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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.

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

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