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Jacobians and Surface integrals

  1. Apr 27, 2009 #1
    Why is it that when we evaluate a surface integral of:

    f(x, y ,z) over dS, where

    x = x(u, v)
    y = y(u, v)
    z = z(u, v)

    dS is equal to ||ru X rv|| dA

    Why don't we use the jacobian here when we change coordinate systems?
     
  2. jcsd
  3. Apr 27, 2009 #2

    HallsofIvy

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    Because you are NOT "changing coordinate systems"- not in the sense of replacing one 3 dimensional coordinate system with another or replacing one 2 dimensional coordinate system with another. The Jacobian is the determinant of an n by n matrix and so requires that you have the same dimension on both sides. That is not the situation when you have a two dimensional surface in a three dimensional space.
     
  4. Apr 27, 2009 #3
    What would be a case then where the jacobian matrix would be used in evaluating a surface integral?

    Thanks for the response.
     
  5. Apr 27, 2009 #4
    Would the Jacobian be used if:

    x = x(u, v, w)
    y = y(u, v, w)
    z = z(u, v, w)

    ?
     
  6. Apr 28, 2009 #5

    HallsofIvy

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    Yes, but of course that's not a "surface integral"- that's changing from one three-dimensional coordinate system to another. You might, after forming the integral over a surface, decide that the integral would be simpler if you chose different coordinates, that is a different parameterization, for the surface. Then you would use the Jacobian to change from one two-dimensional coordinate system to another.
     
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