Mathematical Methods for Physics and Engineering - Matematica.NET

PARTIAL DIFFERENTIATIONAlthough the Helmholtz potential has other uses, in this context it has simplyprovided a means for a quick derivation of the Maxwell relation. The otherMaxwell relations can be derived similarly by using two other potentials, theenthalpy, H = U + PV, and the Gibbs free energy, G = U + PV − ST (seeexercise 5.25).5.12 Differentiation of integralsWe conclude this chapter with a discussion of the differentiation of integrals. Letus 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).∂tAssuming that the second partial derivatives of F(x, t) are continuous, we have∂ 2 F(x, t)∂t∂x= ∂2 F(x, t),∂x∂tand 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. (5.46)∂x ∂xNow consider the definite integralI(x) =∫ t=vt=uf(x, t) dt= F(x, v) − F(x, u),where u and v are constants. Differentiating this integral with respect to x, andusing (5.46), we see thatdI(x) ∂F(x, v)=dx ∂x=u∂F(x, u)∂x−∫ v ∫∂f(x, t) u∂x dt − ∂f(x, t)∂xdt∫ v∂f(x, t)=∂xdt.This is Leibnitz’ rule for differentiating integrals, and basically it states that for178