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Unformatted text preview: Integration also extends to vectorvalued functions componentwise. Given −→ f : [ a,b ] → R 3 and a partition P = { a = t < t 1 < ··· < t n = b } of [ a,b ], we can’t form upper or lower sums, since the “sup” and “inf” of −→ f ( t ) over I j don’t make sense. However we can form (vectorvalued) Riemann sums R ( P , −→ f , { t ∗ j } ) = n s j =1 −→ f ( t ∗ j ) △ t j and ask what happens to these Riemann sums for a sequence of partitions whose mesh size goes to zero. If all such sequences have a common (vector) limit, we call it the defnite integral of −→ f ( t ) over [ a,b ]. It is natural (and straightforward to verify, using Lemma 2.3.5 ) that this happens precisely if...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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