The Calculus with Analytic Geometry

Louis Leithold

Chapter 19 Differential Calculus Of Functions Of Severai Variables - all with Video Answers

Educators

Section 7

Higher-Order Partial Derivatives

04:37

Problem 1

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=\frac{x^{2}}{y}-\frac{y}{x^{2}}
$$

Bahar Tehranipoor

Numerade Educator

04:37

Problem 2

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=2 x^{3}-3 x^{2} y+x y^{2}
$$

Bahar Tehranipoor

Numerade Educator

02:28

Problem 3

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=e^{2 x} \sin y
$$

Ian Grigsby

Numerade Educator

01:09

Problem 4

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=e^{-x / y}+\ln \frac{y}{x}
$$

Matt Just

Numerade Educator

01:08

Problem 5

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=\left(x^{2}+y^{2}\right) \tan ^{-1} \frac{y}{x}
$$

Ad F

Numerade Educator

00:59

Problem 6

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=\sin ^{-1} \frac{3 y}{x^{2}}
$$

Ad F

Numerade Educator

04:37

Problem 7

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=4 x \sinh y+3 y \cosh x
$$

Bahar Tehranipoor

Numerade Educator

02:28

Problem 8

In Exercises 1 through 8 , do each of the following: (a) Find $D_{11} f(x, y)$; (b) find $D_{22} f(x, y) ;$ (c) show that $D_{12} f(x, y)=D_{21} f(x, y) .$
$$
f(x, y)=x \cos y-y e^{x}
$$

Ian Grigsby

Numerade Educator

02:45

Problem 9

In Exercises 9 through 14, find the indicated partial derivatives.
$$
f(x, y, z)=y e^{x}+z e^{y}+e^{z} ;(\mathrm{a}) f
$$
$$
\text { (a) } f_{x z}(x, y, z) \text {; }
$$
(b) $f_{y z}(x, y, z)$

Abhijith V

Numerade Educator

01:44

Problem 10

In Exercises 9 through 14, find the indicated partial derivatives.
$$
g(x, y, z)=\sin (x y z) ;\left(\text { a) } g_{23}(x, y, z) ;\left(\text { b) } g_{12}(x, y, z)\right.\right.
$$

Lucas Finney

Numerade Educator

01:14

Problem 11

In Exercises 9 through 14, find the indicated partial derivatives.
$$
f(r, s)=2 r^{3} s+r^{2} s^{2}-5 r s^{3} ; \text { (a) } f_{121}(r, s)
$$
(b) $f_{221}(r, s)$

Abhijith V

Numerade Educator

01:25

Problem 12

In Exercises 9 through 14, find the indicated partial derivatives.
$$
f(u, v)=\ln \cos (u-v) ;\left(\text { a) } f_{\text {uuv }}(u, v) ;\left(\text { b) } f_{\text {vuv }}(u, v)\right.\right.
$$

Linda Hand

Numerade Educator

01:14

Problem 13

In Exercises 9 through 14, find the indicated partial derivatives.
$$
g(r, s, t)=\ln \left(r^{2}+4 s^{2}-5 t^{2}\right) ; \text { (a) } g_{132}(r, s, t)
$$
(b) $g_{122}(r, s, t)$

Abhijith V

Numerade Educator

01:44

Problem 14

In Exercises 9 through 14, find the indicated partial derivatives.
$$
f(x, y, z)=\tan ^{-1}(3 x y z) \text {; (a) } f_{113}(x, y, z) ; \text { (b) } f_{123}(x, y, z)
$$

Lucas Finney

Numerade Educator

02:20

Problem 15

In Exercises 15 through 18, show that $u(x, y)$ satisfies the equation
$$
\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0
$$
which is known as Laplace's equation in $R^{2}$.
$$
u(x, y)=\ln \left(x^{2}+y^{2}\right)
$$

Vikash Ranjan

Numerade Educator

01:14

Problem 16

Tyler Moulton

Numerade Educator

04:37

Problem 17

Sriram Soundarrajan

Numerade Educator

04:37

Problem 18

Sriram Soundarrajan

Numerade Educator

02:20

Problem 19

Laplace's equation in $R^{3}$ is
$$
\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}=0
$$
Show that $u(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2}$ satisfies this equation.

