I know that if we have functions $f\colon \mathbb{R}^{3}\to \mathbb{R}$ and $\vec{g}\colon \mathbb{R}^{3}\to\mathbb{R}^{3}$ then we can write $$g_{1}f_{x} + g_{2}f_{y} + g_{3}f_{z} = \vec{g}\cdot\nabla f.$$ If we have another function $\vec{h}\colon \mathbb{R}^{3}\to\mathbb{R}^{3}$ is there any similar notation for rewriting $$g_{1}\nabla h_{1} + g_{2}\nabla h_{2} + g_{3}\nabla h_{3} \ ?$$

• Product rules have this term, if I remember correctly. – A---B Aug 30 '17 at 19:45

Yes, you can write it in terms of the tensor product: $[ (\nabla \otimes h)\cdot \vec{g}]_j = \sum_{i=1}^3 (\partial_j h_i)g_i$.
Sure, you could write this as $$g(\vec{x})^TDh$$ where $Dh$ is the total derivative of $h$, which will be the three by three matrix $$\begin{bmatrix}\frac{\partial h_1}{\partial x}&\frac{\partial h_1}{\partial y}&\frac{\partial h_1}{\partial z}\\ \frac{\partial h_2}{\partial x}&\frac{\partial h_2}{\partial y}&\frac{\partial h_2}{\partial z}\\ \frac{\partial h_3}{\partial x}&\frac{\partial h_3}{\partial y}& \frac{\partial h_3}{\partial z} \end{bmatrix}$$ and $g(\vec{x})^T$ is the vector $$\begin{bmatrix}g_1(\vec{x})&g_2(\vec{x})&g_3(\vec{x})\end{bmatrix}$$