Piola transformation

The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola. The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola. Let F : R d → R d {displaystyle F:mathbb {R} ^{d} ightarrow mathbb {R} ^{d}} with F ( x ^ ) = B x ^ + b ,   B ∈ R d , d ,   b ∈ R d {displaystyle F({hat {x}})=B{hat {x}}+b,~Bin mathbb {R} ^{d,d},~bin mathbb {R} ^{d}} an affine transformation. Let K = F ( K ^ ) {displaystyle K=F({hat {K}})} with K ^ {displaystyle {hat {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