If $\operatorname { div } \vec { F } = - 7$ everywhere and S is a smooth surface (oriented outward)enclosing a volume W of size V find $\int _ { S } \vec { F } \cdot \vec { d A }$ .
A) $7 V$
B) $- 7$
C) $- 7 V$
D) $- 7 W$
E)0

Question

Quizplus · Accepted Answer

The divergence theorem states that the surface integral of a vector field $\vec{F}$ over a closed surface $S$ is equal to the volume integral of the divergence of $\vec{F}$ over the volume $V$ enclosed by $S$. Given $\operatorname{div} \vec{F} = -7$, the integral becomes $\int_{S} \vec{F} \cdot \vec{dA} = \int_{V} (-7) dV = -7V$, where $V$ is the volume enclosed by $S$.

If div⁡F⃗=−7\operatorname { div } \vec { F } = - 7divF=−7

If $\operatorname { div } \vec { F } = - 7$