# What is meant by surface divergence of a vector function?

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?

• 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. – StephenG Apr 26 '19 at 14:26
• Sorry about that....Look here – N.G.Tyson Apr 26 '19 at 14:35