$\text { Find the flux of } \mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k } \text { through }$
$S : z = \sqrt { 121 - x ^ { 2 } - y ^ { 2 } } , \iint _ { S } \mathbf { F } \cdot \mathbf { N } d S$ where $\mathbf { N }$ is the upward unit normal vector to $S$.
A) 8
B) 11
C) 1
D) 0
E) 4

Question

$\text { Find the flux of } \mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k } \text { through }$ 
$S : z = \sqrt { 121 - x ^ { 2 } - y ^ { 2 } } , \iint _ { S } \mathbf { F } \cdot \mathbf { N } d S$ where $\mathbf { N }$ is the upward unit normal vector to $S$.
A) 8
B) 11
C) 1
D) 0
E) 4

Quizplus · Accepted Answer

The given surface $S$ is the upper half of a sphere with radius 11 centered at the origin. The divergence of $\mathbf{F}$ is $\nabla \cdot \mathbf{F} = 1 + 1 - 2 = 0$. By the divergence theorem, the flux of a vector field through a closed surface is equal to the integral of its divergence over the volume enclosed by the surface. Since the divergence is zero, the flux of $\mathbf{F}$ through any closed surface that includes $S$ as part of its boundary is zero. Although $S$ is not a closed surface, the flux through the closed surface formed by $S$ and its projection on the $xy$-plane (which does not contribute to the flux since $\mathbf{F}$ is tangent to this plane) is also zero.

Find the flux of F(x,y,z)=xi+yj−2zk through \text { Find the flux of } \mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k } \text { through } Find the flux of F(x,y,z)=xi+yj−2zk through

$\text { Find the flux of } \mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k } \text { through }$