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 welldefined function, due to the pathindependent 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) 