Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .
A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H)
$\frac { 3 } { 2 }$

Question

Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .
A) $0$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$ 
D) $\frac { 1 } { 2 }$ 
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $1$

H)
$\frac { 3 } { 2 }$

Quizplus · Accepted Answer

The vector field $\mathbf{F} = \mathbf{i}$ is constant and its divergence is zero, indicating no net flux through the surface. The only contributions to the surface integral come from the faces perpendicular to the $\mathbf{i}$ direction, where the flux in equals the flux out, resulting in a net flux of $0$.

Let F(x,y,z)=i\mathbf { F } ( x , y , z ) = \mathbf { i }F(x,y,z)=i

Let $\mathbf { F } ( x , y , z ) = \mathbf { i }$