Differentiation of integrals

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 + P V , and the Gibbs free energy, G = U + P V − 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)
= f(x, t).
∂t
Assuming that the second partial derivatives of F(x, t) are continuous, we have
∂2 F(x, t)
∂ 2 F(x, t)
=
,
∂t∂x
∂x∂t
and so we can write
∂ ∂F(x, t)
∂ ∂F(x, t)
∂f(x, t)
.
=
=
∂t
∂x
∂x
∂t
∂x
Integrating this equation with respect to t then gives
∂f(x, t)
∂F(x, t)
=
dt.
∂x
∂x
Now consider the definite integral
I(x) =
(5.46)
t=v
f(x, t) dt
t=u
= F(x, v) − F(x, u),
where u and v are constants. Differentiating this integral with respect to x, and
using (5.46), we see that
∂F(x, v) ∂F(x, u)
dI(x)
=
−
dx
∂x
v∂x
u
∂f(x, t)
∂f(x, t)
dt −
dt
=
∂x
∂x
v
∂f(x, t)
=
dt.
∂x
u
This is Leibnitz’ rule for differentiating integrals, and basically it states that for
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