BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Solutions

Chapter
Section
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Assume that all the given functions have continuous second-order partial derivatives.

49. Show that any function of the form

z = f(x + at) + g(xat)

is a solution of the wave equation

2 z t 2 = a 2 2 z x 2

[Hint: Let u = x + at, v = xat.]

To determine

To show: Any function is of the form z=f(x+at)+g(xat) is satisfies the wave equation 2zt2=a22zx2 .

Explanation

Proof:

The given function is, z=f(x+at)+g(xat) .

Let u=x+atandv=xat .

Thus, the function can be expressed as, z=f(u)+g(v) .

The equation, 2zt2=a22zx2 is to be proved.

The partial derivative, zx is computed as follows,

zx=fuux+gvvx=f'(u)x(x+at)+g'(v)v(xat)=f'(u)(1)+g'(v)(1)=f'(u)+g'(v)

Thus, the partial derivative, zx=f'(u)+g'(v) (1)

Take the partial derivative with respect to x in the equation (1),

2zx2=x[f'(u)+g'(v)]=f''(u).ux+g''(v).vx=f''(u).(1)+g''(v).(1)=f''(u)+g''(v)

Thus, 2zx2=f''(u)+g''(v)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 14 Solutions

Show all chapter solutions add
Sect-14.1 P-11ESect-14.1 P-12ESect-14.1 P-13ESect-14.1 P-14ESect-14.1 P-15ESect-14.1 P-16ESect-14.1 P-17ESect-14.1 P-18ESect-14.1 P-19ESect-14.1 P-20ESect-14.1 P-21ESect-14.1 P-22ESect-14.1 P-23ESect-14.1 P-24ESect-14.1 P-25ESect-14.1 P-26ESect-14.1 P-27ESect-14.1 P-28ESect-14.1 P-29ESect-14.1 P-30ESect-14.1 P-31ESect-14.1 P-32ESect-14.1 P-33ESect-14.1 P-34ESect-14.1 P-35ESect-14.1 P-36ESect-14.1 P-37ESect-14.1 P-38ESect-14.1 P-39ESect-14.1 P-40ESect-14.1 P-41ESect-14.1 P-42ESect-14.1 P-43ESect-14.1 P-44ESect-14.1 P-45ESect-14.1 P-46ESect-14.1 P-47ESect-14.1 P-48ESect-14.1 P-49ESect-14.1 P-50ESect-14.1 P-51ESect-14.1 P-52ESect-14.1 P-53ESect-14.1 P-54ESect-14.1 P-55ESect-14.1 P-56ESect-14.1 P-57ESect-14.1 P-58ESect-14.1 P-59ESect-14.1 P-60ESect-14.1 P-61ESect-14.1 P-62ESect-14.1 P-63ESect-14.1 P-64ESect-14.1 P-65ESect-14.1 P-66ESect-14.1 P-67ESect-14.1 P-68ESect-14.1 P-69ESect-14.1 P-70ESect-14.1 P-71ESect-14.1 P-72ESect-14.1 P-73ESect-14.1 P-74ESect-14.1 P-75ESect-14.1 P-76ESect-14.1 P-77ESect-14.1 P-78ESect-14.1 P-79ESect-14.1 P-80ESect-14.1 P-81ESect-14.2 P-1ESect-14.2 P-2ESect-14.2 P-3ESect-14.2 P-4ESect-14.2 P-5ESect-14.2 P-6ESect-14.2 P-7ESect-14.2 P-8ESect-14.2 P-9ESect-14.2 P-10ESect-14.2 P-11ESect-14.2 P-12ESect-14.2 P-13ESect-14.2 P-14ESect-14.2 P-15ESect-14.2 P-16ESect-14.2 P-17ESect-14.2 