Advanced Calculus of Several Variables (1973)

Part III. Successive Approximations and Implicit Functions

Chapter 5. HIGHER DERIVATIVES

Thus far in this chapter our attention has been confined to mappings, or to first derivatives. This is a brief exercise section dealing with higher derivatives.

Recall (from Section II.7) that the function f : U → , where U is an open subset of ⁿ, is of class if all partial derivatives of f, of order at most k, exist and are continuous on U. Equivalently, f is of class if and only if fand its first-order partial derivatives D₁ f, . . . , D_nf are all of class on U. This inductive form of the definition is useful for inductive proofs.

We say that a mapping F : U → ^m is of class if each of its component functions is of class .

Exercise 5.1 If f and g are functions of class on ⁿ, and , show that the functions f + g, fg, and af are also of class .

The following exercise gives the class chain rule.

Exercise 5.2 If f : ⁿ → ^m and g : ^m → ^l are class mappings, prove that the composition g f : ⁿ → ^l is of class . Use the previous exercise and the matrix form of the ordinary (class ) chain rule.

Recall that we introduced in Section 2 the space _mn of m × n matrices, and showed that the differentiable mapping f : ⁿ → ^m is if and only if its derivative f′ : ⁿ → _mn is continuous.

Exercise 5.3 Show that the differentiable mapping f : ⁿ → ^m is of class on the open set U if and only if f′ : ⁿ → _mn is of class on U.

Exercise 5.4 Denote by _nn the open subset of _nn consisting of all non-singular n × n matrices, and by : _nn → _nn the inversion mapping, (A) = A⁻¹. Show that is of class for every positive integer k. This is simply a matter of seeing that the elements of A⁻¹ are functions of those of A.

We can now establish versions of the inverse and implicit mapping theorems.

Theorem 5.1If, in the inverse mapping theorem, the mapping f is of class in a neighborhood of a, then the local inverse g is of class in a neighborhood of b = f(a).

Theorem 5.2If, in the implicit mapping theorem, the mapping G is of class in a neighborhood of (a, b), then the implicitly defined mapping h is of class in a neighborhood of a.

It suffices to prove Theorem 5.1, since 5.2 follows from it, just as in the case. Recall the formula

from the last paragraph of the proof of Theorem 3.3 (the inverse mapping theorem). We already know that g is of class . If we assume inductively that g is of class , then Exercises 5.2 and 5.4 imply that g′ : ⁿ → _nn is of class , so g is of class by Exercise 5.3.

With the class inverse and implicit mapping theorems now available, the interested reader can rework Section 4 to develop the elementary theory of class manifolds—ones for which the coordinate patches are regular class mappings. Everything goes through with replaced by throughout.