Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

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