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Math Help - Jacobian

  1. #1
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    Jacobian

    Can some one give me a hand on how I can derive the P2sin
    Jacobian in the spherical system.

    Thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MarianaA View Post
    Can some one give me a hand on how I can derive the P2sin
    Jacobian in the spherical system.

    Thanks
    i'm not exactly sure what you're asking here. maybe this will help
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  3. #3
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    Quote Originally Posted by MarianaA View Post
    Can some one give me a hand on how I can derive the P2sin
    Jacobian in the spherical system.

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

    So the Jacobian is:
    \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,
     - \rho ^2 \sin \phi
    But when you change coordinates you need to take  | \ \cdot \ | of the Jacobian which is: \rho^2 \sin \phi.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    When you have the triple integral:
    \iiint_V f(x,y,z) \ dV
    You want to redifine it by letting,
    x = \rho \sin \phi \cos \theta , y = \rho \sin \phi \sin \theta, z = \rho \cos \phi

    So the Jacobian is:
    \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,
     - \rho ^2 \sin \phi
    But when you change coordinates you need to take  | \ \cdot \ | of the Jacobian which is: \rho^2 \sin \phi.
    oh, ok.
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    When you have the triple integral:
    \iiint_V f(x,y,z) \ dV
    You want to redifine it by letting,
    x = \rho \sin \phi \cos \theta , y = \rho \sin \phi \sin \theta, z = \rho \cos \phi

    So the Jacobian is:
    \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,
     - \rho ^2 \sin \phi
    But when you change coordinates you need to take  | \ \cdot \ | of the Jacobian which is: \rho^2 \sin \phi.
    Thanks
    Follow Math Help Forum on Facebook and Google+

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