By Stokes’ theorem
∫∫
S 1
F·dS 1 =
∫∫
S 1
(∇× V)·dS 1 =
∫
C
©V ·dr
and ∫∫
S 2
F·dS 2 =
∫∫
S 2
(∇× V)·dS 2 =
∫
C
©− V·dr,
where the negative sign is due to the relative directions associated with the line
integrals relative to the normals ˆen 1 and ˆen 2 to the respective surfaces S 1 and S 2.
That is, Stokes theorem requires the line integral around the closed curve Cbe in
the positive direction with respect to the normal on the surface. When the above
integrals are added, the result is the net flux through an arbitrary closed surface is
zero.
Two-dimensional Conservative Vector Fields
If corresponding to each point (x, y ) in a region R of the plane z = 0 , there
corresponds a vector
F =F(x, y ) = M(x, y )ˆe 1 +N(x, y )ˆe 2 , (9 .62)
a vector field is said to exist in the region. Further, this field is said to be conservative
if a scalar function of position φ(x, y )exists such that
grad φ=∂φ
∂x
eˆ 1 +∂φ
∂y
ˆe 2 =M(x, y )ˆe 1 +N(x, y )ˆe 2 =F . (9 .63)
The scalar function φis called a potential function for the vector field F . (Again,
note that sometimes F =−grad φis more convenient to use.) The vector F is also
referred to as an irrotational vector field and is derivable from the scalar potential
function φwhich satisfies
∂φ
∂x
=M and ∂φ
∂y
=N.
Differentiating these relations produces
∂^2 φ
∂x ∂y =
∂M
∂y =
∂^2 φ
∂y ∂x =
∂N
∂x (9 .64)
so that a necessary condition that F=Mˆe 1 +Nˆe 2 be a conservative field is that
∂M
∂y
=∂N
∂x
.
An equivalent statement is that curl F = 0.
