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.