Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$ and S is $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ Then the flux of $\text { F }$ through S is

A)8?
B)6?
C)4?
D)2?
E)0

Question

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$ and S is $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ Then the flux of $\text { F }$ through S is

A)8?
B)6?
C)4?
D)2?
E)0

Quizplus · Accepted Answer

The surface S is the upper half of a sphere of radius 1 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 $\mathbf{F}$ through a closed surface enclosing a volume V is equal to the integral over V of the divergence of $\mathbf{F}$. Since the divergence is 0, the flux of $\mathbf{F}$ through any closed surface, including the one formed by S and its projection on the xy-plane, is 0. Since the flux through the flat part (projection on the xy-plane) is also 0 (because $\mathbf{F}$ is tangential to this surface), the flux through S is 0.

Let F(x,y,z)=xi+yj−2zk\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }F(x,y,z)=xi+yj−2zk

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k }$