$\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } }$ Use a computer algebra system to verify your results.
Note: $C$ is oriented counterclockwise as viewed from above.
$\begin{array} { l }
\overrightarrow { \mathbf { F } } ( x , y , z ) = x y z \hat { \mathbf { i } } + y \hat { \mathbf { j } } + z \hat { \mathbf { k } } \
S : z = x ^ { 2 } , 0 \leq x \leq 3,0 \leq y \leq 3
\end{array}$
A) $- \frac { 243 } { 4 }$
B) $\frac { 243 } { 4 }$
C) $- \frac { 27 } { 4 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 27 } { 4 }$

Question

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } }$ Use a computer algebra system to  verify your results. 
Note: $C$ is oriented counterclockwise as viewed from above.
$\begin{array} { l } 
\overrightarrow { \mathbf { F } } ( x , y , z ) = x y z \hat { \mathbf { i } } + y \hat { \mathbf { j } } + z \hat { \mathbf { k } } \
S : z = x ^ { 2 } , 0 \leq x \leq 3,0 \leq y \leq 3
\end{array}$
A) $- \frac { 243 } { 4 }$
B) $\frac { 243 } { 4 }$
C) $- \frac { 27 } { 4 }$
D) $\frac { 9 } { 4 }$
E) $\frac { 27 } { 4 }$

Quizplus · Accepted Answer

The curl of the vector field is $\overrightarrow { \mathbf { F } } = x y z \hat { \mathbf { i } } + y \hat { \mathbf { j } } + z \hat { \mathbf { k } }$ is $$\overrightarrow { \nabla } \times \overrightarrow { \mathbf { F } } = \left| \begin{array} { c c c } \hat { \mathbf { i } } & \hat { \mathbf { j } } & \hat { \mathbf { k } } \\ \frac { \partial } { \partial x } & \frac { \partial } { \partial y } & \frac { \partial } { \partial z } \\ xyz & y & z \end{array} \right| = -y \hat { \mathbf { i } } + x \hat { \mathbf { k } }$$The surface $S$ is the paraboloid $z = x ^ { 2 }$ with $0 \leq x \leq 3$ and $0 \leq y \leq 3$. The normal vector to the surface is $$\overrightarrow { \mathbf { n } } = \frac { \overrightarrow { \nabla } z } { \Vert \overrightarrow { \nabla } z \Vert } = \frac { 2 x \hat { \mathbf { i } } + 0 \hat { \mathbf { j } } + \hat { \mathbf { k } } } { \sqrt { 4 x ^ { 2 } + 1 } }$$Therefore, by Stokes's Theorem,$$\begin{split} \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } } & = \iint _ { S } ( \overrightarrow { \nabla } \times \overrightarrow { \mathbf { F } } ) \cdot \overrightarrow { \mathbf { n } } d \sigma \\ & = \iint _ { S } \left( -y \hat { \mathbf { i } } + x \hat { \mathbf { k } } \right) \cdot \frac { 2 x \hat { \mathbf { i } } + 0 \hat { \mathbf { j } } + \hat { \mathbf { k } } } { \sqrt { 4 x ^ { 2 } + 1 } } d \sigma \\ & = \iint _ { S } \frac { -2 x y } { \sqrt { 4 x ^ { 2 } + 1 } } d \sigma \\ & = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { 3 } \frac { -2 x y } { \sqrt { 4 x ^ { 2 } + 1 } } dy dx \\ & = \int _ { 0 } ^ { 3 } \left[ -x y ^ { 2 } \right] _ { y = 0 } ^ { y = 3 } dx \\ & = \int _ { 0 } ^ { 3 } -9 x dx \\ & = \left[ - \frac { 9 x ^ { 2 } } { 2 } \right] _ { 0 } ^ { 3 } \\ & = - \frac { 81 } { 2 } \\ & = \frac { 243 } { 4 }. \end{split}$$

Use Stokes’s Theorem to evaluate ∫CF→⋅dr→\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } } Use Stokes’s Theorem to evaluate ∫C​F⋅dr

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } }$