Page 43 - Vector Analysis

§2.3 Definition of Derivatives and the Matrix Representation of Derivatives 39

Definition 2.21. Let tekukn=1 be the standard basis of Rn, U Ď Rn be an open set, a P U

and f : U Ñ R be a function. The partial derivative of f at a with respect to xj, denoted

by B f (a), is the limit lim f (a + hej) ´ f (a)
hÑ0 h
B xj

if it exists. In other words, if a = (a1, ¨ ¨ ¨ , an), then

B f (a) = lim f (a1, ¨ ¨ ¨ , aj´1, aj + h, aj+1, ¨ ¨ ¨ , an) ´ f (a1, ¨ ¨ ¨ , an) .
B xj hÑ0 h

Theorem 2.22. Suppose U Ď Rn is an open set and f : U Ñ Rm is differentiable at a P U .
Then the partial derivatives B fi (a) exists for all i = 1, ¨ ¨ ¨ m and j = 1, ¨ ¨ ¨ n, and the matrix

B xj

representation of the linear transformation Df (a) (with respect to the standard basis of Rn

and Rm) is given by


B f1 (a) B f1 (a)
[] =  ¨¨¨  or [] = B fi (a) .
Df (a) B x1 ... B xn Df (a) ij B xj

... ¨¨¨ ...

B fm (a) B fm (a)
B x1 B xn

Proof. Since U is open and a P U, there exists r ą 0 such that B(a, r) Ď U. By the
differentiability of f at a, there is L P B(Rn, Rm) such that for any given ε ą 0, there exists
0 ă δ ă r such that

}f (x) ´ f (a) ´ L(x ´ a)}Rm ď ε}x ´ a}Rn whenever x P B(a, δ) .

In particular, for each i = 1, ¨ ¨ ¨ , m,

ˇ fi (a + hej ) ´ fi(a) ˇ ›f (a + hej ) ´ f (a) ›
ˇ h (Lej )i ˇˇ › h
´ ď › ´ Lej › ď ε @ 0 ă |h| ă δ, h P R ,
ˇ ›Rm

where (Lej)i denotes the i-th component of Lej in the standard basis. As a consequence,

for each i = 1, ¨ ¨ ¨ , m,

lim fi(a + hej ) ´ fi(a) = (Lej )i exists
h
hÑ0

and by definition, we must have (Lej )i = B fi (a). Therefore, Lij = B fi (a). ˝

B xj B xj