Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$.
$\mathbf { F } ( x , y , z ) = 7 \cos x \mathbf { i } + 5 e ^ { y } \mathbf { j } + 2 x y \mathbf { k }$
$C$ is the curve obtained by intersecting the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ with the hyperbolic paraboloid $z = x ^ { 2 } - y ^ { 2 }$, oriented in a counterclockwise direction when viewed from above
A) $- 2 \pi$
B) 0
C) $\pi$
D) 1

Question

Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$.
$\mathbf { F } ( x , y , z ) = 7 \cos x \mathbf { i } + 5 e ^ { y } \mathbf { j } + 2 x y \mathbf { k }$ 
$C$ is the curve obtained by intersecting the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ with the hyperbolic paraboloid $z = x ^ { 2 } - y ^ { 2 }$, oriented in a counterclockwise direction when viewed from above
A) $- 2 \pi$
B) 0
C) $\pi$
D) 1

Quizplus · Accepted Answer

The boundary of the surface is the circle $x^2+y^2=1$ in the $xy$-plane, oriented counterclockwise when viewed from above. The normal vector to the surface is $$\mathbf{n}=\frac{\nabla f}{\Vert \nabla f \Vert}=\frac{2x\mathbf{i}-2y\mathbf{j}+1\mathbf{k}}{\sqrt{4x^2+4y^2+1}}=\frac{2x\mathbf{i}-2y\mathbf{j}+\mathbf{k}}{3}$$So, by Stokes' Theorem,$$\begin{split} \oint_C \mathbf{F}\cdot d\mathbf{r} &=\iint_S (\text{curl } \mathbf{F})\cdot \mathbf{n} d\sigma \\ &=\iint_S \left(\frac{\partial}{\partial x}[2xy]-\frac{\partial}{\partial y}[5e^y]\right)\frac{2x\mathbf{i}-2y\mathbf{j}+\mathbf{k}}{3} d\sigma \\ &=\iint_S (2y)\frac{2x\mathbf{i}-2y\mathbf{j}+\mathbf{k}}{3} d\sigma \\ &=\iint_S \frac{4xy}{3} d\sigma \\ &=\iint_R \frac{4r^2}{3} r \ dr d\theta \quad (\text{where } x=r\cos\theta, y=r\sin\theta) \\ &=\int_0^{2\pi} \int_0^1 \frac{4r^3}{3} \ dr \ d\theta \\ &=\int_0^{2\pi} \left[\frac{r^4}{3}\right]_0^1 d\theta \\ &=\int_0^{2\pi} \frac{1}{3} d\theta \\ &=\frac{1}{3}[2\pi-0] \\ &=\frac{2\pi}{3} \end{split}$$

Use Stokes' Theorem to Evaluate ∮CF⋅dr\oint _ { C } \mathbf { F } \cdot d \mathbf { r }∮C​F⋅dr

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