The Divergence Theorem expresses a basic conservation law of fundamental importance in mathematical modeling of physics:
where D is a domain in 3d with boundary B with outward unit normal,
- is a given flux vector depending on (x,y,z),
- is the outward unit normal to B,
- is the flux in the normal direction,
- is the divergence of q also denoted div(q) representing a flux source,
- ds is the element of surface area on B,
and (1) expresses that the totality of what is produced in D from the source (left) is equal to what flows through the boundary B (right), that is, nothing is lost = conservation.
In 1d (1) is simply one way of formulating the Fundamental Theorem of Calculus as
with here D the interval [a, b] and nx(b) = 1 and nx(a) = -1.
Let us prove (1) in the special case of 2d and q =(qx,0,0) in which case (1) reads
- . (3)
To prove (3) assume D is is given as the set of point (x,y) with f1(y) ≤ x ≤ f2(y) with a≤y≤b with two bounding curves x=f2(y) and x=f1(y). In this case by (2)
because dy = nx*ds for x=f2(y) and dy = -nx*ds for x=f1(y) (motivate!). We see that in (3) an integral of dqx/dx with respect to dx is expressed in terms of end-point values of qx on B as in (2). See Divergence Theorem Square.
Recall that Poisson’s equation results from with or as in Heat Equation 1d.
Notice this variant of the Divergence Theorem of particular significance for electromagnetics based on Maxwell’s equations:
is called the rotation of q denoted by rot(q) and measures “how much q rotates”. Compare Stokes Theorem Disc.