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

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

$${\displaystyle p:L^{2}({\hat {K}})^{d}\rightarrow L^{2}(K)^{d},\quad {\hat {q}}\mapsto p({\hat {q}})(x):={\frac {1}{|\det(B)|}}\cdot B{\hat {q}}({\hat {x}})}$$ is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant.^[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.^[2]

• Piola–Kirchhoff stress tensor
• Raviart–Thomas basis functions
• Raviart–Thomas Element

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