Page 50 - Vector Analysis

46 CHAPTER 2. Differentiation of Functions of Several Variables

Example 2.37. Let f : R2 Ñ R be defined by

$ xy(x2 ´ y2) if (x, y) ‰ (0, 0) ,
&
f (x, y) = x2 + y2

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

Then $ x4y + 4x2y3 ´ y5

& (x2 + y2)2 if (x, y) ‰ (0, 0) ,
if (x, y) = (0, 0) ,
fx(x, y) =

%0

and $ x5 ´ 4x3y2 ´ xy4

& (x2 + y2)2 if (x, y) ‰ (0, 0) ,

fy(x, y) =

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

It is clear that fx and fy are continuous on R2; thus f is differentiable on R2. However,

fxy(0, 0) = lim fx(0, k) ´ fx(0, 0) = ´1 ,
k
kÑ0

while

fyx(0, 0) = lim fy(h, 0) ´ fy(0, 0) = 1;
h
hÑ0

thus the Hessian matrix of f at the origin is not symmetric.

Definition 2.38. Let U Ď Rn be open, and f : U Ñ Rm be a vector-valued function. The
function f is said to be of class C 2 if f P C 1(U ; Rm) and the second partial derivatives

B 2fi exists and is continuous in U for all 1 ď i ď m and 1 ď j, k ď n. The collection of

B xjB xk

all C 2-functions f : U Ñ Rm is denoted by C 2(U ; Rm).
In general, the function f is said to be of class C k if f P C k´1(U ; Rm) and the k-th order

partial derivatives B kf exists and is continuous in U for all 1 ď i ď m and

B xik B xik´1 ¨ ¨ ¨ B xi1

1 ď i1, ¨ ¨ ¨ , ik ď n. The collection of all C k-functions f : U Ñ Rm is denoted by C k(U ; Rm).

A function is said to be smooth or of class C 8 if it is of class C k for all positive

integer k.

Corollary 2.39. Let U Ď Rn be open, and f P C 2(U ; R). Then

B 2f (a) = B 2f (a) @ a P U and 1 ď i, j ď n .
B xiB xj B xjB xi