
572 Appendix 1
Each integral on the right-hand side requires for its execution more than knowledge of the limits.
In fact, the three integrals on the right-hand side are not completely dened because in the rst
integral, for example, we do not the know value of x
2
and x
3
in A
1
:
IAxx
P
P
1112
1
2
=
∫
(, ,)d .
What is needed is a statement, such as x
2
= f(x
1
), x
3
= g(x
1
), that species x
2
, x
3
for each value
of x
1
(so specify the path of integration). The integrand now reduces to A
1
(x
1
, x
2
, x
3
) = A
1
(x
1
, f(x
1
),
g(x
1
))= B
1
(x
1
) so that the integral I
1
becomes well dened. Therefore, in general, a line integral is
a path-dependent integral. However