Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = e ^ { - 7 x } \mathbf { i } + e ^ { 4 y } \mathbf { j } + e ^ { 5 x } \mathbf { k }$ and C is the boundary of the part of the plane $8 x + y + 8 z = 8$ in the first octant.
A) 69
B) 16
C) 49
D) 0
E) 23

Question

Quizplus · Accepted Answer

Stoke's theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the field over the surface bounded by the curve. The curl of $\mathbf{F}$ is zero, so the integral is zero.

Use Stoke's Theorem to Evaluate ∫CF⋅dr\int _ { C } \mathbf { F } \cdot d \mathbf { r }∫C​F⋅dr

Use Stoke's Theorem to Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$