gradient of tensor multiplied by vector, Laplacian of tensor multiplied by vector

Question

What identies can be applied to multiply out the terms in

$\nabla\left( T\cdot\mathbf{v}\right)$

and

$\nabla^{2}\left( T\cdot\mathbf{v}\right)$

where $T$ is a tensor and $\mathbf{v}$ a vector?

Thanks!

Answer 1

Assume $T$ is a second-rank tensor. We can proceed componentwise in an arbitrary Cartesian orthonormal system. By using Einstein notation, definitions and the product rule give \begin{aligned} \left[\nabla(T\cdot {\bf v})\right]_{ij} = (T_{ik}v_k)_{,j} = v_k T_{ik,j} + T_{ik}v_{k,j} = ({\bf v}^\top\!\cdot\nabla T^\top + T\cdot\nabla {\bf v})_{ij} \, . \end{aligned} One can proceed similarly for the other formula.