Piola transformation

The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola.

Definition
Let $$ F: \mathbb{R}^d \rightarrow \mathbb{R}^d$$ with $$ F( \hat{x}) = B \hat{x} +b, ~ B \in \mathbb{R}^{d,d}, ~ b \in \mathbb{R}^{d} $$ an affine transformation. Let $$ K=F(\hat{K}) $$ with  $$ \hat{K} $$ a domain with Lipschitz boundary. The mapping

$$ p: L^2( \hat{K} )^d \rightarrow L^2(K)^d, \quad \hat{q} \mapsto p(\hat{q})(x) := \frac{1}{|\det(B)|} \cdot B \hat{q} (\hat{x}) $$ 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.