Piola transformation

Definition

Let F : R d → R d with F ( x ^ ) = B x ^ + b ,   B ∈ R d , d ,   b ∈ R d an affine transformation. Let K = F ( K ^ ) with K ^ a domain with Lipschitz boundary. The mapping

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