Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PARTIAL DIFFERENTIATION

Although the Helmholtz potential has other uses, in this context it has simply

provided a means for a quick derivation of the Maxwell relation. The other

Maxwell relations can be derived similarly by using two other potentials, the

enthalpy,H=U+PV,andtheGibbs free energy,G=U+PV−ST(see

exercise 5.25).

5.12 Differentiation of integrals

We conclude this chapter with a discussion of the differentiation of integrals. Let

us consider the indefinite integral (cf. equation (2.30))

F(x, t)=

∫

f(x, t)dt,

from which it follows immediately that

∂F(x, t)

∂t

=f(x, t).

Assuming that the second partial derivatives ofF(x, t) are continuous, we have

∂^2 F(x, t)

∂t∂x

=

∂^2 F(x, t)

∂x∂t

,

and so we can write

∂

∂t

[

∂F(x, t)

∂x

]

=

∂

∂x

[

∂F(x, t)

∂t

]

=

∂f(x, t)

∂x

.

Integrating this equation with respect totthen gives

∂F(x, t)

∂x

=

∫

∂f(x, t)

∂x

dt. (5.46)

Now consider the definite integral

I(x)=

∫t=v

t=u

f(x, t)dt

=F(x, v)−F(x, u),

whereuandvare constants. Differentiating this integral with respect tox,and

using (5.46), we see that

dI(x)

dx

=

∂F(x, v)

∂x

−

∂F(x, u)

∂x

=

∫v

∂f(x, t)

∂x

dt−

∫u

∂f(x, t)

∂x

dt

=

∫v

u

∂f(x, t)

∂x

dt.

This isLeibnitz’ rulefor differentiating integrals, and basically it states that for