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1. Differentiability, differentials, linear approximation and all that stuff.
 

Summary: 1. Differentiability, differentials, linear approximation and all
that stuff.
Suppose n is a positive integer, A Rn
,
f : A R
and a A.
Definition 1.1. We say f is differentiable at a if there exists a vector w Rn
such that
(1) lim
xa
|f(x) - f(a) - (x - a) w|
|x - a|
= 0.
The vector w is immediately seen to be unique; it is called the gradient of f
at a and is written
f(a).
If we let
A(x) = f(a) + (x - a) f(a) and e(x) = f(x) - A(x)
for x A we find that (1) is equivalent to
(2) lim

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics