
318 The Principles of Thermodynamics
It is difficult to establish this directly from the definition of the Jacobian given above.
If the independent variables are thought of as coordinates of a two-dimensional man-
ifold, the invariant area element is given by dA =
√
g dxdy where g is the determinant
of the inverse metric on the manifold. Under changes of coordinates from (x,y) to,
say (a,b), this changes according to
g(x,y)=
∂
(x,y)
∂
(a,b)
−2
g
(a,b) (16.18)
The area dA being a geometrical quantity does not depend on the choice of coordi-
nates and one gets the important relation
dxdy =
∂
(x,y)
∂
(a,b)
·dadb (16.19)
Considering the sequence of transformations (