Gauss' Divergence Theorem

Gauss’ Divergence Theorem is an integral theorem that relates the flux integral of a vector field onto a surface to the triple integral of the solid region enclosed. It is a generalisation of the simple divergence theorem to three dimensions.

%%🖋 Edit in Excalidraw, and the dark exported image%%

Definition §

Gauss' Divergence Theorem ^theorem

Let $\Omega$ be a solid region in ${\mathbb{R}}^3$ bounded by a smooth (potentially piecewise), closed, orientable surface $S=\partial \Omega$. Let $\vec{F}$ be a $C^1$ vector field on an open region containing $\Omega$ (that is, the domain of the vector field is a superset of $\Omega$)

Gauss’ Divergence Theorem states:

$${\iint }_S\vec{F}\cdot \text{d}\vec{S}={\iiint }_{\Omega }\nabla \cdot \vec{F}\mathrm{d}V$$

$${\iint }_S\vec{F}\cdot \hat{n}\text{d}S={\iiint }_{\Omega }\nabla \cdot \vec{F}\mathrm{d}x\mathrm{dy}\mathrm{dz}$$

Finding Volume §

One of the simplest applications for the theorem is finding the volume of complex solid regions , assuming that their wrapping surface is easy to calculate.

If we take $\vec{F}=(x,y,z)$ (such that $\Vert F\Vert =\sqrt{x^2+y^2+z^2}$ being the distance from origin), then:

$${\iiint }_{\Omega }\nabla \cdot \vec{F}\mathrm{d}V={\iiint }_{\Omega }1+1+1\text{d}V=3{\iiint }_{\Omega }1\text{d}V=3\times \text{Volume of Ω}$$

So then:

$$V_{\Omega }=\frac{1}{3}{\iint }_S\vec{F}\cdot \text{d}\vec{S}$$

Applications §

• Gauss’ Law