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As the title says, is it merely a convenience to write the divergence as a dot product? Is there an intuition on the relationship between the geometric interpretation of the divergence and that of the dot product?

I ask this also because the cross product is closely related to rotations, and the symbol is used for calculating the curl which is also related to rotations.

I tried to find a connection but I guess I need help on this one.

João Pedro
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  • To be clear, you are talking about $\nabla \cdot v$? You can imagine $\nabla$ as $[\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}]$ and $v$ as $[v_x,v_y,v_z]$. If you dot product these you get $\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}$ which is how divergence is defined. – JMoravitz Apr 26 '23 at 03:39
  • See my answer here. For any $L_x(v)$ linear in $v$ we can define $L_x(\nabla)$, and this can be interpreted as the average change of $L_x(v)$ in a small neighborhood of $x$ and all directions $v$. The divergence of $f(x)$ is the case $L_x(v) = v\cdot f(x)$. – Nicholas Todoroff Apr 28 '23 at 04:18

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