around an arbitrary closed path is zero. Note that Stokes’ theorem, with ∇× F= 0 ,
implies that the line integral around an arbitrary simple closed path is zero.
To show that statement 2 implies statement 1, let F = grad φ=∇φ. In this case
it is readily verified that
curl F =∇× F=∇×∇ φ=
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 ˆe 3
∂
∂x
∂
∂y
∂
∂z
∂φ
∂x
∂φ
∂y
∂φ
∂z
∣∣
∣∣
∣∣=^0. (9 .44)
The relation (9.41) establishes that in a conservative vector field the line integral
between any two points is independent of the path of integration. In this case one
can write ∫ P(x,y,z)
P 0 (x 0 ,y 0 ,z 0 )
F·dr =φ(x, y, z )−φ(x 0 , y 0 , z 0 ), (9 .45)
and this line integral is independent of the path of integration which joins the two
end points. The function φcan be evaluated from F by selecting the special path of
integration which is the piecewise smooth curve constructed from straight line seg-
ments parallel to the coordinate axes. This special path of integration is illustrated
in figure 9-3.
Figure 9-3. Straight line segments connecting end points of integration.
Along the sectionally continuous straight-line paths of integration illustrated in fig-
ure 9-3 the line integral (9.45) can be expressed in the component form as
∫P
P 0
F·dr =
∫P
P 0
F 1 (x, y, z )dx +F 2 (x, y, z)dy +F 3 (x, y, z )dz =φ(x, y, z)−φ(x 0 , y 0 , z 0 ).
