Page 68 - Vector Analysis

64 CHAPTER 2. Differentiation of Functions of Several Variables

In particular, if x = x0 + tv with v being a unit vector in Rn and 0 ă |t| ă δ, then

ˇ f (x0 + tv) ´ f (x0) ´ ˇ = ˇˇf (x0 + tv) ´ f (x0) ´ (Df )(x0)(tv)ˇˇ
ˇ t (Df )(x0)(v)ˇˇ |t|

ˇ

= ˇˇf (x) ´ f (x0) ´ (Df )(x0)(x ´ x0)ˇˇ ď ε ă ε ;
|t| 2

thus (Dvf )(x0) = (Df )(x0)(v). ˝

Remark 2.76. When v P Rn but 0 ă }v}Rn ‰ 1, we let v = v. Then the direction

}v}Rn
derivatives of a function f : U Ď Rn Ñ R at a P U in the direction v is

(Dvf )(a) = lim f (a + tv) ´ f (a) .
t
tÑ0

Making a change of variable s = t . Then

}v}Rn

(Df )(x0)(v) = }v}Rn(Df )(x0)(v) = }v}Rn lim f (a + tv) ´ f (a) = lim f (a + sv) ´ f (a) .
t s
tÑ0 sÑ0

We sometimes also call the value (Df )(x0)(v) the “directional derivative” of f in the “direc-
tion” v.

Example 2.77. The existence of directional derivatives of a function f at x0 in all directions
does not guarantee the differentiability of f at x0. For example, let f : R2 Ñ R be given as
in Example 2.44, and v = (v1, v2) P R2 be a unit vector. Then

(Dvf )(0) = lim f (tv1, tv2) ´ f (0, 0) = v31 .
t
tÑ0

However, f is not different[iable at (0, 0). We ]also note that in this example, (Dvf )(0) ‰
(Jf )(0)v, where (Jf )(0) = B f (0, 0) B f (0, 0) is the Jacobian matrix of f at (0, 0).

Bx By

Example 2.78. The existence of directional derivatives of a function f at x0 in all directions
does not even guarantee the continuity of f at x0. For example, let f : R2 Ñ R be given by

$ xy2 if (x, y) ‰ (0, 0) ,
&
f (x, y) = x2 + y4

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

and v = (v1, v2) P R2 be a unit vector. Then if v1 ‰ 0,

(Dvf )(0) = lim f (tv1, tv2) ´ f (0, 0) = lim t3v1v22 = v22
t t(t2v21 + t4v24) v1
tÑ0 tÑ0