Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Jacobian in change of coordinates

  1. Aug 11, 2007 #1


    User Avatar
    Gold Member

    So in the change of coordinates equation for multiple integrals, we have the Jacobian which allows us to change our bounds of integration.

    It's the connection between multivariable calculus and linear algebra.

    is this possible since the linear transformation allows a one-to-one-mapping between one coordinate system and the other coordinate system? And is this precisely the reason why we can take the Jacobian of a multiple integral?
  2. jcsd
  3. Aug 11, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I don't understand your question, but I will just say that it might be insightful for you to know that if S is a region of R^n and T is a linear map from R^n to R^n, then the volume of T(S) equals the volume of S times the determinant of T (as a matrix). (And also that a linear map sends little cubes to little parallelepipedes)
  4. Aug 11, 2007 #3


    User Avatar
    Homework Helper

    hard to see what you are really asking but it is true that linear analysis has a lot to do with Jacobian and Integration. It comes from the realm of Measure theory where you get all those results for Lebesgue integral on R^n and product measures...
  5. Aug 11, 2007 #4


    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    in linear algebra a determinant measures the volume change under a linear map.

    in non linear calculus, integrating the (different) determinant functions at different points measures volume change under a non linear map.
    Last edited: Aug 11, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Jacobian in change of coordinates