# Tensor product of vector with a tensor

I'm reading a paper describing transformation of gradient of a vector $$\mathbf u$$ (velocity vector) when I came across the following: $$\nabla \mathbf u = \mathbf q$$ after transformation is, $$\nabla_x \cdot (\mathbf u \otimes j \mathbf G^{-1}) = j\mathbf{q}$$ where, $$u \in \mathbb R^3$$, $$\mathbf G$$ is a $$3\times 3$$ matrix, j is a scalar (here, determinant of matrix $$G$$), $$\mathbf q$$ tensor or matrix (auxiliary variable role).

How do I go about expanding the tensor product?

To simplify notation, we will let $$\mathbf H$$ stand for $$j \mathbf G^{-1}$$. Then the components of the tensor product $$\mathbf u \otimes \mathbf H$$ are
$$\left(\mathbf u \otimes \mathbf H\right)^{ij}_k = \mathbf u^i \mathbf H^j_k$$
• If you write vectors $\mathbf v^k$ as column vectors then in $\mathbf H^j_k$ we have $j$ as the row index and $k$ as the column index. Then with Einstein summation convention we have $(\mathbf Hv)^j=\mathbf H^j_k \mathbf v^k$. – gandalf61 Mar 14 at 9:11