9.8 Line integrals 279
the graphical representation of z(x, y)discussed in Section 9.2, the line integral is
the change in ‘height’ above the xy-plane on the representative surface of z, and this
cannot depend on the path between the end points.
EXAMPLE 9.26Independence of path for an exact differential
Volume is a function of pressure and temperature, V 1 = 1 V(p, T), and the total
differential volume is (see Example 9.11 for fixed amount of substance)
As in Example 9.25, we consider the pathsC
11 + 1 C
2andC
31 + 1 C
4shown in Figure 9.10.
(i) PathC 1 = 1 C
11 + 1 C
2:
The change in volume along pathC
1is at constant pressurep 1 = 1 p
1(dp 1 = 1 0), so that
PathC
2is at constantT 1 = 1 T
2(dT 1 = 1 0),and
Therefore
(ii) PathC 1 = 1 C
31 + 1 C
4. PathC
3is at constantp
2and pathC
4is at constantT 1 = 1 T
1.
Then, as above,
Therefore∆V
B→A1 + 1 ∆V
A→B1 = 10 and the change in volume around the closed path
is zero.*
0 Exercises 62, 63
=−VpT Vp T(,) (, )
11 2 2∆= −
+−
→V Vp T Vp T VpT Vp
BA(,)(,) (,)(
21 22 11 2,, )T
1
=−Vp T VpT(,) (,)
22 11∆= −
+−
→VVpTVpTVpTVp
AB(,)(,) (,)(
12 11 22 1,, )T
2
∆=
∂
∂
=
∂
∂
=
=
V
V
p
dp
V
p
dp
T
p
p
TT
21
2
2
ZZ
C
2VVpT Vp T VpT
p
p
(, ) ( , ) ( , )
222121
2
=−
∆=
∂
∂
=
∂
∂
=
=
V
V
T
dT
V
T
dT
p
T
T
pp
11
2
1
ZZ
C
1
VVpT VpT VpT
T
T
(,) (,) (,)
112111
2
=−
dV
V
T
dT
V
p
dp
pT
=
∂
∂
∂
∂
*We note however that the line integral (9.48) around a closed path C is zero in general only if the differential
Fdx 1 + 1 Gdyis exact at all points on and withinthe closed path; further consideration of this point is beyond the
scope of this book.