Question

Use Stokes' Theorem to evaluate |c F * dr where C is oriented counterclockwise as viewed from above.F(x, y, z) = xyi + 5zj + 7yk, where C is the curve of intersection of the plane x + z = 3 and the cylinder x2 + y2 = 144.

Answer 1

Compute the curl of $\vec F$:

$\vec F(x,y,z)=xy\,\vec\imath+5z\,\vec\jmath+7y\,\vec k\implies\nabla\times\vec F(x,y,z)=2\,\vec\imath-x\,\vec k$

By Stoke's theorem, the (line) integral of $\vec F$ along $C$ is equivalent to the (surface) integral (or flux) of $\nabla\times\vec F$ across $S$, the oriented surface with boundary $C$.

Take $S$ to be the part of the plane $x+z=3$ within the cylinder $x^2+y^2=144$. Parameterize $S$ by

$\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(3-u\cos v)\,\vec k$

with $0\le u\le12$ and $0\le v\le2\pi$. Take the normal vector to $S$ to be

$\vec s_u\times\vec s_v=u\,\vec\imath+u\,\vec k$

Then the integral is

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^{2\pi}\int_0^{12}(2\,\vec\imath-u\cos v\,\vec k)\cdot(u\,\vec\imath+u\,\vec k)\,\mathrm du\,\mathrm dv$

$=\displaystyle\int_0^{2\pi}\int_0^{12}(2u-u^2\cos v)\,\mathrm du\,\mathrm dv=\boxed{288\pi}$

Answer 2

After evaluating the given integral the value obtained is 288$\pi$ and this can be determined by using the Stokes theorem.

Given :

F(x, y, z) = xyi + 5zj + 7yk, where C is the curve of intersection of the plane x + z = 3 and the cylinder x2 + y2 = 144.

Parameterize S as given below:

$\rm \bar{s}(u,v)=u\;cosv\;i+u\;sinv\;j+(3-ucosv)\;k$

where the value of $\rm 0\leq u\leq 12$ and $\rm 0\leq v\leq 2\pi$.

Now, the integral can be evaluated as given below:

$\rm \int_C\bar{F}.d\bar{r}=\int \int_S(\bigtriangledown \times \bar{F}).(\bar{s}_u\times (\bar{s}_v)\;du\;dv$

$\rm \int_C\bar{F}.d\bar{r}=\int^{2\pi}_0 \int^{12}_0(2i-u\;cosv\;k).(u\;i+u\;k)\;du\;dv$

$\rm \int_C\bar{F}.d\bar{r}=\int^{2\pi}_0 \int^{12}_0(2u-u^2cosv)\;du\;dv$

$\rm \int_C\bar{F}.d\bar{r}=288\pi$

After evaluating the given integral the value obtained is 288$\pi$ and this can be determined by using the Stokes theorem.

For more information, refer to the link given below:

https://brainly.com/question/14093731