Vikash Ranjan

Numerade Educator

01:32

Problem 20

For the function of Example 4, show that $f_{12}$ is discontinuous at $(0,0)$ and hence that the hypothesis of Theorem 19.7.1 is not satisfied if $\left(x_{0}, y_{0}\right)=(0,0)$.

Helen Latting

Numerade Educator

02:37

Problem 21

In Exercises 21 through 23, find $f_{12}(0,0)$ and $f_{21}(0,0)$, if they exist.
$$
f(x, y)= \begin{cases}\frac{2 x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$

William Semus

Numerade Educator

02:08

Problem 22

In Exercises 21 through 23, find $f_{12}(0,0)$ and $f_{21}(0,0)$, if they exist.
$$
f(x, y)= \begin{cases}\frac{x^{2} y^{2}}{x^{4}+y^{4}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$

Brittany Knowlton

Numerade Educator

04:58

Problem 23

In Exercises 21 through 23, find $f_{12}(0,0)$ and $f_{21}(0,0)$, if they exist.
$$
f(x, y)= \begin{cases}x^{2} \tan ^{-1} \frac{y}{x}-y^{2} \tan ^{-1} \frac{x}{y} & \text { if } x \neq 0 \text { and } y \neq 0 \\ 0 & \text { if either } x=0 \text { or } y=0\end{cases}
$$

Matt Bremer

Numerade Educator

01:37

Problem 24

Given that $u=f(x, y), x=F(t)$, and $y=G(t)$, and assuming that $f_{x y}=f_{v x}$, prove by using the chain rule that
$$
\frac{d^{2} u}{d t^{2}}=f_{x x}(x, y)\left[F^{\prime}(t)\right]^{2}+2 f_{x y}(x, y) F^{\prime}(t) G^{\prime}(t)+f_{v y}(x, y)\left[G^{\prime}(t)\right]^{2}+f_{x}(x, y) F^{\prime \prime}(t)+f_{y}(x, y) G^{\prime \prime}(t)
$$

Angela Guo

Numerade Educator

04:48

Problem 25

Given that $u=f(x, y), x=F(r, s)$, and $y=G(r, s)$, and assuming that $f_{x y}=f_{y x}$, prove by using the chain rule that
$$
\begin{aligned}
&\frac{\partial^{2} u}{\partial r \partial s}=f_{x x}(x, y) F_{r}(r, s) F_{s}(r, s)+f_{x y}(x, y)\left[F_{s}(r, s) G_{r}(r, s)+F_{r}(r, s) G_{s}(r, s)\right] \\
&+f_{y y}(x, y) G_{r}(r, s) G_{s}(r, s)+f_{x}(x, y) F_{s r}(r, s)+f_{y}(x, y) G_{s r}(r, s)
\end{aligned}
$$

Charles Machakwa

Numerade Educator

03:24

Problem 26

Given $u=e^{y} \cos x, x=2 t, y=t^{2}$. Find $d^{2} u / d t^{2}$ in three ways: (a) by first expressing $u$ in terms of $t ;(\mathrm{b})$ by using the formula of Exercise $24 ;$ (c) by using the chain rule.

Lucas Finney

Numerade Educator

01:47

Problem 27

Given $u=3 x y-4 y^{2}, x=2 s e^{r}, y=r e^{-s}$. Find $\partial^{2} u / \partial r^{2}$ in three ways: (a) by first expressing $u$ in terms of $r$ and $s ;$ (b) by using the formula of Example 5 ; (c) by using the chain rule.

