lecture3

en of rn where e1 1 0 0 e2 0 1 0

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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.

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