Evaluating the Curl Using Stokes' Theorem

[Sai-]

Homework Statement

Use Stokes' Theorem to evaluate \int\int curl \vec{F}\bullet d\vec{S} where \vec{F}(x,y,z) = <e^{z^{2}},4z-y,8xsin(y)> and S is the portion of the paraboloid
z = 4-x^{2}-y^{2} above the xy plane.

Homework Equations

Stokes Thm:\int\int curl \vec{F}\bullet d\vec{S} = \int \vec{F}\bullet d\vec{r}

\vec{F}(x,y,z) = <e^{z^{2}},4z-y,8xsin(y)>

S: z = 4-x^{2}-y^{2} above the z = 0.

The Attempt at a Solution

C: \vec{r}(t) = <2cos(t), 2sin(t), 0> where 0\leq t\leq2\pi
\vec{r}'(t) = <-2sin(t), 2cos(t), 0>

\vec{F}(\vec{r}(t)) = <e^{0}, (4(0) - 2sin(t), 8(2(cos(t))sin(2cos(t))>
\vec{F}(\vec{r}(t)) = <1, -2sin(t), 16cos(t)sin(2cos(t))>

\int <1, -2sin(t), 16cos(t)sin(2cos(t))> \bullet <-2sin(t), 2cos(t), 0> dt from 0 to 2pi
=\int -2sin(t)-2sin(t)2cos(t) dt from 0 to 2pi
=0

 

[LCKurtz]

Homework Statement

Use Stokes' Theorem to evaluate \int\int \nabla \times \vec{F}\cdot d\vec{S} where \vec{F}(x,y,z) = <e^{z^{2}},4z-y,8x\sin(y)> and S is the portion of the paraboloid
z = 4-x^{2}-y^{2} above the xy plane.

Homework Equations

Stokes Thm:\int\int \nabla \times \vec{F}\cdot d\vec{S} = \int \vec{F}\cdot d\vec{r}

\vec{F}(x,y,z) = <e^{z^{2}},4z-y,8x\sin(y)>

S: z = 4-x^{2}-y^{2} above the z = 0.

The Attempt at a Solution

C: \vec{r}(t) = <2\cos(t), 2\sin(t), 0> where 0\leq t\leq2\pi
\vec{r}'(t) = <-2\sin(t), 2\cos(t), 0>

\vec{F}(\vec{r}(t)) = <e^{0}, (4(0) - 2\sin(t), 8(2(\cos(t))\sin(2\cos(t))>
\vec{F}(\vec{r}(t)) = <1, -2\sin(t), 16\cos(t)\sin(2\cos(t))>

\int <1, -2\sin(t), 16\cos(t)\sin(2\cos(t))> \cdot <-2\sin(t), 2\cos(t), 0> dt from 0 to 2pi
=\int_0^{2\pi} -2\sin(t)-2\sin(t)2\cos(t) dt [STRIKE]from 0 to 2pi[/STRIKE]
=0

Some latex pointers. If you will put a backslash in front of any function in tex it prints in a much nicer font. Also you can use \cdot instead of \bullet and \nabla\times for curl. I have done that in my quote so you can see the difference. Also you can put the limits on the integral as I illustrate in one line above.

A problem like this should give some orientation for the surface, although with an answer of zero it doesn't matter much. Your work looks correct unless I have overlooked something.

 

[Sai-]

Some latex pointers. If you will put a backslash in front of any function in tex it prints in a much nicer font. Also you can use \cdot instead of \bullet and \nabla\times for curl. I have done that in my quote so you can see the difference. Also you can put the limits on the integral as I illustrate in one line above.

A problem like this should give some orientation for the surface, although with an answer of zero it doesn't matter much. Your work looks correct unless I have overlooked something.

Thank you, I will remember those hints and tips for next time!

And thanks for looking at my work, I think it is correct too; I just can't afford to miss any points on any problem, its dead week and I need all the points I can get.