P-18ESect-14.2 P-19ESect-14.2 P-20ESect-14.2 P-21ESect-14.2 P-22ESect-14.2 P-23ESect-14.2 P-24ESect-14.2 P-25ESect-14.2 P-26ESect-14.2 P-27ESect-14.2 P-28ESect-14.2 P-29ESect-14.2 P-30ESect-14.2 P-31ESect-14.2 P-32ESect-14.2 P-33ESect-14.2 P-34ESect-14.2 P-35ESect-14.2 P-36ESect-14.2 P-37ESect-14.2 P-38ESect-14.2 P-39ESect-14.2 P-40ESect-14.2 P-41ESect-14.2 P-42ESect-14.2 P-43ESect-14.2 P-44ESect-14.2 P-45ESect-14.2 P-46ESect-14.3 P-1ESect-14.3 P-2ESect-14.3 P-3ESect-14.3 P-4ESect-14.3 P-5ESect-14.3 P-6ESect-14.3 P-7ESect-14.3 P-8ESect-14.3 P-9ESect-14.3 P-10ESect-14.3 P-11ESect-14.3 P-12ESect-14.3 P-13ESect-14.3 P-14ESect-14.3 P-15ESect-14.3 P-16ESect-14.3 P-17ESect-14.3 P-18ESect-14.3 P-19ESect-14.3 P-20ESect-14.3 P-21ESect-14.3 P-22ESect-14.3 P-23ESect-14.3 P-24ESect-14.3 P-25ESect-14.3 P-26ESect-14.3 P-27ESect-14.3 P-28ESect-14.3 P-29ESect-14.3 P-30ESect-14.3 P-31ESect-14.3 P-32ESect-14.3 P-33ESect-14.3 P-34ESect-14.3 P-35ESect-14.3 P-36ESect-14.3 P-37ESect-14.3 P-38ESect-14.3 P-39ESect-14.3 P-40ESect-14.3 P-41ESect-14.3 P-42ESect-14.3 P-43ESect-14.3 P-44ESect-14.3 P-45ESect-14.3 P-46ESect-14.3 P-47ESect-14.3 P-48ESect-14.3 P-49ESect-14.3 P-50ESect-14.3 P-51ESect-14.3 P-52ESect-14.3 P-53ESect-14.3 P-54ESect-14.3 P-55ESect-14.3 P-56ESect-14.3 P-57ESect-14.3 P-58ESect-14.3 P-59ESect-14.3 P-60ESect-14.3 P-61ESect-14.3 P-62ESect-14.3 P-63ESect-14.3 P-64ESect-14.3 P-65ESect-14.3 P-66ESect-14.3 P-67ESect-14.3 P-68ESect-14.3 P-69ESect-14.3 P-70ESect-14.3 P-71ESect-14.3 P-72ESect-14.3 P-73ESect-14.3 P-74ESect-14.3 P-75ESect-14.3 P-76ESect-14.3 P-77ESect-14.3 P-78ESect-14.3 P-79ESect-14.3 P-80ESect-14.3 P-81ESect-14.3 P-82ESect-14.3 P-83ESect-14.3 P-84ESect-14.3 P-85ESect-14.3 P-86ESect-14.3 P-87ESect-14.3 P-88ESect-14.3 P-89ESect-14.3 P-90ESect-14.3 P-91ESect-14.3 P-92ESect-14.3 P-93ESect-14.3 P-94ESect-14.3 P-95ESect-14.3 P-96ESect-14.3 P-97ESect-14.3 P-98ESect-14.3 P-99ESect-14.3 P-100ESect-14.3 P-101ESect-14.3 P-102ESect-14.3 P-103ESect-14.3 P-104ESect-14.4 P-1ESect-14.4 P-2ESect-14.4 P-3ESect-14.4 P-4ESect-14.4 P-5ESect-14.4 P-6ESect-14.4 P-7ESect-14.4 P-8ESect-14.4 P-11ESect-14.4 P-12ESect-14.4 P-13ESect-14.4 P-14ESect-14.4 P-15ESect-14.4 P-16ESect-14.4 P-17ESect-14.4 P-18ESect-14.4 P-19ESect-14.4 P-20ESect-14.4 P-21ESect-14.4 P-22ESect-14.4 P-23ESect-14.4 P-24ESect-14.4 P-25ESect-14.4 P-26ESect-14.4 P-27ESect-14.4 P-28ESect-14.4 P-29ESect-14.4 P-30ESect-14.4 P-31ESect-14.4 P-32ESect-14.4 