Conservative fields
Consider a vector field
. In general, the line integral
depends on the path
taken between the end points,
and
.
However, for some special vector fields the integral is path independent. Such fields
are called conservative fields. It can be shown that if
is a
conservative field then
for some scalar field
.
The proof of this is straightforward. Keeping
fixed, we have
 |
(A.82) |
where
is a well-defined function, due to the path-independent nature of the
line integral. Consider moving the position of the end point by an infinitesimal
amount
in the
-direction. We have
 |
(A.83) |
Hence,
 |
(A.84) |
with analogous relations for the other components of
. It follows that
 |
(A.85) |