Modern Thermodynamics
- Chapter 1
42
Appendix 1.1: Partial Derivatives
Derivatives of many variables
When a variable such as energy U(T,V, Nk) is a function of many variables V, T and Nk, its
partial derivative with respect to each variables is defined by holding all other variables constant. Thus,
for example, if
U(T,V,N)
=
5
2
NRT a
N2
V
then the partial derivatives are:
∂U
∂T


V,N
=
5
2
NR
(A1.1.1)
∂U
∂NV,T


=
5
2
RT a
2N
V
(A1.1.2)
∂U
∂VN,T


= a
N2
V2
(A1.1.3)
The subscripts indicate the variables that are held constant during the differentiation. In cases where the
variables being held constant are understood, the subscripts are often dropped. The change in U, i.e. the
differential dU, due to changes in N, V and T is given by:
dU =
∂U
∂T


V,N
dT +
∂U
∂VT,N


dV +
∂U
∂NV,T


dN
(A1.1.4)
For functions of many variables, there is a second derivative corresponding to every pair of variables:
∂2U
∂T∂V
,
∂2U
∂N∂V
,
∂2U
∂T2
.. etc. For the "cross derivatives" such as
∂2U
∂T∂V
that are derivatives with respect
to two different variables, the order of differentiation does not matter. That is
∂2U
∂T∂V
=
∂2U
∂V∂T
(A1.1.5)
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