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22.3 Measures on Volumes and the Divergence Theorem

Suppose we have a volume V in three dimensions that has piecewise locally planar boundaries. (This means that its boundary can be broken up into a finite number of pieces each of which looks planar at small distances. We have defined an additive integral of a function f over it (and can do the same for a function over "hyper"-volume in any number of dimensions).)

Can we define an integral over its boundary that is additive on volumes?

The answer is yes.

The surface of V at any point at which the boundary is locally planar can be characterized by an "outward normal" direction. If we integrate the outward normal component of a vector v ⟶ over this surface, and join together adjacent volumes, what is the outward normal to one will be the inward normal to the other where they meet, and contributions from mutual parts of the boundary will cancel out.

This means that the integral of the outward normal of a continuous vector field v ⟶ over the boundary, δ V of the volume V will be additive on volumes. We write it as

∯ δ V ( v ⟶ · n ⟶ ) d S

and call it the outward "flux" of v ⟶ through the surface of V .

We can now address the same question we examined for areas in the last section: if we let V be an infinitesimal axis parallel rectangular volume, with sides d x , d y and d z , and lower corner at ( x , y , z ) , what will this integral be?

The rectangular volume V has six outer surfaces which comprise its front and back in each of the three axis directions. The outward normals point positively on the front faces and negatively on the back ones.

Exactly as in the two dimensional case, the contribution from the two faces normal to the x direction becomes a difference of v ⟶ between its values at x + d x and its value at x , integrated with respect to y and z variables on these faces.

This contribution, d y d z ( v x ( x + d x , y , z ) − v x ( x , y , z ) ) can be rewritten as d x d y d z ∂ v x ∂ x .

In an identical manner, the contributions from the faces normal to the y and z directions can be written as d x d y d z ∂ v y ∂ y and d x d y d z ∂ v z ∂ z respectively, and they sum together to form d x d y d z ( ∇ ⟶ · v ⟶ ) .

We find then, that for a small volume V as indicated, we have

∯ δ V ( v ⟶ · n ⟶ ) d S = ∭ V ( ∇ ⟶ · v ⟶ ) d τ

Again, the same result can be obtained for tilted parallelepiped regions or prisms, and again we can use additivity to deduce the same result for any volume with piecewise locally planar boundary and any piecewise continuous vector field defined on it.

This result is again extremely important. It is called the Divergence Theorem, and also is known as Gauss's Theorem.

It is more or less obvious that, with suitable definitions, there is a similar theorem in any higher dimension. Again it represents a higher dimensional version of the Fundamental Theorem of Calculus: the derivative in the divergence on the right can be integrated over, yielding differences at extreme points of the integration which convert the three dimensional integral on the right into a two dimensional one over the boundary of V on the left.

Exercises:

22.1 Consider the vector field E ⟶ defined by E ⟶ = ρ ⟶ ρ 3 .
Compute the integral of its flux through the surface of a sphere of radius R centered about the origin. Also compute its divergence. (We have done this earlier.) Can you explain this?

22.2 What is the integral of the flux of E ⟶ around a circle of radius 1 centered about the point ( 0 , 0 , 2 ) ?

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