# Simplify $c_i=a_j\frac{\partial b_j}{\partial x_i}$

Consider two 3-D vector fields $$\vec{a}$$ and $$\vec b$$. Then define a new vector field $$\vec c$$ as

$$c_i = a_j\frac{\partial b_j}{\partial x_i},$$

i.e.,

$$\vec c = \left( \vec a \cdot \frac{\partial\vec b}{\partial x}, \ \vec a \cdot \frac{\partial\vec b}{\partial y}, \ \vec a \cdot \frac{\partial\vec b}{\partial z} \right).$$

The question is: is there any way to simplify the definition of $$\vec c$$? I thought that

$$\vec c = \vec a \cdot \nabla \vec b,$$

but this is

$$c_i = a_j \frac{\partial b_i}{\partial x_j},$$

not the one what I expect.

• I think that $c = (\nabla b)a$ works, since $\nabla b$ is a $3\times 3$ matrix – whpowell96 Apr 21 at 1:46
• Using $(\nabla\vec b)_{ij} = \partial b_j/\partial x_i$ it works. Thanks! – Jeongu Kim Apr 21 at 2:34