Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Sevcral Variables 103andaua udu -2 dxl+...+-'l-dx,, .- ax, ax,,In (2.44), du1, . . . , du, are arbitrary increments Au,, . . . , Arc,,, whereas in (2.45)they are functions of the arbitrary increments dxl(= Axl), . . . , d.r,(= Ax, ). However,we know from Section 2.8 that the relationships are the same no matter howwe interpret the differentials. We can write these equations in matrix form:If we eliminate the vector col (du . . . , du,) in these equations, we obtainand so on. From these equations we can read off ayl /axl. ay,/a.rz, . . . . Clearly, theresults are the same as (2.40) or (2.43). Thus (2.47) can be termed the general chainrule in differential form.The preceding development can be carried out even more concisely in terms ofthe notations of Section 2.7: The given functions are really vector functionsIf we take differentials, we obtainand henceso thatThis last equation is the same as (2.43); the previous one is the same as (2.47).We saw in Section 2.7 that a Jacobian matrix such as y, is the matrix of the lineartransformation approximating the given, in general nonlinear, mapping y = f(x).