My book says:

If there is a surface discontinuity in a vector field $\vec{E}$, we enclose it in a thin transitional layer (of width $h$) and apply divergence theorem. If $\hat{n}_1$ and $\hat{n}_2$ are outward normal vectors to the surface:

$$\lim_\limits{h \to 0} \int_V \nabla \cdot \vec{E}\ dV = \oint_S (\vec{E}_1.\hat{n}_1 + \vec{E}_2.\hat{n}_2)\ dS = \oint_S \text{divs}\ \vec{E}\ dS$$

I do understand that the book calls (or defines):

$\text{divs}\ \vec{E}=\vec{E}_1.\hat{n}_1 + \vec{E}_2.\hat{n}_2$

What I don't understand is the relation between $(\text{divs}\ \vec{v}=\vec{v}_1.\hat{n}_1 + \vec{v}_2.\hat{n}_2)$ and $(\text{div}\ \vec{v})$. Why are they both called the divergence?

  • $\begingroup$ Typically we expect people to give the book title, author and page and equation reference numbers where possible. It won't hurt and it can help sometimes. $\endgroup$ – StephenG Apr 26 '19 at 14:26
  • 1
    $\begingroup$ Sorry about that....Look here $\endgroup$ – N.G.Tyson Apr 26 '19 at 14:35

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