132 Functions of Several VariablesWe say that F is a potential vector field with potential / if F = V/
for some scalar field f.UF = V/, then, by (3.1.9),F(r(t))-r'(t) = ftf(r(t)),
and the value of the integral (3.1.16) depends only on the starting and
ending points r(a), r(b) of the curve and not on the curve itself:Vf-dr = f(r(b))-f(r(a)). (3.1.18)In particular, §c V/ • dr = 0 for every simple closed curve C.
The alternative name for a potential vector field is conservative, which
comes from the physical interpretation of the line integral. If F is a force
acting on a point mass moving along C, then Jc F-dr is the work done by the
force along the curve C. By the Second Law of Newton, F(r(t)) = mr "(t),
so that F(r(t)) • r'(t) = (l/2)md\r'(t)\^2 /dt = d£K(t)/dt, where £K is
the kinetic energy; verify this. For a potential force field F, we write
F = -VV, where V is a scalar function, called the potential energy.
Then
J F-dr = £K(b) - SK(a) = V(r(a)) - V(r(b)),so that £K{O) + V(r(a)) — £*:(&) + V(r(b)), which means that the total
energy £# + V is conserved as the point mass moves along C.
In general, we say that a vector field F has the path independence
property in a domain G if §c F • dr = 0 for every simple closed piece-wise
smooth curve C in G. The following result should be familiar from a course
in multi-variable calculus.
Theorem 3.1.1 A continuous vector field has the path independence
property in a domain G if and only if the field has a potential.
EXERCISE 3.1.13. c Prove the above theorem. Hint: in one direction, use
(3.1.18). In another direction, verify that, as stated, the path independence prop-
erty implies that, for every two points A,B inG the value of the integral fcF-dr
does not depend on the particular choice of the curve C in G connecting the
points A and B. Then fix a point A in G and, for each point P in G define
f{P) = fcF • dr, where C is a simple piece-wise smooth curve starting at A and
ending at P. Path independence makes f a well-defined function of P. Verify
that V/ = F.
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