Let $\mathbf { F } ( x , y , z ) = x y \mathbf { i } + y ^ { 2 } \mathbf { j } + 2 \mathbf { k }$ and C be the boundary of $z = x ^ { 2 } + y ^ { 2 }$ below z = 1.Using Stokes' Theorem, $\int _ { C } \mathbf { F } \cdot \mathbf { T } d s$ is

A)3$\pi$
B)2$\pi$
C)$\pi$
D)0
E)-$\pi$

Question

Let

\mathbf { F } ( x , y , z ) = x y \mathbf { i } + y ^ { 2 } \mathbf { j } + 2 \mathbf { k }

and C be the boundary of

z = x ^ { 2 } + y ^ { 2 }

below z = 1.Using Stokes' Theorem,

\int _ { C } \mathbf { F } \cdot \mathbf { T } d s

is

A)3$\pi$
B)2$\pi$
C)$\pi$
D)0
E)-$\pi$

Quizplus · Accepted Answer

Stokes' Theorem relates a surface integral over a surface $S$ to a line integral over the boundary curve $C$ of the surface. It states that $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = \int _ { S } \nabla \times \mathbf { F } \cdot d \mathbf { S }$, where $\nabla \times \mathbf { F }$ is the curl of $\mathbf { F }$. Given $\mathbf { F } ( x , y , z ) = x y \mathbf { i } + y ^ { 2 } \mathbf { j } + 2 \mathbf { k }$, the curl of $\mathbf { F }$ is $\nabla \times \mathbf { F } = \left| \begin{array} { c c c } \mathbf { i } & \mathbf { j } & \mathbf { k } \ \partial / \partial x & \partial / \partial y & \partial / \partial z \ x y & y ^ { 2 } & 2 \end{array} \right| = (0 - 0) \mathbf { i } - (0 - 0) \mathbf { j } + (0 - 0) \mathbf { k } = \mathbf { 0 }$.Since the curl of $\mathbf { F }$ is zero, the integral of $\mathbf { F }$ over any closed curve $C$, according to Stokes' Theorem, is also zero. Therefore, the correct answer is D) 0.

Let F(x,y,z)=xyi+y2j+2k\mathbf { F } ( x , y , z ) = x y \mathbf { i } + y ^ { 2 } \mathbf { j } + 2 \mathbf { k }F(x,y,z)=xyi+y2j+2k

Let $\mathbf { F } ( x , y , z ) = x y \mathbf { i } + y ^ { 2 } \mathbf { j } + 2 \mathbf { k }$