Calc 3 Lecture Notes Section 14.7 Page 3 of 4

Section 14.7: The Divergence Theorem

Big idea: The Divergence Theorem (or Gauss’s Theorem) is like Green’s Theorem in that it relates an integral of a vector field over the boundary of a geometric object to an integral of the divergence of the vector field over the interior of the geometric object.

Big skill: You should be able to use Green’s Theorem to go between surface integrals over the boundary of a closed surface and volume integrals of the divergence of the field.

Recall Green’s Theorem from section 14.4:

For a vector field with curl and path parameterized by so that (Where is the unit tangent vector to the curve), Green’s theorem can be re-written as:

But what happens if we take the dot product of F with the unit normal vector as we perform the line integral around the closed curve? Well, we can show (given that ):


Theorem 7.1: The Divergence Theorem:

If F is a vector field whose components have continuous first partial derivatives over some region bounded by the closed surface ¶Q with an exterior unit normal vector N, then


Practice:

  1. Show that the divergence theorem holds true for the hemispherical solid of radius 1 in the vector field (this is the last example from section 14.6’s notes)

  1. Show that the divergence theorem holds true for the hemispherical solid of radius 1 in the vector field . Interpret the non-zero answer in terms of a source for the field lines.