P-33ESect-14.4 P-34ESect-14.4 P-35ESect-14.4 P-36ESect-14.4 P-37ESect-14.4 P-38ESect-14.4 P-39ESect-14.4 P-40ESect-14.4 P-41ESect-14.4 P-42ESect-14.4 P-43ESect-14.4 P-44ESect-14.4 P-45ESect-14.4 P-46ESect-14.5 P-1ESect-14.5 P-2ESect-14.5 P-3ESect-14.5 P-4ESect-14.5 P-5ESect-14.5 P-6ESect-14.5 P-7ESect-14.5 P-8ESect-14.5 P-9ESect-14.5 P-10ESect-14.5 P-11ESect-14.5 P-12ESect-14.5 P-13ESect-14.5 P-14ESect-14.5 P-15ESect-14.5 P-16ESect-14.5 P-17ESect-14.5 P-18ESect-14.5 P-19ESect-14.5 P-20ESect-14.5 P-21ESect-14.5 P-22ESect-14.5 P-23ESect-14.5 P-24ESect-14.5 P-25ESect-14.5 P-26ESect-14.5 P-27ESect-14.5 P-28ESect-14.5 P-29ESect-14.5 P-30ESect-14.5 P-31ESect-14.5 P-32ESect-14.5 P-33ESect-14.5 P-34ESect-14.5 P-35ESect-14.5 P-36ESect-14.5 P-37ESect-14.5 P-38ESect-14.5 P-39ESect-14.5 P-40ESect-14.5 P-41ESect-14.5 P-42ESect-14.5 P-43ESect-14.5 P-44ESect-14.5 P-45ESect-14.5 P-46ESect-14.5 P-47ESect-14.5 P-48ESect-14.5 P-49ESect-14.5 P-50ESect-14.5 P-51ESect-14.5 P-53ESect-14.5 P-55ESect-14.5 P-56ESect-14.5 P-57ESect-14.5 P-58ESect-14.5 P-59ESect-14.6 P-1ESect-14.6 P-2ESect-14.6 P-3ESect-14.6 P-4ESect-14.6 P-5ESect-14.6 P-6ESect-14.6 P-7ESect-14.6 P-8ESect-14.6 P-9ESect-14.6 P-10ESect-14.6 P-11ESect-14.6 P-12ESect-14.6 P-13ESect-14.6 P-14ESect-14.6 P-15ESect-14.6 P-16ESect-14.6 P-17ESect-14.6 P-18ESect-14.6 P-19ESect-14.6 P-20ESect-14.6 P-21ESect-14.6 P-22ESect-14.6 P-23ESect-14.6 P-24ESect-14.6 P-25ESect-14.6 P-26ESect-14.6 P-27ESect-14.6 P-28ESect-14.6 P-29ESect-14.6 P-30ESect-14.6 P-31ESect-14.6 P-32ESect-14.6 P-33ESect-14.6 P-34ESect-14.6 P-35ESect-14.6 P-36ESect-14.6 P-37ESect-14.6 P-38ESect-14.6 P-39ESect-14.6 P-40ESect-14.6 P-41ESect-14.6 P-42ESect-14.6 P-43ESect-14.6 P-44ESect-14.6 P-45ESect-14.6 P-46ESect-14.6 P-47ESect-14.6 P-48ESect-14.6 P-49ESect-14.6 P-50ESect-14.6 P-51ESect-14.6 P-52ESect-14.6 P-53ESect-14.6 P-54ESect-14.6 P-55ESect-14.6 P-56ESect-14.6 P-57ESect-14.6 P-58ESect-14.6 P-59ESect-14.6 P-60ESect-14.6 P-61ESect-14.6 P-62ESect-14.6 P-63ESect-14.6 P-64ESect-14.6 P-65ESect-14.6 P-66ESect-14.6 P-67ESect-14.6 P-68ESect-14.6 P-69ESect-14.6 P-70ESect-14.7 P-1ESect-14.7 P-2ESect-14.7 P-3ESect-14.7 P-4ESect-14.7 P-5ESect-14.7 P-6ESect-14.7 P-7ESect-14.7 P-8ESect-14.7 P-9ESect-14.7 P-10ESect-14.7 P-11ESect-14.7 P-12ESect-14.7 P-13ESect-14.7 P-14ESect-14.7 P-15ESect-14.7 P-16ESect-14.7 P-17ESect-14.7 P-18ESect-14.7 P-19ESect-14.7 P-20ESect-14.7 P-21ESect-14.7 P-22ESect-14.7 P-23ESect-14.7 P-24ESect-14.7 