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As I understand it, the relationship between the second order tensor $\bf T$ over a reference configuration and the same tensor in a deformed configuration $\bf T^\prime$ is given defined as follows:

$ {\bf T} := {\bf T^\prime} \, \textrm{cof} \, {\bf F} $

Where $\bf F$ is the deformation gradient. The above relation is presented as a definition without proof in every text that I've come across, but it looks like a relationship that ought to have a proof, or some intuition ought to be given for the relation.

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The Piola transform appears to be based on Nanson's formula for which there is a proof.

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