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 $\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S$ , where a is the constant vector.
A) 6
B) 5
C) 7
D) 8
E) 3

Question

Quizplus · Accepted Answer

By the Divergence Theorem, we have
$$\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S = \iiint _ { V } 3 \mathbf { a } \cdot \nabla d V$$
Since a is a constant vector, we can take it outside the integral and use the fact that $\nabla\cdot\mathbf{a}=0$ (since a is a constant vector) to get
$$\iiint _ { V } 3 \mathbf { a } \cdot \nabla d V = 0$$
Therefore, the answer is 0, which corresponds to choice E.

Assuming That S Satisfies the Conditions of the Divergence Theorem ∬53andS\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S∬5​3andS

Assuming That S Satisfies the Conditions of the Divergence Theorem $\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S$