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Consider the continuously differentiable vector field in ${\mathbb R}^3$: $$ F:{\mathbb R}^3\to{\mathbb R}^3,\qquad F(x,y,z)=(U,V,W) $$ where $$ U,V,W:{\mathbb R}^3\to{\mathbb R}. $$ According to the wikipedia article for divergence, the divergence of the vector field $F$ at the point $p\in {\mathbb R}^3$ is defined as $$ \text{div} F(p):=\lim_{V\to\{p\}}\iint\limits_{S(V)}\frac{F\cdot n}{|V|}dS. $$ This formula is of much clearer meaning in physics. Here is my question:

Is it because the divergence theorem that people call $\nabla \cdot F$ divergence? Or is there any direct calculation in physics that leads to the quantity $\nabla\cdot F$?

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I'm not sure about the reason for the name, but physically the divergence should tell you the rate at which the density enters or leaves a given space – Deven Ware Sep 14 '11 at 15:58
up vote 4 down vote accepted

Look up "Divergence" at

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