Hi there,

SymPy currently has Integral.transform 
<http://docs.sympy.org/latest/modules/integrals/integrals.html#sympy.integrals.Integral.transform>
 
to change the variable for one-dimensional integrals.

This could be generalized to multivariate integrals by calculating the 
Jacobian matrix of the transformation and getting its determinant to 
determine the change of integration measure.

The main problem is about the integration region. Specifically two cases:

   1. case where the integration limits do not contain other integration 
   variables.
   2. case where the do

The second case is way more complicated than the first.


My question is whether there is a general algorithm to determine the shape 
of the integration region for multivariate integrals after a generic 
(diffeomorphic) coordinate transformation?

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