title: Piola transformation
text: # 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.
  
  