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Unformatted text preview: Lecture 3 2
Diﬀerentiation in n dimensions We are setting out to generalize to n dimensions the notion of diﬀerentiation in one
dimensional calculus. We begin with a review of elementary onedimensional calculus.
Let I ⊆ R be an open interval, let f : I → R be a map, and let a ∈ I .
Deﬁnition 2.1. The derivative of f at a is
f (a + t) − f (a)
t f � (a) = lim (2.1) provided that the limit exists. If the limit exists, then f is diﬀerentiable at a.
There are half a dozen or so possible reasonable generalizations of the notion
of derivative to higher dimensions. One candidate generalization which you have
probably already encountered is the directional derivative.
Deﬁnition 2.2. Given an open set U in Rn , a map f : U → Rm , a point a ∈ U , and
a point u ∈ Rn , the directional derivative of f in the direction of u at a is
f (a + tu) − f (a)
t Du f (a) = lim (2.2) provided that the limit exists.
In particular, we can calculate...
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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