The Divergence Theorem

This is a second, very, very important statement of the Fundamental Theorem:

$$\int_{\mathcal{V}/S} (\deldot \vV) d\tau = \oint_S \vV \cdot \hn dA$$

In this expression $\mathcal{V}/S$ should be read in your mind as ``over the open volume $\mathcal{V}$ bounded by the closed surface $S$ '', and $\vV$ is an arbitrary vector quantity, typically a vector field like $\vE$ or $\vB$ or a vector current density such as $\vJ$ . Note well that the right hand side you should be reading as ``the flux of the vector function $\vV$ out through the closed surface S''.

You might also see this written as:

$$\int_\mathcal{V} (\deldot \vV) d\tau = \oint_{\partial \mathcal{V}} \vV \cdot \hn dA$$

where $\partial \mathcal{V}$ is read as ``the surface bounding the volume $\mathcal{V}$ ''. This is slightly more compact notation, but a student can easily be confused by what appears to be a partial differential in the surface limits.

A simple consequence of the divergence theorem is:

$$\int_{\mathcal{V}/S} \grad f d\tau = \oint_S f\ \hn dA = \oint_S f\ d\va$$

Proof: Assume

$$\vA = f\hc$$

then

$$\deldot \vA = (\hc \cdot \grad) f + f (\deldot \hc) = (\hc \cdot \grad) f$$

so that

$$\int_{\mathcal{V}/S} \deldot \vA d\tau = \int_{\mathcal{V}/S} (\hc\cdot \grad)f d\tau = \oint_s \vA \cdot \hn dA = \oint_s \hc f \cdot \hn dA$$

Since $\hc$ is constant and arbitrary, we can factor it out from the integral:

$$\hc \cdot \int_{\mathcal{V}/S} \grad f d\tau = \hc \cdot \oint_s f \hn dA$$

Since this has to be true for any nonzero $\hc$ , we can essentially divide out the constant and conclude that:

$$\int_{\mathcal{V}/S} \grad f d\tau = \oint_s f \hn dA$$

Q.E.D.

You should prove on your own (using exactly the same sort of reasoning) that:

$$\int_{\mathcal{V}/S} \curl \vA\ d\tau = \oint_s \hn \times \vA\ dA$$

There thus is one such theorem for $\grad$ (acting on any scalar $f$ ), $\deldot \vA$ (acting on any vector function $\vA$ ) or $\curl \vA$ acting on any vector function $\vA$ . We can use all of these forms in integration by parts.