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.