The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics.
It is named after Gabrio Piola.
d
with
an affine transformation.
a domain with Lipschitz boundary.
The mapping
det (
is called Piola transformation.
The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.
[1] 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.
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