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Unformatted text preview: the directional derivatives in the direction of the
standard basis vectors e1 , . . . , en of Rn , where
e1 = (1, 0, . . . , 0),
e2 = (0, 1, 0, . . . , 0),
.
.
.
en = (0, . . . , 0, 1). (2.3)
(2.4)
(2.5)
(2.6) Notation. The directional derivative in the direction of a standard basis vector ei of
Rn is denoted by
∂
Di f (a) = Dei f (a) =
f (a).
(2.7)
∂xi
We now try to answer the following question: What is an adequate deﬁnition of
diﬀerentiability at a point a for a function f : U → Rm ? 1 • Guess 1: Require that ∂f
(a)
∂xi exists. However, this requirement is inadequate. Consider the function deﬁned by
�
0, if (x1 , x2 ) lies on the x1 axis or the x2 axis,
f (x1 , x2 ) = (2.8)
1, otherwise. Then, both ∂f
∂f
(0) = 0 and
(0) = 0, (2.9)
∂x1
∂x2 but the function f is not diﬀerentiable at (0, 0) along any other direction. • Guess 2: Require that all directional derivatives exist at a.
Unfortunately, this requirement is still inadequate. For example (from Munkres
chapter...
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This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.
 Fall '04
 unknown
 Calculus

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