6.8 The divergence theorem

We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study. The divergence theorem has many uses in physics; in particular, the divergence theorem is used in the field of partial differential equations to derive equations modeling heat flow and conservation of mass. We use the theorem to calculate flux integrals and apply it to electrostatic fields.

Overview of theorems

Before examining the divergence theorem, it is helpful to begin with an overview of the versions of the Fundamental Theorem of Calculus we have discussed:

1. The Fundamental Theorem of Calculus :
   
   ${\displaystyle {\int }_a^b{f}^{\prime }\left(x\right)dx=f\left(b\right)-f\left(a\right)}.$
   
   This theorem relates the integral of derivative $f^{\prime }$ over line segment $\left[a,b\right]$ along the x -axis to a difference of $f$ evaluated on the boundary.
2. The Fundamental Theorem for Line Integrals :
   
   ${\displaystyle {\int }_C\nabla f·d\text{r}=f\left(P_1\right)-f\left(P_0\right)},$
   
   where $P_0$ is the initial point of C and $P_1$ is the terminal point of C . The Fundamental Theorem for Line Integrals    allows path C to be a path in a plane or in space, not just a line segment on the x -axis. If we think of the gradient as a derivative, then this theorem relates an integral of derivative $\nabla f$ over path C to a difference of $f$ evaluated on the boundary of C .
3. Green’s theorem, circulation form :
   
   ${\displaystyle {\iint }_D(Q_x-P_y)dA}={\displaystyle {\int }_C\text{F}·d\text{r}}.$
   
   Since $Q_x-P_y=\text{curl}\text{F}·\text{k}$ and curl is a derivative of sorts, Green’s theorem    relates the integral of derivative curl F over planar region D to an integral of F over the boundary of D .
4. Green’s theorem, flux form :
   
   ${\displaystyle {\iint }_D(P_x+Q_y)dA}={\displaystyle {\int }_C\text{F}·\text{N}ds}.$
   
   Since $P_x+Q_y=\text{div}\text{F}$ and divergence is a derivative of sorts, the flux form of Green’s theorem relates the integral of derivative div F over planar region D to an integral of F over the boundary of D .
5. Stokes’ theorem :
   
   ${\displaystyle {\iint }_S\text{curl}\text{F}·d\text{S}}={\displaystyle {\int }_C\text{F}·d\text{r}}.$
   
   If we think of the curl as a derivative of sorts, then Stokes’ theorem    relates the integral of derivative curl F over surface S (not necessarily planar) to an integral of F over the boundary of S .

Stating the divergence theorem

The divergence theorem follows the general pattern of these other theorems. If we think of divergence as a derivative of sorts, then the divergence theorem    relates a triple integral of derivative div F over a solid to a flux integral of F over the boundary of the solid. More specifically, the divergence theorem relates a flux integral of vector field F over a closed surface S to a triple integral of the divergence of F over the solid enclosed by S .