Multivariate Calculus - Bruce E. Shapiro

Lecture 11
Gradients and the Directional
Derivative
We will learn in this section that the concepts of differentiability and local linearity
are equivalent, and we will generalize our definitions of the derivative and
and partial derivative from one-dimensional objects or slopes to generalized slopes
in higher dimensional spaces.
Definition 11.1 A function f(x) : R ↦→ R is said to be locally linear at x = a
if it can be approximated by a line in some neighborhood of a, i.e., if there is some
constant m such that
f(a + h) = f(a) + hm + hɛ(h) (11.1)
where
lim ɛ(h) = 0
h→0
The number m is called the slope of the line.
Suppose that a function f(x) is locally linear at a. Solving equation 11.1 for ɛ(h),
ɛ(h) =
f(a + h) − f(a)
h
− m
and therefore
( )
f(a + h) − f(a)
0 = lim ɛ(h) = lim
− m
h→0 h→0 h
= f ′ (a) − m
The last equality only makes sense if the limit exists, which we know it does because
we have assumed that f(x) is locally linear, and therefore lim h→0 ɛ(h) exists and
equals zero. But the first term in the final limit is the derivative f ′ (a), and we say
the f(x) is differentiable if and only of the derivative f ′ (a), defined by this limit,
exists. Therefore we have proven that a function is locally linear if and only
if it is differentiable. Our immediate goal is to extend this result to multivariate
functions.
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