Session 13

Divergence Theorem

The Divergence Theorem expresses a basic conservation law of fundamental importance in mathematical modeling of physics:

  • \int_D \nabla\cdot q\,dxdydz =\int_B n\cdot q\, ds   (1)

where D is a domain in 3d with boundary B with outward unit normal,

  • q = (qx,qy,qz) is a given flux vector depending on (x,y,z),
  • n = (nx,ny,nz) is the outward unit normal to B,
  • n\cdot q= nx*qx + ny*qy + nz*qz is the flux in the normal direction,
  • \nabla\cdot q = \frac{\partial qx}{\partial x}+ \frac{\partial qy}{\partial y}+\frac{\partial qz}{\partial z} 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

  • \int_a^b \frac{dq}{dx}dx = q(b) - q(a) = nx(b)*q(b) +nx(a)*q(a)  (2)

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

  • \int_D \frac{\partial qx}{\partial x}dxdy =\int_B qx*nx*ds.  (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)

  • \int_D \frac{\partial qx}{\partial x}dxdy=\int_a^b qx(f2(y),y)*dy-\int_a^b qx(f1(y),y)*dy=\int_B qx*nx*ds,

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 -\Delta u =f results from \nabla q = f with q=-\nabla u or \Delta u = div(grad (u)) as in Heat Equation 1d.

Notice this variant of the Divergence Theorem of particular significance for  electromagnetics based on Maxwell’s equations:

  • \int_D \nabla\times q\,dxdydz =\int_B n\times q\, ds   (4)

with

  • \nabla\times q =(\frac{\partial qz}{\partial y}-\frac{\partial qy}{\partial z},-\frac{\partial qz}{\partial x}+\frac{\partial qx}{\partial z},\frac{\partial qy}{\partial x}-\frac{\partial qx}{\partial y})
  • n\times q = (ny*qz - nz*qy, -nx*qz+nz*qx, nx*qy-ny*qx).

\nabla\times q is called the rotation of q denoted by rot(q) and measures “how much q rotates”. Compare Stokes Theorem Disc.