Real and Complex Analysis (Rudin)

156 REAL AND COMPLEX ANALYSIS
If f satisfies the hypotheses of the theorem, so does -f; therefore (6)
holds with -fin place off, and these two inequalities together give (1). IIII
Differentiable Transformations
7.22 Definitions Suppose V is an open set in Rk, T maps V into Rk, and
x E V. If there exists a linear operator A on Rk (i.e., a linear mapping of Rk
into Rk, as in Definition 2.1) such that
lim I T(x + h) - T(x) - Ah I = 0
h .... O Ihl
(where, of course, h E Rk), then we say that T is differentiable at x, and define
(1)
T'(x) = A. (2)
The linear operator T'(x) is called the derivative of T at x. (One shows
easily that there is at most one linear A that satisfies the preceding requirements;
thus it is legitimate to talk about the derivative of T.) The term differential
is also often used for T'(x}.
The point of (1) is of course that the difference T(x + h) - T(x) is
approximated by T'(x)h, a linear function of h.
Since every real number ex gives rise to a linear operator on Rl (mapping
h to exh), our definition of T'(x) coincides with the usual one when k = 1.
When A: Rk_ Rk is linear, Theorem 2.20(e) shows that there is a number
~(A) such that
for all measurable sets E c Rk. Since
m(A(E» = ~(A)m(E) (3)
A'(x) = A (4)
and since every differentiable transformation T can be locally approximated
by a constant plus a linear transformation, one may conjecture that
m(T(E» ,..., ~(T'(x»
m(E)
for suitable sets E that are close to x. This will be proved in Theorem 7.24,
and furnishes the motivation for Theorem 7.26.
Recall that ~(A) = I det A I was proved in Sec. 2.23. When T is differentiable
at x, the determinant of T'(x) is called the Jacobian of T at x, and is
denoted by J .,.(x). Thus
~(T'(x» = I J .,.(x) I. (6)
The following lemma seems geometrically obvious. Its proof depends on the
Brouwer fixed point theorem. One can avoid the use of this theorem by imposing
(5)