In Section 6.7 we determined that the divergence of the gravitational field in free space vanishes: (∇ · g) = 0. This was sufficient to determine the gravitational field in N dimensions as expression (6.49). However, that expression is not quite satisfactory, because it contains an unknown constant A. In fact, at this point we have no idea how this constant is related to the mass M that causes the gravitational field! The reason for this is simple: to derive the gravitational field in (6.49) we have only used the field equation (6.45) for free space (where ρ = 0). However, if we want to find the relation between the mass and the resulting gravitational field, we must also use the field equation (∇ · g) = −4πGρ at places where mass is present. More specifically, we have to integrate the field equation in order to find the total effect of the mass. The theorem of Gauss gives us an expression for the volume integral of the divergence of a vector field.

8.1 Statement of Gauss’ theorem

Section 6.2 introduced the divergence as the flux per unit volume. In fact, expression (6.8) gives us the outward flux dΦv through an infinitesimal volume dV: dΦv = (∇ · v)dV. We can integrate this expression to find the total flux through the surface S, which encloses the total volume V:

∲Sv · dS = ∫V(∇ · v) dV (in three dimensions)

In deriving this, equation (6.8) has been used to express the total flux on the left-hand side. This expression is called the theorem of Gauss, or the divergence theorem.

We did not use the dimensionality of the space to derive (8.1); the relation holds in any number of dimensions. You may recognize the one-dimensional version of (8.1). In one dimension the vector v has only one component vx; hence, (∇ · v) = δxvx.