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# The divergence theorem is a remarkable result, relating

## Problem 13.33 Chapter 13

Classical Mechanics | 0th Edition

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Classical Mechanics | 0th Edition

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Problem 13.33

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.)

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