Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-$F = \sin y i + x z j + 5 z k$; D: the thick sphere $4 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 9$
A) $135 \pi$
B) $\frac { 76 } { 3 } \pi$
C) $380 \pi$
D) $\frac { 380 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The Divergence Theorem states that the outward flux of a vector field $F$ across the boundary of a region $D$ is equal to the triple integral over $D$ of the divergence of $F$. The divergence of $F = \sin y i + x z j + 5 z k$ is $\nabla \cdot F = \frac{\partial}{\partial x}(\sin y) + \frac{\partial}{\partial y}(x z) + \frac{\partial}{\partial z}(5 z) = 0 + x + 5$. To find the outward flux, we integrate this divergence over the volume of the region $D$, which is the thick sphere between radii $2$ and $3$. The volume element in spherical coordinates is $dV = r^2 \sin \phi dr d\theta d\phi$, and the limits for $r$ are from $2$ to $3$, for $\theta$ from $0$ to $2\pi$, and for $\phi$ from $0$ to $\pi$. The integral becomes $\int_{0}^{2\pi} \int_{0}^{\pi} \int_{2}^{3} (5 + r \cos \phi) r^2 \sin \phi dr d\phi d\theta$. Solving this integral gives $\frac { 380 } { 3 } \pi$.

Using the Divergence Theorem, Find the Outward Flux of F F=sin⁡yi+xzj+5zkF = \sin y i + x z j + 5 z kF=sinyi+xzj+5zk

Using the Divergence Theorem, Find the Outward Flux of F $F = \sin y i + x z j + 5 z k$