Lecture Notes in Advanced Thermodynamics

2.2 New variable sets and functions
2.2.1 Change of variables
introduction of a new variable
– given a function f(x, y)
– the variable x is changed to z with the substitution x = x(z, y)
– the new function is
ˆf(y, z) := f ( x(y, z), z ) (9)
derivatives respect to the new variable
– (partial) differentiating (9) respect to z
– rearranging:
∂ z ˆf|y = ∂ x f| y · ∂ z x| y
∂ x f| y = ∂ z ˆf| y
∂ z x| y
(10)
– special case z = f
∂ x f| y = 1
∂ f x| y
derivatives respect to the remaining variable
– (partial) differentiating (9) respect to y:
– using (10):
– rearranging:
– special case z = f
∂ y ˆf|z = ∂ x f| y · ∂ y x| z + ∂ y f| x
∂ y ˆf|z = ∂ z ˆf| y
∂ z x| y
· ∂ y x| z + ∂ y f| x
∂ y f| x = ∂ y ˆf|z − ∂ yx| z
∂ z x| y
· ∂ z ˆf|y (11)
∂ y f| x = −∂ y x| f · ∂ x f| y
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