Page 42 - Vector Analysis

38 CHAPTER 2. Differentiation of Functions of Several Variables

Remark 2.18. Adopting the standard basis of Rn and Rm, a linear transformation L :
Rn Ñ Rm has a matrix representation [L]mˆn such that L(x) = [L]mˆn[x]n for all x P Rn. In
the following, we will always use the standard basis for Rn and Rm and use L and L(x) to
denote [L]mˆm and [L]mˆn[x]n, respectively, if L is a linear transformation from Rn to Rm
and x P Rn.

Proposition 2.19. Let U Ď Rn be an open set, and f : U Ñ Rm be differentiable at x0 P U .
Then (Df )(x0), the derivative of f at x0, is uniquely determined by f .

Proof. Suppose L1, L2 P B(Rn, Rm) are derivatives of f at x0. Let ε ą 0 be given and
e P Rn be a unit vector; that is, }e}Rn = 1. Since U is open, there exists r ą 0 such that
B(x0, r) Ď U. By Definition 2.15, there exists 0 ă δ ă r such that

}f (x) ´ f (x0) ´ L1(x ´ x0)}Rm ă ε and }f (x) ´ f (x0) ´ L2(x ´ x0)}Rm ă ε
}x ´ x0}Rn 2 }x ´ x0}Rn 2

if 0 ă }x ´ x0}Rn ă δ. Letting x = x0 + λe with 0 ă |λ| ă δ, we have

}L1e ´ L2e}Rm = 1 ´ x0) ´ L2(x ´ x0)}Rm
|λ| }L1(x

ď 1 (››f (x) ´ f (x0) ´ L1(x ´ x0)››Rm + ››f (x) ´ f (x0) ´ L2(x ´ x2)››Rm )
|λ|

= ››f (x) ´ f (x0) ´ L1(x ´ x0)››Rm + ››f (x) ´ f (x0) ´ L2(x ´ x0)››Rm
}x ´ x0}Rn }x ´ x0}Rn

εε
ă + = ε.
22

Since ε ą 0 is arbitrary, we conclude that L1e = L2e( for all unit vectors e P) Rn which
guarantees that L1 = L2 (since if x ‰ 0, L1x = }x}RnL1 ) ( = L2x). ˝
x = }x}Rn L2 x
}x}Rn }x}Rn

Example 2.20. (Df )(x0) may not be unique if the domain of f is not open. For example,

let A = ␣(x, y) ˇ 0 ď x ď 1, y = 0( be a subset of R2, and f : A Ñ R be given by f (x, y) = 0.
ˇ

Fix x0 = (a, 0) P A, then both of the linear maps

L1(x, y) = 0 and L2(x, y) = ay @ (x, y) P R2

satisfy Definition 2.15 since

lim ˇˇf (x, 0) ´ f (a, 0) ´ L1(x ´ a, 0)ˇˇ = lim ˇˇf (x, 0) ´ f (a, 0) ´ L2(x ´ a, 0)ˇˇ = 0.
››(x, 0) ´ (a, 0)››R2 ››(x, 0) ´ (a, 0)››R2
(x,0)Ñ(a,0) (x,0)Ñ(a,0)