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Change of Variables in Tripple Integrals

  1. Aug 14, 2012 #1
    1. The problem statement, all variables and given/known data

    In double integrals, the change of variables is fairly easy to understand. With u = constant and v = constant, along line KL v = constant so dv = 0. Therefore the only contributing variable to ∂x and ∂y is ∂v.


    3. The attempt at a solution

    However, in tripple integrals, you're simply adding one more w = constant that gives the 3rd dimension (the height to the initial flat-surfaced parallelogram before).

    So as you move along a line PQ (which is formed by say u = constant), then won't BOTH v and w contribute to ∂x, ∂y and ∂z since only u is constant?
     

    Attached Files:

  2. jcsd
  3. Aug 14, 2012 #2
    Are you aware of the Jacobian determinant?
     
  4. Aug 14, 2012 #3
    In 3D, the object defined by a constant coordinate is a surface, not a line. z = 0 defines a plane, for instance, doesn't it?

    They're saying that along PQ two coordinates are constant because that's where two surfaces of (different) constant coordinates intersect.

    In short, for 3D:

    No coordinates constant: a volume
    1 coordinate constant: a plane
    2 coordinates constant: a line
    All 3 coordinates constant: a point
     
  5. Aug 14, 2012 #4
    Hmm I think I got it! For example, u = constant family of curves define the x-y plane, while v = constant define the y-z plane etc... the line is at intersection of the 2 planes where they cut one another.

    Therefore, since the line lies on both plane x-y and y-z, v and u must be constant!
     
  6. Aug 14, 2012 #5
    Am I right to say that in the new coordinate surfaces u, v and w they need not necessarily be at right angles to one another as in x, y and z?
     
    Last edited: Aug 14, 2012
  7. Aug 14, 2012 #6
    In the picture attached there is a parallelepiped. My question is, what happens if plane ABCD is not vertical, like it slants into the page then the corresponding shape will not be of a parallelpiped? As in it slants into the page, making an angle with AC.
     

    Attached Files:

  8. Aug 14, 2012 #7
    That's correct. The coordinate system need not be orthogonal.



    The parallelipiped is still a parallelpiped as long as the corresponding surface opposite ABCD slants the same way.
     
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