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I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after having done the computations. The topic is a velocity induced by a dipole of forces in a Stokes flow, and given $U_S (r,k)$ the velocity induced at position $r$ by a point force placed in the origin with orientation $k$ the velocity induced by a certain dipole is

$$ - (k \cdot \nabla ) U_S (r,k) .$$

Now, how should I intepret $(k \cdot \nabla ) U$? I would see $k$ as a column vector, $\nabla$ a row vector, the product is then a square matrix and $(k \cdot \nabla ) U_S$ remains a vector. This is the only way in which the above expression makes sense to me, but I wanted to be sure.

Qmechanic
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tommy1996q
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1 Answers1

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Let's work in 3 dimenstions.

$k \cdot\nabla = k_{x} \partial_{x} + k_{y} \partial_{y} + k_{z} \partial_{z}$.

Then $(k \cdot \nabla) v$ has three components: $k_{x} \partial_{x} v_{x} + k_{y} \partial_{y} v_{x} + k_{z} \partial_{z} v_{x}$, $k_{x} \partial_{x} v_{y} + k_{y} \partial_{y} v_{y} + k_{z} \partial_{z} v_{y}$ and $k_{x} \partial_{x} v_{z} + k_{y} \partial_{y} v_{z} + k_{z} \partial_{z} v_{z}$.

saad
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