180 3 Real Analysis
The Gauss–Ostrogradsky (Divergence) Theorem.LetSbe a smooth, orientable sur-
face that encloses a solid regionVin space. If
−→
Fis a continuously differentiable vector
field onV, then
∫∫
S−→
F ·−→ndS=∫∫∫
Vdiv−→
FdV,where−→n is the outward unit normal vector to the surfaceS,dSis the area element on
the surface, anddVis the volume element inside ofV.
We recall that for a vector field−→
F =(F 1 ,F 2 ,F 3 ), the divergence isdiv−→
F =∇·
−→
F =
∂F 1
∂x+
∂F 2
∂y+
∂F 3
∂z,
while the curl is
curl−→
F =∇×
−→
F =
∣∣
∣∣
∣∣
∣
−→
i−→
j−→
k
∂
∂x∂
∂y∂
∂z
F 1 F 2 F 3∣∣
∣∣
∣∣
∣
=
(
∂F 3
∂y−
∂F 2
∂z)
−→
i +(
∂F 1
∂z−
∂F 3
∂x)
−→
j +(
∂F 2
∂x−
∂F 1
∂y)
−→
k.The quantity
∫∫
S−→
F·−→ndSis called the flux of−→
Facross the surfaceS.
Let us illustrate the use of these theorems with some examples. We start with an
encouraging problem whose solution is based on Stokes’ theorem.
Example.Compute
∮
Cydx+zdy+xdz,whereCis the circlex^2 +y^2 +z^2 =1,x+y+z=1, oriented counterclockwise when
seen from the positive side of thex-axis.
Solution.By Stokes’ theorem,
∮
Cydx+zdy+xdz=∫∫
Scurl−→
F ·−→ndS,whereS is the disk that the circle bounds. It is straightforward that curl
−→
F =
(− 1 ,− 1 ,− 1 ), while−→n, the normal vector to the planex+y+z=1, is equal to
(√^13 ,√^13 ,√^13 ). Therefore,