Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ $\mathbf { F } ( x , y , z ) = 7 x y \mathbf { i } + 7 e ^ { x } \mathbf { j } + 7 x y ^ { 2 } \mathbf { k }$ S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.
A) 0
B) 12
C) 16
D) 49
E) 1

Question

Quizplus · Accepted Answer

The surface integral of the curl of a vector field over a closed surface is zero, as per Stokes' Theorem. The given surface is closed, so the integral evaluates to zero.

Use Stokes' Theorem to Evaluate ∬Scurl⁡F⋅dS\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }∬S​curlF⋅dS

Use Stokes' Theorem to Evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$