P-25ESect-14.7 P-26ESect-14.7 P-27ESect-14.7 P-28ESect-14.7 P-29ESect-14.7 P-30ESect-14.7 P-31ESect-14.7 P-32ESect-14.7 P-33ESect-14.7 P-34ESect-14.7 P-35ESect-14.7 P-36ESect-14.7 P-37ESect-14.7 P-38ESect-14.7 P-39ESect-14.7 P-40ESect-14.7 P-41ESect-14.7 P-42ESect-14.7 P-43ESect-14.7 P-44ESect-14.7 P-45ESect-14.7 P-46ESect-14.7 P-47ESect-14.7 P-48ESect-14.7 P-49ESect-14.7 P-50ESect-14.7 P-51ESect-14.7 P-52ESect-14.7 P-53ESect-14.7 P-54ESect-14.7 P-55ESect-14.7 P-56ESect-14.7 P-57ESect-14.7 P-58ESect-14.7 P-59ESect-14.7 P-60ESect-14.8 P-1ESect-14.8 P-2ESect-14.8 P-3ESect-14.8 P-4ESect-14.8 P-5ESect-14.8 P-6ESect-14.8 P-7ESect-14.8 P-8ESect-14.8 P-9ESect-14.8 P-10ESect-14.8 P-11ESect-14.8 P-12ESect-14.8 P-13ESect-14.8 P-14ESect-14.8 P-15ESect-14.8 P-16ESect-14.8 P-17ESect-14.8 P-18ESect-14.8 P-19ESect-14.8 P-20ESect-14.8 P-21ESect-14.8 P-22ESect-14.8 P-23ESect-14.8 P-24ESect-14.8 P-25ESect-14.8 P-27ESect-14.8 P-28ESect-14.8 P-29ESect-14.8 P-30ESect-14.8 P-31ESect-14.8 P-32ESect-14.8 P-33ESect-14.8 P-34ESect-14.8 P-35ESect-14.8 P-36ESect-14.8 P-37ESect-14.8 P-38ESect-14.8 P-39ESect-14.8 P-40ESect-14.8 P-41ESect-14.8 P-42ESect-14.8 P-43ESect-14.8 P-44ESect-14.8 P-45ESect-14.8 P-46ESect-14.8 P-49ESect-14.8 P-50ECh-14 P-1RCCCh-14 P-2RCCCh-14 P-3RCCCh-14 P-4RCCCh-14 P-5RCCCh-14 P-6RCCCh-14 P-7RCCCh-14 P-8RCCCh-14 P-9RCCCh-14 P-10RCCCh-14 P-11RCCCh-14 P-12RCCCh-14 P-13RCCCh-14 P-14RCCCh-14 P-15RCCCh-14 P-16RCCCh-14 P-17RCCCh-14 P-18RCCCh-14 P-19RCCCh-14 P-1RQCh-14 P-2RQCh-14 P-3RQCh-14 P-4RQCh-14 P-5RQCh-14 P-6RQCh-14 P-7RQCh-14 P-8RQCh-14 P-9RQCh-14 P-10RQCh-14 P-11RQCh-14 P-12RQCh-14 P-1RECh-14 P-2RECh-14 P-3RECh-14 P-4RECh-14 P-5RECh-14 P-6RECh-14 P-7RECh-14 P-8RECh-14 P-9RECh-14 P-10RECh-14 P-11RECh-14 P-12RECh-14 P-13RECh-14 P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

Prove the identity. 10. cosh x sinh r = ex

Single Variable Calculus: Early Transcendentals, Volume I

Find the inverse function of f(x)x2+2forx0.

Calculus: An Applied Approach (MindTap Course List)

In Exercises 2340, find the indicated limit. 37. limx32x4+x2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Evaluate the limit. limxe2xe2xln(x+1)

Single Variable Calculus: Early Transcendentals

A definite integral for the area of the region bounded by y = 2 − x2 and y = x2 is:

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The graph of is: a) b) c) d)

Study Guide for Stewart's Multivariable Calculus, 8th