Page 44 - Vector Analysis

40 CHAPTER 2. Differentiation of Functions of Several Variables

Definition 2.23. Let U Ď Rn be an open set, and f : U Ñ Rm. The matrix

 B f1 ¨¨¨ B f1  B f1 (x) ¨¨¨ B f1 (x) 
... B xn  (x) ”  ... 
(Jf )(x) ”  B x1 B x1 B xn
¨¨¨ ... ¨¨¨
... ... ...
B fm
B fm B fm (x) B fm (x)

B x1 B xn B x1 B xn

is called the Jacobian matrix of f at x (if each entry exists).

Remark 2.24. A function f might not be differential even if the Jacobian matrix Jf exists;
however, if f is differentiable at x0, then (Df )(x) can be represented by (Jf )(x); that is,
[(Df )(x)] = (Jf )(x).

Example 2.25. Let f : R2 Ñ R3 be given by f (x1, x2) = (x21, x13x2, x14x22). Suppose that f

is differentiable at x = (x1, x2), then

 2x1 
3x12x2 0
[] = 
(Df )(x) x13  .
2x41x2
4x31x22

Remark 2.26. For each x P A, Df (x) is a linear transformation, but Df in general is not

linear in x.

Example 2.27. Let f : R2 Ñ R be given by

# xy if (x, y) ‰ (0, 0) ,
f (x, y) = x2 + y2

0 if (x, y) = (0, 0) .

Then Bf (0, 0) = B f (0, 0) = 0; thus if f is differentiable at (0, 0), then (Df )(0, 0) = [ ]
0 0.
Bx By
However,

ˇ[ ] [] ˇ = |xy| = |xy| ax2 + y2 ;
ˇf (x, y) ´ f (0, 0) ´ 0 0 x ˇ x2 + y2
ˇ y ˇ (x2 y2) 3
+ 2

thus f is not differentiable at (0, 0) since |xy| cannot be arbitrarily small even if x2+y2

(x2 + y2) 3
2

is small.

Example 2.28. Let f : R2 Ñ R be given by

$ x if y = 0 ,
&
f (x, y) = y if x = 0 ,
% 1 otherwise .