Use Divergence Theorem to evaluate $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$ and find the outward flux of $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + z ^ { 2 } \mathbf { k }$ through the surface $S$ of the solid bounded by the planes $x = 0 , x = 12 , y = 0 , y = 12 , z = 0$ and $z = 12$.
A) 6,912
B) 576
C) 62,208
D) 20,736
E) 5,184

Question

Use Divergence Theorem to evaluate $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$  and find the outward flux of $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + z ^ { 2 } \mathbf { k }$ through the surface $S$ of the solid bounded by the planes $x = 0 , x = 12 , y = 0 , y = 12 , z = 0$ and $z = 12$.
A) 6,912
B) 576
C) 62,208
D) 20,736
E) 5,184

Quizplus · Accepted Answer

The divergence of $\mathbf{F}$ is $\nabla \cdot \mathbf{F} = 2x + 2y + 2z$. The volume integral over the cube from 0 to 12 for each variable gives $\int_0^{12} \int_0^{12} \int_0^{12} (2x + 2y + 2z) \, dx \, dy \, dz = 62,208$.

Use Divergence Theorem to Evaluate ∬SF⋅Nds\iint_{S} \mathbf{F} \cdot \mathbf{N} d s∬S​F⋅Nds

Use Divergence Theorem to Evaluate $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$