In Exercises 60–62 you will establish that the matrix D f(a) of partial derivatives...

In Exercises 60–62 you will establish that the matrix D f(a) of partial derivatives of the component functions of f is uniquely determined by the limit equation in Definition 3.8.

Let X be an open set in Rⁿ, let a ∈ X, and let f: X ⊆ Rⁿ → R^m. Suppose that A is an m × n matrix such that

In this problem you will establish that A = Df(a).

(a) Define F: X ⊆ Rⁿ → R^m by

Identify the ith component function Fi (x) using component functions of f and parts of the matrix A.

(b) Note that under the assumptions of this problem and Exercise 60, we have that lim_x→a F(x) = 0.

First argue that, for i = 1, . . . ,m, we have lim_x→a Fi (x) = 0. Next, argue that

lim _x→a Fi (x) = 0 implies lim _h→0 Fi (a + he j ) = 0, where e j denotes the standard basis vector (0, . . . , 1, . . . , 0) for Rⁿ.

(c) Use parts(a) and (b) to show that where ai j denotes the i jth entry of A. (Hint: Break

into cases where h > 0 and where h < 0.)