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Consider the following volume enclosed by a surface we will call $S$ .
Now we will embed $S$ in a vector field:
We will cut the the object into two volumes that are enclosed by surfaces we will call ${S}_{1}$ and ${S}_{2}$ .
Again we embed it in the same vectorfield. It is clear that flux through ${S}_{1}$ + ${S}_{2}$ is equal to flux through $S\text{.}$ This is because the flux through one side of the plane is exactly opposite to theflux through the other side of the plane: So we see that $${\oint}_{S}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{a}={\oint}_{{S}_{1}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{1}}+{\oint}_{{S}_{2}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{2}}\text{.}$$ We could subdivide the surface as much as we want and so for $n$ subdivisions the integral becomes:$${\oint}_{S}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{a}=\sum _{i=1}^{n}{\oint}_{{S}_{i}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{i}}\text{.}$$ What is ${\oint}_{{S}_{i}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{i}}$ .? We can subdivide the volume into a bunch of littlecubes:
To first order (which is all that matters since we will take the limit of a smallvolume) the field at a point at the bottom of the box is $${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}$$ where we have assumed the middle of the bottom of the box is the point $(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z)$ . Through the top of the box $(x+\frac{\Delta x}{2},y+\frac{\Delta y}{2},z+\Delta z)$ you get $${F}_{z}+\frac{\Delta x}{2}\frac{\partial {F}_{z}}{\partial x}+\frac{\Delta y}{2}\frac{\partial {F}_{z}}{\partial y}+\Delta z\frac{\partial {F}_{z}}{\partial z}$$ Through the top and bottom surfaces you get Flux Top - Flux bottom $$$$Which is $$\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}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{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}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{i}}=\stackrel{\u20d7}{\nabla}\cdot \stackrel{\u20d7}{F}\Delta {V}_{i}$$ $$\begin{array}{c}{\oint}_{S}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{a}=\sum _{i=1}^{n}{\oint}_{{S}_{i}}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{{a}_{i}}\\ =\sum _{i=1}^{n}\stackrel{\u20d7}{\nabla}\cdot \stackrel{\u20d7}{F}\Delta {V}_{i}\end{array}$$
So in the limit that $\Delta {V}_{i}\to 0$ and $n\to \infty $ $${\oint}_{S}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{a}={\oint}_{V}\stackrel{\u20d7}{\nabla}\cdot \stackrel{\u20d7}{F}dV$$
This result is intimately connected to the fundamental definition of the divergence which is $$\stackrel{\u20d7}{\nabla}\cdot \stackrel{\u20d7}{F}\equiv \underset{V\to 0}{{lim}}\frac{1}{V}{\oint}_{S}\stackrel{\u20d7}{F}\cdot d\stackrel{\u20d7}{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.
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