Find the flux of the vector field F across the surface S in the indicated direction.
-$\mathbf { F } = \frac { z ^ { 2 } } { 9 } \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 9$; direction is outward
A) $\frac { 9 } { 2 } \pi$
B) $- \frac { 9 } { 2 } \pi$
C) $- \frac { 9 } { 2 }$
D) $\frac { 9 } { 2 }$

Question

Quizplus · Accepted Answer

The flux of the vector field $\mathbf{F} = \frac{z^2}{9}\mathbf{k}$ across the surface $S$ (the upper hemisphere of $x^2 + y^2 + z^2 = 9$) in the outward direction can be calculated using the formula for flux through a surface: $\Phi = \int \int_S \mathbf{F} \cdot d\mathbf{S}$. Since $\mathbf{F}$ is in the direction of $\mathbf{k}$ and the surface is a hemisphere, we can simplify the calculation by considering the projection of the hemisphere onto the $xy$-plane and using polar coordinates. The radius of the hemisphere is 3, and the projection is a circle of radius 3 in the $xy$-plane. The flux through the hemisphere is equivalent to the flux through the circle in the $xy$-plane, scaled by the z-component of $\mathbf{F}$, which is $\frac{z^2}{9}$. Integrating over the circle with radius 3 gives us the total flux as $\frac{9}{2}\pi$, considering the outward direction and the symmetry of the hemisphere.

Find the Flux of the Vector Field F Across the Surface