Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 7 - 2 x - y$ lying above the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \left\langle z , y ^ { 2 } , 3 x \right\rangle$
A) $- 16$
B) $- 8$
C) $- 4$
D) $- 2$

Question

Quizplus · Accepted Answer

Stokes' theorem relates the line integral of a vector field over a closed curve to the surface integral of the curl of the vector field over the surface bounded by the curve. The curl of $\vec{F}$ is $\nabla \times \vec{F} = \langle -2, -3, 1 \rangle$. The surface is the plane $z = 7 - 2x - y$ above the square, with a normal vector $\langle -2, -1, 1 \rangle$. The surface integral $\iint_S (\nabla \times \vec{F}) \cdot d\vec{S}$ evaluates to $-8$.

Use Stokes' Theorem to Evaluate ∫CF⃗⋅dr⃗\int _ { C } \vec { F } \cdot d \vec { r }∫C​F⋅dr

Use Stokes' Theorem to Evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$