Piola transformation

The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola.
Definition.
Let formula_1 with formula_2 an affine transformation. Let formula_3 with formula_4 a domain with Lipschitz boundary. The mapping
formula_5
is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.
Note: for a more general definition in the context of tensors and elasticity, as well as a proof of the property that the Piola transform conserves the flux of tensor fields across boundaries, see Ciarlet's book.