Page 67 - Vector Analysis

§2.8 Directional Derivatives and Gradient Vectors 63

can be solved for y and z as continuously differentiable function of x for x near ´1 and (y, z)
near (1, 1). Furthermore, if we write (y, z) = g(x) for x near ´1, then

[ xey + ez yez yez ]´1 [ ]
zey xez + ey ey
g1(x) = ez .

2.8 Directional Derivatives and Gradient Vectors

Definition 2.72 (Directional Derivatives). Let f be real-valued and defined on a neighbor-
hood of x0 P Rn, and let v P Rn be a unit vector. Then

(Dvf )(x0) ” d ˇ f (x0 + tv) = lim f (x0 + tv) ´ f (x0)
dt ˇ t
ˇt=0 tÑ0

is called the directional derivative（方向導數）of f at x0 in the direction v.

Remark 2.73. Let tej unj=1 be the standard basis of Rn. Then the partial derivative Bf (x0 )
B xj

(if it exists) is the directional derivative of f at x0 in the direction ej.

Remark 2.74. Let f be a real-valued differentiable function defined on a neighborhood
of x0 P Rn, and let v P Rn be a unit vector. For a curve γ : (´δ, δ) Ñ Rn satisfying that
γ(0) = x0 and γ 1(0) = v, the chain rule shows that

dˇ ˝ γ)(t) = (Df )(x0)(v) = (Dvf )(x0) .
ˇ (f

dt ˇt=0

In other words, for a differentiable function f in a neighborhood of x0, the derivative

dˇ (f ˝ γ) is independent of γ as long as γ(0) = x0 and γ 1(0) = v. Therefore, direc-
ˇ

dt ˇt=0

tional derivative of a differential function f at x0 in the direction v can also be defined by

the value dˇ (f ˝ γ)(t), where γ : (´δ, δ) Ñ Rn is any curve satisfying γ(0) = x0 and
ˇ

γ 1(0) = v. dt ˇt=0

Theorem 2.75. Let U Ď Rn be open, and f : U Ñ R be differentiable at x0. Then the
directional derivative of f at x0 in the direction v is (Df )(x0)(v).

Proof. Since f is differentiable at x0, @ ε ą 0, Q δ ą 0 such that

ˇˇf (x) ´ f (x0) ´ (Df )(x0)(x ´ x0)ˇˇ ď ε whenever }x ´ x0}Rn ă δ.
2 }x ´ x0}Rn