looking recently at the higher dimensional fokker planck equation enter image description here

i noticed it can be written pleasingly as $$ \partial_t p = - \nabla\cdot(p \underline{\mu}) + \nabla\cdot(\nabla\cdot(p \underline{\underline{D}})) $$ using the definition of the divergence of a tensor from https://www.wikiwand.com/en/Divergence (which also nicely avoids the question of which of the two conventions to take as the outcome is the same either way after the second div operation)

i'm used to thinking about the divergence of vector fields (in gravitational and fluid dynamics) but haven't directly previously though about divergences of matrices, or their divergences

is there a name for this second order divergence of a matrix, does it have intuitive prperties like the divergence of a vector field and does it occur in other places?


1 Answer 1


Let $\,A = pD\,$ then the rightmost term (in index notation) becomes $$ \def\n{\nabla} \def\p{\partial} \def\H{\n\n} \p_j\p_kA_{jk} $$ Using the Hessian operator $\H=\n\!\otimes\!\n$ and the double-dot product this becomes $$ \H:A $$ Be careful not to confuse $\H$ with the Laplacian operator $\,\Delta=\n\!\cdot\!\n$


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