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

Question

Quizplus · Accepted Answer

Stoke's theorem states that $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = \iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ for any closed curve $C$ that is the boundary of a smooth surface $S$.

In this case, we want to find the value of $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$, where $C$ is the boundary of the part of the plane $8 x + y + 8 z = 8$ in the first octant.

To use Stoke's theorem, we need to find the curl of $\mathbf F$: 
$$\operatorname{curl} \mathbf F = \nabla \times \mathbf F = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ e^{-7x} & e^{4y} & e^{5z} \end{vmatrix} = (20e^{5z} - 7e^{-7x}) \mathbf i - 5e^{5z} \mathbf j + 4e^{4y} \mathbf k$$

Now, we need to find a surface $S$ whose boundary is $C$ and on which we can evaluate the curl of $\mathbf F$. We can take the part of the plane $8 x + y + 8 z = 8$ in the first octant as our surface.

The boundary of this surface is the curve $C$, so we can apply Stoke's theorem: 
$$\int _ { C } \mathbf { F } \cdot d \mathbf { r } = \iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S } = \iint _{S} ((20e^{5z} - 7e^{-7x}) \mathbf i - 5e^{5z} \mathbf j + 4e^{4y} \mathbf k) \cdot d\mathbf S$$

We can parameterize the surface $S$ using the variables $y$ and $z$: 
$$\mathbf r (y,z) = \langle \frac{8-8z-y}{8}, y, z \rangle$$
where $0 \leq y \leq 8 - 8z$ and $0 \leq z \leq \frac{1}{8}$.

The surface normal is given by: 
$$\frac{\partial \mathbf r}{\partial y} \times \frac{\partial \mathbf r}{\partial z} = \langle 1, 0, -\frac{8}{8-8z-y} \rangle \times \langle 0, 1, 0 \rangle = \langle \frac{8}{8-8z-y}, 0, 1 \rangle$$

Therefore, we have: 
$$\int _ { C } \mathbf { F } \cdot d \mathbf { r } = \iint _{S} ((20e^{5z} - 7e^{-7x}) \mathbf i - 5e^{5z} \mathbf j + 4e^{4y} \mathbf k) \cdot \langle \frac{8}{8-8z-y}, 0, 1 \rangle dydz$$

Switching the order of integration and evaluating, we get: 
\begin{align*}
\int_0^{1/8} \int_0^{8-8z} \left[(20e^{5z} - 7e^{-7x})\cdot \frac{8}{8-8z-y} + 4e^{4y}\cdot 1 \right] dy dz = 0
\end{align*}

Thus, the value of the integral is $\boxed{0}$.

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 $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$