Thread: transforming from spherical coordinates

1. transforming from spherical coordinates

This question is born from a question relating to Stokes' theorem, where I'm evaluating the surface integral.
The area I'm confused with is where I need to transform the follwing into speherical polar coordinates:

$
\int\int_{S} (\nabla \times F)\cdot \bold{\hat{n}}\ dS \ = \frac{1}{2}\int\int_S(-yz-3z)\ dS$

where

$x=2\cos\phi\sin\theta\ y=2\sin\phi\sin\theta\ z=2\cos\theta$

how would I transform the above integral into one wrt $d\theta\ d\phi$?

2. I think we must know what surface $S$ is.
Out of curiosity, your " $\frac{1}{2}\int\int_S(-yz-3z)\ dS$" shouldn't be $dA$ instead of $dS$?

3. Originally Posted by bigdoggy
This question is born from a question relating to Stokes' theorem, where I'm evaluating the surface integral.
The area I'm confused with is where I need to transform the follwing into speherical polar coordinates:

$
\int\int_{S} (\nabla \times F)\cdot \bold{\hat{n}}\ dS \ = \frac{1}{2}\int\int_S(-yz-3z)\ dS$

where

$x=2\cos\phi\sin\theta\ y=2\sin\phi\sin\theta\ z=2\cos\theta$

how would I transform the above integral into one wrt $d\theta\ d\phi$?
Use Jacobian matrix.

4. I think we must know what surface S is.
Out of curiosity, your " $\frac{1}{2}\int\int_S(-yz-3z)\ dS$" shouldn't be dA instead of dS?
The surface is the hemisphere $x^2 + y^2 + z^2 = 4$ and the boundary curve is the intersection of the hemisphere with the plane $z=0.$

Use Jacobian matrix.
I've looked on Wikipedia, Jacobian matrix and determinant - Wikipedia, the free encyclopedia, but how would I go from the Jacobian to $dS = 4\sin\theta d\phi d\phi$ ??

I'm trying to understand how to convert the integral from the surface integral to one which can be evaluated using spherical polar's.[incidentally I know the answer is $dS = 4\sin\theta d\phi d\phi$ but cant figure out how to get to it!]

5. On this page,http://en.wikipedia.org/wiki/Spheric...rdinate_system shows integration of sherical coordinates,namely the surface element:

$dS=r^2\sin\theta d\phi d\theta$

I'm hoping someone can explain how to come to this...?