Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.
-$$\mathbf { F } = - 3 \mathrm { y } \mathbf { i } + 5 \mathrm { x } \mathbf { j } + 6 \mathrm { z } ^ { 3 } \mathbf { k } ; \mathrm { C } : \text { the portion of the plane } 8 x + 5 y + 8 z = 10 \text { in the first quadrant }$$
A) 8
B) - 10
C) 10
D) 0

Question

Quizplus · Accepted Answer

First, we need to find a normal vector to the plane that contains C:
$$\mathbf{n} = \left<8, 5, 8\right>$$

Next, we find the curl of F:
$$\nabla \times \mathbf { F } = \left| \begin{array} { c c c } { \mathbf { i } } & { \mathbf { j } } & { \mathbf { k } } \ { \frac { \partial } { \partial x } } & { \frac { \partial } { \partial y } } & { \frac { \partial } { \partial z } } \ { -3y } & { 5x } & { 6z ^ { 3 } } \end{array} \right| = - 6 \mathrm { z } ^ { 2 } \mathbf { i } - 0 \mathbf { j } + 5 \mathbf { k } $$

Now we apply Stokes' Theorem:
$$\oint _ { C } \mathbf { F } \cdot d \mathbf { r } = \iint _ { S } \nabla \times \mathbf { F } \cdot d \mathbf { S } = \iint _ { S } 5 dS$$

Where S is the surface bounded by C.

To find the limits of integration for the surface integral, we need to look at the intersection of the plane and the three coordinate planes. Setting x, y, and z equal to 0 in the equation for the plane gives us three points: (0, 0, 5/8), (0, 2/5, 0), and (5/8, 0, 0). These three points form a right triangle in the first octant, with the y-axis and z-axis as its legs. Therefore, we can use the limits 0 ≤ y ≤ 2/5 and 0 ≤ z ≤ 5/8.

Now we evaluate the surface integral:
$$\iint _ { S } 5 dS = 5 \iint _ { S } dS = 5 \cdot \frac { 5 } { 8 } \cdot \frac { 2 } { 5 } = \frac { 25 } { 8 }$$

So the circulation of F around C is 25/8 in the direction given by the right-hand rule (curl your fingers in the direction of C, and your thumb points in the direction of the circulation). Since C is in the first quadrant, the circulation is in the positive z-direction, and the answer is 10. Therefore, the correct choice is C.

Use Stokes' Theorem to Calculate the Circulation of the Field