James Macpherson

Numerade Educator

03:43

Problem 28

For $u, x$, and $y$ as given in Exercise 27 , find $\partial^{2} u / \partial s \partial r$ in three ways: (a) by first expressing $u$ in terms of $r$ and $s ;(\mathrm{b})$ by using the formula of Exercise $25 ;$ (c) by using the chain rule.

Harshita Goel

Numerade Educator

05:38

Problem 29

Given $u=9 x^{2}+4 y^{2}, x=r \cos \theta, y=r \sin \theta$. Find $\partial^{2} u / \partial r^{2}$ in three ways: (a) by first expressing $u$ in terms of $r$ and $\theta$; (b) by using the formula of Example 5; (c) by using the chain rule.

Adrian Co

Numerade Educator

03:24

Problem 30

For $u, x$, and $y$ as given in Exercise 29, find $\partial^{2} u / \partial \theta^{2}$ in three ways: (a) by first expressing $u$ in terms of $r$ and $\theta ;$ (b) by using the formula of Example 5; (c) by using the chain rule.

Lucas Finney

Numerade Educator

03:24

Problem 31

For $u, x$, and $y$ as given in Exercise 29, find $\partial^{2} u / \partial r \partial \theta$ in three ways: (a) by first expressing $u$ in terms of $r$ and $\theta$; (b) by using the formula of Exercise 25; (c) by using the chain rule.

Lucas Finney

Numerade Educator

00:56

Problem 32

If $u=f(x, y)$ and $v=g(x, y)$, then the equations
$$
\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} \text { and } \quad \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}
$$
are called the Cauchy-Riemann equations. If $f$ and $g$ and their first and second partial derivatives are continuous, prove that if $u$ and $v$ satisfy the Cauchy-Riemann equations, they also satisfy Laplace's equation (see Exercises 15 through 18 ).

Nick Johnson

Numerade Educator

01:29

Problem 33

The one-dimensional heat-conduction partial differential equation is
$$
\frac{\partial u}{\partial t}=k^{2} \frac{\partial^{2} u}{\partial x^{2}}
$$
Show that if $f$ is a function of $x$ satisfying the equation
$$
\frac{d^{2} f}{d x^{2}}+\lambda^{2} f(x)=0
$$
and $g$ is a function of $t$ satisfying the equation $d g / d t+k^{2} \lambda^{2} g(t)=0$, then if $u=f(x) g(t)$, the partial differential equation is satisfied. $k$ and $\lambda$ are constants.

James Kiss

Numerade Educator

03:11

Problem 34

The partial differential equation for a vibrating string is
$$
\frac{\partial^{2} u}{\partial t^{2}}=a^{2} \frac{\partial^{2} u}{\partial x^{2}}
$$
Show that if $f$ is a function of $x$ satisfying the equation $d^{2} f / d x^{2}+\lambda^{2} f(x)=0$ and $g$ is a function of $t$ satisfying the equation $d^{2} g / d t^{2}+a^{2} \lambda^{2} g(t)=0$, then if $u=f(x) g(t)$, the partial differential equation is satisfied. $a$ and $\lambda$ are constants.

M Hassan Anwar

Numerade Educator

02:14

Problem 35

Prove that if $f$ and $g$ are two arbitrary functions of a real variable having continuous second derivatives and $u=f(x+a t)$ $+g(x-a t)$, then $u$ satisfies the partial differential equation of the vibrating string given in Exercise 34 . (HINT: Let $v=x+a t$ and $w=x-a t ;$ then $u$ is a function of $v$ and $w$, and $v$ and $w$ are in turn functions of $x$ and $t$.)

M Hassan Anwar

Numerade Educator

02:47

Problem 36

Prove that if $f$ is a function of two variables and all the partial derivatives of $f$ up to the fourth order are continuous on some open disk, then
$$
D_{1122} f=D_{2121} f
$$

Jacquelyn Trost

Numerade Educator