partial x} $$ for $f(x, y, z)=z e^{x y}+y z^{3} x^{2}$

Lucas Finney

Numerade Educator

Problem 19

Verify that $f_{x z w}=f_{\text {sur } x}$ for $f(x, y, z, w)=e^{x y z} \sin (x w)$

Lucas Finney

Numerade Educator

Problem 20

If $f(x, y, z, w)$ is of class $C^{3},$ show that $f_{x z w}=f_{\varepsilon w x}$

Lucas Finney

Numerade Educator

Problem 21

Evaluate all first and second partial derivatives of the following functions:
(a) $f(x, y)=x \arctan (x / y)$
(b) $f(x, y)=\cos \sqrt{x^{2}+y^{2}}$
(c) $f(x, y)=\operatorname{cxp}\left(-x^{2}-y^{2}\right)$

Lucas Finney

Numerade Educator

Problem 22

I.et $w=f(x, y)$ be a function of two variables and let $x=u+v, y=u-v .$ Show that $$ \frac{\partial^{2} w}{\partial u \partial v}=\frac{\partial^{2} w}{\partial x^{2}}-\frac{\partial^{2} w}{\partial y^{2}}$$

Aman Gupta

Numerade Educator

Problem 23

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a $C^{2}$ function and let $c(t)$ be a $C^{2}$ curve in $\mathbb{R}^{2}$. Write a formula for the second derivative $\left(d^{2} / d t^{2}\right)((f \circ \mathbf{c})(t))$ using the chain rule twice.

Dushyant Barot

Numerade Educator

Problem 24

Let $f(x, y, z)=e^{x z} \tan (y z)$ and let $x=g(s, t)$ $y=h(s, t), z=k(s, t),$ and define the function $m(s, t)=f(g(s, t), h(s, t), h(s, t)) .$ Find a formula for $m_{s t}$ using the chain rule and verify that your answer is symmctric in $s$ and $t$

Victoria Dollar

Victoria Dollar

Numerade Educator

Problem 25

A function $u=f(x, y)$ with continuous second partial derivatives satisfying I aplace's equation $$ \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$ is called a harmonic function. Show that the function $u(x, y)=x^{3}-3 x y^{2}$ is harmonic.

Wendi Zhao

Numerade Educator

Problem 26

Which of the following functions are harmonic? (See Exercise $25 .)$
(a) $f(x, y)=x^{2}-y^{2}$
(b) $f(x, y)=x^{2}+y^{2}$
(c) $f(x, y)=x y$
(d) $f(x, y)=y^{3}+3 x^{2} y$
(e) $f(x, y)=\sin x \cosh y$
(f) $f(x, y)=e^{x} \sin y$

Dilip Paruchuri

Dilip Paruchuri

Numerade Educator

Problem 27

(a) Is the function $f(x, y, z)=x^{2}-2 y^{2}+z^{2}$ harmonic? What about $f(x, y, z)=x^{2}+y^{2}-z^{2} ?$
(b) I aplace's equation for functions of $n$ variables is $$ \frac{\partial^{2} f}{\partial x_{1}^{2}}+\frac{\partial^{2} f}{\partial x_{2}^{2}}+\ldots+\frac{\partial^{2} f}{\partial x_{n}^{2}}=0 $$
Find an example of a function of $n$ variables that is harmonic, and show that your example is harmonic.

Wendi Zhao

Numerade Educator

Problem 28

Show that the following functions are harmonic:
(a) $f(x, y)=\arctan \frac{x}{x}$
(b) $f(x, y)=\log \left(x^{2}+y^{2}\right)$

Nick Johnson

Numerade Educator

Problem 29

Let $f$ and $g$ be $C^{2}$ functions of one variable. Sct $\phi=f(x-t)+g(x+t)$
(a) Prove that $\phi$ satisfics the wave cquation:
$a^{2} \phi / a t^{2}=a^{2} \phi / \partial x^{2}$
(b) Skctch the graph of $\phi$ against $t$ and $x$ if $f(x)=x^{2}$ and $g(x)=0$

Urvashi Arora

Numerade Educator

Problem 30

(a) Show that function $g(x, t)=2+e^{-t} \sin x$ satisfics the heat equation: $g_{t}=g_{x x} .$ [Ilere $g(x, t)$ represcats the tempcrature in a mctal rod at position $x \text { and time } t .]$
(b) Sketch the graph of $g$ for $t \geq 0 .$ (HiNT: Look at scetions by the plancs $t=0, t=1,$ and $t=2 .$
(c) What happens to $g(x, t)$ as $t \rightarrow \infty ?$ Interpret this limit in terms of the bchavior of heat in the rod.

Aman Gupta

Numerade Educator

Problem 31

Show that Newton's potential $V=-G m M / r$ satisfies Laplace's equation
$\frac{\partial^{2} V}{\partial x^{2}}+\frac{\partial^{2} V}{\partial y^{2}}+\frac{\partial^{2} V}{\partial z^{2}}=0 \quad$ for $\quad(x, y, z) \neq(0,0,0)$

Urvashi Arora

Numerade Educator

Problem 32

Let
$f(x, y)=\left\{\begin{array}{ll}x y\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right), & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{array}\right.$
(sce Figure $3.1 .4)$
(a) If $(x, y) \neq(0,0),$ calculate $\partial f / \partial x$ and $a f / \partial y$
(b) Show that $(a f / a x)(0,0)=0=(a f / \partial y)(0,0)$
(c) Show that $\left(\partial^{2} f / \partial x \partial y\right)(0,0)=1$
$\left(a^{2} f / a y \partial x\right)(0,0)=-1$
(d) What went wrong? Why are the mixed partials not cqual? (GRAPH CAN'T COPY)

Brittany Knowlton

Brittany Knowlton

Numerade Educator