Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 139To evaluate the expressions (d/dt)(az/ax) and (d/dt)(az/ay), one uses (2.130)again, this time applied to azlax and az/ay rather than to z.One thus finds the rule:This is a new chain rule.Similarly, if z = f (x, y), x = g(u, v), y = h(u, v), so that (2.34) holds, onehas-- az az ax az ay - -- + --,au ax au ay aua2z - az a2x- --+---+--+---- a (az) ax az a2y a (az) ayau2 ax au2 au ax au ayau2 au ay a ~ 'Applying (2.132) again, one finds(2.132)i *so thatIt should be remarked that (2.133) is a special case of (2.13 l), since v is treated asa constant throughout.Rules for a2z/au av, a2z/av2 and for higher derivatives can be formed, analogousto (2.13 1) and (2.133). These rules are of importance, but in most practicalcases it is better to use only the chain rules (2.33), (2.34), (2.35), applying repeatedlyif necessary. One reason for this is that simplifications are obtained if the derivativesoccurring are expressed in terms of the right variables, and a complete descriptionof all possible cases would be too involved to be useful.The variations possible can be illustrated by the following example, which concernsonly functions of one variable.Let y = f (x) and x = e'. Then