Section 2.4 · Examples · 3 of 5
Example 2.4.3: Divergence theorem for a sphere
Divergence theorem for a sphere
#Verify the divergence theorem for the vector field and the sphere of radius centred at the origin shown in the figure.
The region is a ball and the field is radial, so spherical coordinates make both sides one-line integrals. The spherical divergence (2.3.54) must be used, not .
On the sphere , and . Hence
(Since is constant in magnitude on the sphere and normal to it everywhere, this is simply times the area .)
With , , the spherical divergence (2.3.54) gives
Then
Here the field is smooth everywhere, including the origin ( has Cartesian components , , with , which are polynomials), so the theorem applies. Observe that the divergence grows outwards: most of the "source strength" sits in the outer layers of the ball. Using the naive expression would have given only of the correct answer, . Figure 2.4.6 repeats the check for every radius.