Calc 3 Lecture Notes
Section 14.7
Page 1 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:
 N M 
R Mdx  Ndy  R  x  y  dA
For a vector field F  x, y   M  x, y  , N  x, y  , 0 with curl   F  0, 0, N x  M y and path
parameterized by r  x  t  , y  t  , 0 so that dr  dx, dy,0  Tds (Where T 
x ', y ', 0
r'
is the
unit tangent vector to the curve), Green’s theorem can be re-written as:
 F  dr   F  Tds    F   kdA
R
R
R
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  F  Nds     FdA (given that
R
N
y ',  x ', 0
r'
):
R
Calc 3 Lecture Notes
Section 14.7
Page 2 of 4
Theorem 7.1: The Divergence Theorem:
If F is a vector field whose components have continuous first partial derivatives over some
region Q  3 bounded by the closed surface Q with an exterior unit normal vector N, then
 F  NdS     FdV
Q
Q
Calc 3 Lecture Notes
Section 14.7
Page 3 of 4
Practice:
1. Show that the divergence theorem holds true for the hemispherical solid of radius 1 in the
vector field F  0,0,1 (this is the last example from section 14.6’s notes)
Calc 3 Lecture Notes
Section 14.7
Page 4 of 4
2. Show that the divergence theorem holds true for the hemispherical solid of radius 1 in the
vector field F  x, y, z . Interpret the non-zero answer in terms of a source for the field
lines.