Mathematics of Physics and Engineering

124 Functions of Several Variables

The function / is called dif f erentiable at the point A if / is de-
fined in some neighborhood of A and there exists a vector a so that

lim \f(B)-f(A)-a.AB\={

\AB\-^0 \AB\

Alternatively, for every unit vector u,

lha\KBt)-f(A)-ta.u\^

t-^o t

where ABt = t u. If / is differentiable at the point A, then the correspond-
ing vector a in (3.1.3) is denoted by V/(J4) or grad/(A) and is called
the gradient of / at A. We say that the function / is continuously
differentiable in a domain G if the vector field V/(^4) is defined at
all points A in G and the vector field V/ is continuous in G. Similarly, a
vector field F is called continuously differentiable in G if, for every
unit vector n, the scalar field F • n is continuously differentiable in G.
A scalar or vector field is called smooth if it is continuously differentiable
infinitely many times.

EXERCISE 3.1.2. (a)c Verify that if f is differentiable at the point A,

then f is differentiable at A in every direction u, and

Duf{A) = Vf{A)-u. (3.1.5)

(b)A What does it mean for a vector field F to be three times continuously

differentiable in a domain G?

By (3.1.5) and the definition of the dot product (1.2.1), page 9, we have

-\\Vf(A)\\<Dzf(A)<\\Vf(A)\\,

which shows that the most rapid rate of increase of / at the point A is in

the direction of V/(^4), and the most rapid rate of decrease of / at the

point A is in the direction of — Vf(A).

Given a point A, a direction at A can be defined by another point B with

the unit vector UQ = -A^m in the direction of AB. Then (3.1.3) becomes

f(B) = f(A) + (Vf(A) • uB) \AB\ + s(B), (3.1.6)

where lim|AB|_ 0 |e(JB)|/|AB| = 0. The number V/(A) • UB is

the (directional) derivative of / at point A in the direction

of point B.

(3.1.3)

(3.1.4)