F
Contour Integration
We begin by recalling some relevant results from the calculus of real functions
of two real variables x and y, which we shall phrase in ‘physical’ terms. Consider a particle moving in the xy-plane subject to a force F = (P (x, y), Q(x, y))
whose x- and y-components P and Q vary throughout the plane. Suppose the
particle moves, under the action of the force, around a closed path C in the
xy-plane. Then the total work done by the force on the particle, W C , will be
given by the integral
∮
∮
W C =
F · dr =
P dx + Q dy
(F.1)
C
C
∮
where the
sign means that the integration path is closed. Using Stokes’
theorem, we can rewrite (F.1) as a surface integral
∫ ∫
W C =
curl F · dS
(F.2)
S
where S is any surface bounded by C (as a butterfly net is bounded by the rim).
Taking S to be the area in the xy-plane enclosed by C, we have dS = dx dy k
and
∫ ∫ (
)
∂Q ∂P
W C =
−
dx dy.
(F.3)
∂x
∂y
S
A mathematically special, but physically common, case is that in which F
is a ‘conservative force’, derivable from a potential function V (x, y) (in this
two-dimensional example) such that
∂V
∂V
P (x, y) = −
and
Q(x, y) = −
(F.4)
∂x
∂y
the minus signs being the usual convention. In that case, it is clear that
∂P
∂Q
=
(F.5)
∂y
∂x
and hence W C in (F.3) is zero. The condition (F.5) is, in fact, both necessary
and sufficient for W C = 0.
There can, however, be surprises. Consider, for example, the potential
V (x, y) = − tan
−1 y/x.
(F.6)
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