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Math Help - Change of coordinate system when integrating

  1. #1
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    Change of coordinate system when integrating

    Suppose we have two coordinate systems: (x_1, x_2, x_3) (representing the usual Cartesian coordinates, for example) and (X_1, X_2, X_3) (representing cylindrical polars, say). I think it's true that:

    \int f(x_1, x_2, x_3) dx_1 = \int \sum_{i=1}^3 f(x_1(X_1,X_2,X_3), x_2(X_1,X_2,X_3), x_3(X_1,X_2,X_3)) \frac{\partial x_1}{\partial X_i}  dX_i ,

    but I can't rigorously prove this. (It feels right based on the chain rule but I can't write a formal proof, and haven't found anything online.) Can anyone help please?

    On a similar note, for functions f_1, f_2, f_3 is it true that

     \int \sum_i \sum_j f_i(x_1, x_2, x_3) \frac{\partial f_i}{\partial x_j}   dx_j = \int \sum_i \sum_j f_i(x_1(X_1,X_2,X_3), x_2(X_1,X_2,X_3), x_3(X_1,X_2,X_3)) \frac{\partial f_i}{\partial X_j}   dX_j ?

    I'm less sure that this second formula holds in general.
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  2. #2
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    Re: Change of coordinate system when integrating

    Both are correct,
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