Find the flux of the curl of field F through the shell S.
-$\mathbf { F } = x ^ { 2 } \mathbf { i } + 6 x \mathbf { j } + 9 \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16$
A) $16 \pi$
B) 16
C) $96 \pi$
D) $\pi$

Question

Quizplus · Accepted Answer

First, we need to find the curl of the given field F:

$$\operatorname{curl}\mathbf{F}=\left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^2 & 6x & 9\end{array}ight|=\left(0-0ight)\mathbf{i}+\left(9-0ight)\mathbf{j}+\left(2x-6ight)\mathbf{k}=9\mathbf{j}+\left(2x-6ight)\mathbf{k}$$
Next, we need to find the unit normal vector to the upper hemisphere S. Since S is the upper hemisphere, the unit normal vector is simply the positive z-axis unit vector, i.e., $$\mathbf{n}=\begin{pmatrix}0 \ \,0 \ \,1\end{pmatrix}$$
The flux of the curl of F through S is given by the surface integral

$$\iint_S\left(\operatorname{curl}\mathbf{F}ight)\cdot\mathbf{n}\,\mathrm{d}S$$
Substituting the values, we get

$$\iint_S\left(\operatorname{curl}\mathbf{F}ight)\cdot\mathbf{n}\,\mathrm{d}S=\iint_S\left(9\mathbf{j}+\left(2x-6ight)\mathbf{k}ight)\cdot\begin{pmatrix}0 \ \,0 \ \,1\end{pmatrix}\,\mathrm{d}S=\iint_S \left(2x-6ight)\,\mathrm{d}S$$
To evaluate this integral, we need to parameterize the surface S. Since S is the upper hemisphere, we can use spherical coordinates:

$$x=r\sin	heta\cos\phi,\qquad y=r\sin	heta\sin\phi,\qquad z=r\cos	heta$$

where

$$0\leq	heta\leq\frac{\pi}{2},\qquad 0\leq\phi\leq2\pi,\qquad 0\leq r\leq4$$

Note that the equation of S in spherical coordinates is simply $r=4,\,0\leq	heta\leq\frac{\pi}{2},\,0\leq\phi\leq2\pi$. Thus,

$$\iint_S \left(2x-6ight)\,\mathrm{d}S=\int_{0}^{2\pi}\int_{0}^{\frac{\pi}{2}}\left[2\left(4\sin	heta\cos\phiight)-6ight]\left(4^{2}\sin	hetaight)\,\mathrm{d}	heta\,\mathrm{d}\phi=96\pi$$

Therefore, the answer is $\boxed{	extbf{(C) }96\pi}$.

Find the Flux of the Curl of Field F Through F=x2i+6xj+9k;S\mathbf { F } = x ^ { 2 } \mathbf { i } + 6 x \mathbf { j } + 9 \mathbf { k } ; \mathrm { S }F=x2i+6xj+9k;S

Find the Flux of the Curl of Field F Through $\mathbf { F } = x ^ { 2 } \mathbf { i } + 6 x \mathbf { j } + 9 \mathbf { k } ; \mathrm { S }$