Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find $\iint _ { s } 4 \mathbf { a } \cdot \mathbf { n } d S \text {, }$ where a is the constant vector.
A) 8
B) 6
C) 5
D) 7
E) 0

Question

Quizplus · Accepted Answer

The Divergence Theorem relates the flux of a vector field through a closed surface to the divergence of the field inside the volume enclosed by the surface. For a constant vector $\mathbf{a}$, its divergence is zero because the divergence of a constant vector (which has no spatial variation) is always zero. Therefore, according to the Divergence Theorem, the integral of a constant vector field over a closed surface is zero.

Assuming That S Satisfies the Conditions of the Divergence Theorem ∬s4a⋅ndS, \iint _ { s } 4 \mathbf { a } \cdot \mathbf { n } d S \text {, }∬s​4a⋅ndS,

Assuming That S Satisfies the Conditions of the Divergence Theorem $\iint _ { s } 4 \mathbf { a } \cdot \mathbf { n } d S \text {, }$