The divergence theorem is a remarkable result, relating the surface integral that gives the flow of v out of a closed surface S to the volume integral of V v. Occasionally it is easy to evaluate both of these integrals and one can check the validity of the theorem. More often, one of the integrals is much easier to evaluate than the other, and the divergence theorem then gives one a slick way to evaluate a hard integral. The following exercises illustrate both of these situations. (a) Let v = kr, where k is a constant and let S be a sphere of radius R centered on the origin. Evaluate the left side of the divergence theorem (13.56) (the surface integral). Next calculate V v and use this to evaluate the right side of (13.56) (the volume integral). Show that the two agree. (b) Now use the same velocity v, but let S be a sphere not centered on the origin. Explain why the surface integral is now hard to evaluate directly, but don't actually do it. Instead, find its value by doing the volume integral. (This second route should be no harder than before.)

# The divergence theorem is a remarkable result, relating

## Problem 13.33 Chapter 13

Classical Mechanics | 0th Edition

- 2901 Step-by-step solutions solved by professors and subject experts
- Get 24/7 help from StudySoup virtual teaching assistants

Classical Mechanics | 0th Edition

Get Full Solutions
15

3

Problem 13.33

Step-by-Step Solution:

Step 1 of 3

Step 2 of 3

Step 3 of 3

#### People also purchased

#### Related chapters

×

Log in to StudySoup

Get Full Access to
Classical Mechanics - 0 Edition - Chapter 13 - Problem 13.33

Join StudySoup for FREE

Get Full Access to
Classical Mechanics - 0 Edition - Chapter 13 - Problem 13.33

Already have an account?
Login here

Forgot password?
Reset your password here

I don't want to reset my password

Need help? Contact support

We're here to help

Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or support@studysoup.com

Password Reset Request Sent
An email has been sent to the email address associated to your account.
Follow the link in the email to reset your password.
If you're having trouble finding our email please check your spam folder

Incorrect Password
The password used to log in with this account is incorrect

Forgot password? Reset it here