Question

If S is the unit sphere oriented outward and F is a vector field satisfying F · dA S = 0 , then divF = 0 at all points inside S.

a. true
b. false

Answer 1

The statement is completely true. The divergence of a vector field is zero at all points inside a closed surface with zero flux.

Step-by-step explanation:

In this question, we are given that the unit sphere S is oriented outward and the vector field F satisfies F · dAS = 0. We are asked to determine if divF = 0 at all points inside S. The divergence of a vector field measures the rate at which the field flows away from or toward a point.

If the flux of the vector field through a surface is zero, it implies that the field is neither flowing in nor flowing out of the surface. For a closed surface like the unit sphere S, divF = 0 at all points inside the sphere because the flux through the entire surface is zero.

Therefore, the answer is a. true.