1. Jacobian

Can some one give me a hand on how I can derive the P2sinø
Jacobian in the spherical system.

Thanks

2. Originally Posted by MarianaA
Can some one give me a hand on how I can derive the P2sinø
Jacobian in the spherical system.

Thanks
i'm not exactly sure what you're asking here. maybe this will help

3. Originally Posted by MarianaA
Can some one give me a hand on how I can derive the P2sinø
Jacobian in the spherical system.

Thanks
When you have the triple integral:
$\displaystyle \iiint_V f(x,y,z) \ dV$
You want to redifine it by letting,
$\displaystyle x = \rho \sin \phi \cos \theta , y = \rho \sin \phi \sin \theta, z = \rho \cos \phi$

So the Jacobian is:
$\displaystyle \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)} = \det \left( \begin{array}{ccc}\sin \phi \cos \theta & - \rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & - \rho \sin \phi \end{array} \right)$
This simplifies to,
$\displaystyle - \rho ^2 \sin \phi$
But when you change coordinates you need to take $\displaystyle | \ \cdot \ |$ of the Jacobian which is: $\displaystyle \rho^2 \sin \phi$.

4. Originally Posted by ThePerfectHacker
When you have the triple integral:
$\displaystyle \iiint_V f(x,y,z) \ dV$
You want to redifine it by letting,
$\displaystyle x = \rho \sin \phi \cos \theta , y = \rho \sin \phi \sin \theta, z = \rho \cos \phi$

So the Jacobian is:
$\displaystyle \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)} = \det \left( \begin{array}{ccc}\sin \phi \cos \theta & - \rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & - \rho \sin \phi \end{array} \right)$
This simplifies to,
$\displaystyle - \rho ^2 \sin \phi$
But when you change coordinates you need to take $\displaystyle | \ \cdot \ |$ of the Jacobian which is: $\displaystyle \rho^2 \sin \phi$.
oh, ok.

5. Originally Posted by ThePerfectHacker
When you have the triple integral:
$\displaystyle \iiint_V f(x,y,z) \ dV$
You want to redifine it by letting,
$\displaystyle x = \rho \sin \phi \cos \theta , y = \rho \sin \phi \sin \theta, z = \rho \cos \phi$

So the Jacobian is:
$\displaystyle \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)} = \det \left( \begin{array}{ccc}\sin \phi \cos \theta & - \rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & - \rho \sin \phi \end{array} \right)$
This simplifies to,
$\displaystyle - \rho ^2 \sin \phi$
But when you change coordinates you need to take $\displaystyle | \ \cdot \ |$ of the Jacobian which is: $\displaystyle \rho^2 \sin \phi$.
Thanks