I need the inverse coordinate form of the 3 dimensional Poisson's equation i:e transformation from the zeta,eta,zi plane to the x,y,z plane. While the Poisson equation in it's normal form can give me the transformation from the x,y,z plane to the zeta,eta,zi plane I also need the reverse to make my grid. I have been trying to solve it but it is so long that I have almost given up. The books I could find here in Japan are no help either. If someone has it could you kindly mail it to me at rajesh6167@hotmail.com. Would be thankful and help my research move forward as well. Thanks, Rajesh

It seems like you are involved in matrix manipulations for coordinate transformations.

The inverse of matrix is its transpose and such a matrix is called orthogonal matrix. From matrix algebra, the determinant of a product of two matrices is equal to the product of the determinants of the two matrices and is equal to the determinant of an identity matrix which is unity. The determinant of a transposed matrix is equal to the determinant of the matrix.

In fluid dynamics, for example Poission's equations are used to model non-linear dynamics problems such as visualizing the turbulent flows of waves, ripples of water in river, sea or ocean.