Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.
-$\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }$; C: the portion of the paraboloid $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = \mathrm { z }$ cut by the cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$
A) $- 432 \pi$
B) $- 216 \pi$
C) $432 \pi$
D) $216 \pi$

Question

Quizplus · Accepted Answer

Stokes' Theorem relates the circulation of a vector field around a closed curve to a surface integral over a surface bounded by the curve. The theorem states that $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$, where $C$ is the closed curve, $S$ is any surface bounded by $C$, and $d\mathbf{S}$ is a vector normal to the surface $S$ with magnitude equal to the area of an infinitesimal piece of $S$.Given $\mathbf{F} = -9y^3\mathbf{i} + 9x^3\mathbf{j} + 8z^3\mathbf{k}$, we first find the curl of $\mathbf{F}$, $\nabla \times \mathbf{F}$.$\nabla \times \mathbf{F} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -9y^3 & 9x^3 & 8z^3 \end{array} \right| = (0 - 0)\mathbf{i} - (0 - 0)\mathbf{j} + (27x^2 - (-27y^2))\mathbf{k} = 27(x^2 + y^2)\mathbf{k}$.The surface $S$ is the portion of the paraboloid $x^2 + y^2 = z$ inside the cylinder $x^2 + y^2 = 4$, and $z$ ranges from 0 to 4 (the height of the paraboloid at the edge of the cylinder). The vector $d\mathbf{S}$ can be taken as $d\mathbf{S} = \mathbf{k} \, dA$, where $dA$ is the differential area element in the $xy$-plane.Substituting $\nabla \times \mathbf{F}$ and $d\mathbf{S}$ into the surface integral gives:$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S 27(x^2 + y^2) \, dA$.Since $x^2 + y^2 = 4$ within the cylinder, the integral simplifies to:$\iint_S 27 \cdot 4 \, dA = 108 \iint_S dA$.The area $A$ of the circle $x^2 + y^2 = 4$ is $\pi r^2 = \pi \cdot 2^2 = 4\pi$, so:$108 \iint_S dA = 108 \cdot 4\pi = 432\pi$.However, the direction of $d\mathbf{S}$ is upwards (positive $z$-direction), which is the same direction as the circulation direction given by the right-hand rule for the curve $C$. Therefore, the circulation is positive, but since the question does not specify the orientation of $C$ relative to the direction of $d\mathbf{S}$, and the calculation directly leads to a positive result, the correct answer based on the calculation is $432\pi$, which seems to contradict the provided options. Given the options and the typical interpretation of Stokes' Theorem, there might have been an oversight in interpreting the question's requirements or a mistake in the calculation steps as outlined. The correct application of Stokes' Theorem indeed leads to $432\pi$, but the options suggest a misunderstanding or misinterpretation. Given the context and typical expectations for such a problem, a reevaluation of the question's details or the assumptions made during the calculation might be necessary.

Use Stokes' Theorem to Calculate the Circulation of the Field F=−9y3i+9x3j+8z3k\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }F=−9y3i+9x3j+8z3k

Use Stokes' Theorem to Calculate the Circulation of the Field $\mathbf { F } = - 9 \mathrm { y } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 3 } \mathbf { j } + 8 \mathrm { z } ^ { 3 } \mathbf { k }$