Mathematical Tools for Physics

9—Vector Calculus 1 275

9.8 What is the area of the spherical cap on the surface of a sphere of radiusR: 0 ≤θ≤θ 0?
(b) Does the result have the correct behavior for both small and largeθ 0?

(c) What iare the surface integrals over this cap of the vector field~v=ˆrv 0 cosθsin^2 φ? Consider both

∫

~v.dA~

and

∫

~v×dA~.

9.9 A rectangular area is specified parallel to thex-yplane atz=dand 0 < x < a,a < y < b. A vector field
is~v=

(

xAxyzˆ +ˆyByx^2 +ˆzCx^2 yz^2

)

Evaluate the two integrals over this surface

∫

~v.dA,~ and

∫

dA~×~v

9.10 For the vector field~v=Arn~r, compute the integral over the surface of a sphere of radiusRcentered at
the origin:

∮

~v.dA~.
Compute the integral over the volume of this same sphere

∫

d^3 r∇.~v.

9.11 The velocity of a point in a rotating rigid body is~v=ω×~r. See problem7.5. Compute its divergence and
curl. Do this in rectangular, cylindrical, and spherical coordinates.

9.12 Fill in the missing steps in the calculation of Eq. ( 25 ).

9.13 Paralleling the calculation in section9.6for the divergence in cylindrical coordinates, compute the curl in
cylindrical coordinates,∇×~v. Ans: Eq. ( 28 ).

9.14 Another way to get to Eq. ( 35 ) is to work with Eq. ( 34 ) directly and to write the functionρ(r)explicitly
as two cases:r < Randr > R. Multiply Eq. ( 34 ) byr^2 and integrate it from zero tor, being careful to handle
the integral differently when the upper limit is< Rand when it is> R.

r^2 gr(r) =− 4 πG

∫r

0

dr′r′^2 ρ(r′)

Note: This is not simply reproducing that calculation that I’ve already done. This is doing it a different way.