3.4 Gauss' theorem

Gauss' theorem

Consider the following volume enclosed by a surface we will call $S$ .

We will cut the the object into two volumes that are enclosed by surfaces we will call $S_1$ and $S_2$ .

$${\oint }_S\vec{F}\cdot d\vec{a}=\sum _{i=1}^n{\oint }_{S_i}\vec{F}\cdot d\overrightarrow{a_i}\text{.}$$ What is ${\oint }_{S_i}\vec{F}\cdot d\overrightarrow{a_i}$ .? We can subdivide the volume into a bunch of littlecubes:

Which is $$\Delta x\Delta y\Delta z\frac{\partial F_z}{\partial z}=\Delta V\frac{\partial F_z}{\partial z}$$

Likewise you get the same result in the other dimensionsHence $${\oint }_{S_i}\vec{F}\cdot d\overrightarrow{a_i}=\Delta V_i\left[\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\right]$$

or $${\oint }_{S_i}\vec{F}\cdot d\overrightarrow{a_i}=\vec{\nabla }\cdot \vec{F}\Delta V_i$$ $${\displaystyle \begin{array}{c}{\oint }_S\vec{F}\cdot d\vec{a}=\sum _{i=1}^n{\oint }_{S_i}\vec{F}\cdot d\overrightarrow{a_i}\\ =\sum _{i=1}^n\vec{\nabla }\cdot \vec{F}\Delta V_i\end{array}}$$

So in the limit that $\Delta V_i\to 0$ and $n\to \infty$ $${\oint }_S\vec{F}\cdot d\vec{a}={\oint }_V\vec{\nabla }\cdot \vec{F}dV$$

This result is intimately connected to the fundamental definition of the divergence which is $$\vec{\nabla }\cdot \vec{F}\equiv \underset{V\to 0}{lim}\frac{1}{V}{\oint }_S\vec{F}\cdot d\vec{a}$$ where the integral is taken over the surface enclosing the volume $V$ . The divergence is the flux out of a volume, per unit volume, in the limit ofan infinitely small volume. By our proof of Gauss' theorem, we have shown that the del operator acting on a vector field captures this definition.