Engineering Calculus Notes 182

Engineering Calculus Notes 182 - Integration also extends...

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170 CHAPTER 2. CURVES whose components are the linearizations of the component functions of −→ p ( t ) : if −→ p ( t ) = x ( t ) −→ ı + y ( t ) −→  , then the linearization at t = t 0 is T t 0 −→ p ( t ) = ( T t 0 x ( t )) −→ ı + ( T t 0 y ( t )) −→ = ( x ( t 0 ) + x ( t 0 ) t ) −→ ı + ( y ( t 0 ) + y ( t 0 ) t ) −→  . A component-by-component analysis Exercise ?? easily gives Remark 2.3.18. The vector-valued functions T t 0 −→ p ( t ) and −→ p ( t ) have frst-order contact at t = t 0 : lim t t 0 −→ p ( t ) T t 0 −→ p ( t ) | t t 0 | = −→ 0 . When we interpret −→ p ( t ) as describing the motion of a point in the plane or in space, we can interpret T t 0 −→ p ( t ) as the constant-velocity motion which would result, according to Newton’s First Law of motion, if all the forces making the point follow −→ p ( t ) were instantaneously turned o± at time t = t 0 . If the velocity v p ( t 0 ) is a nonzero vector, then T t 0 −→ p ( t ) traces out a line with direction vector v p ( t 0 ), which we call the tangent line to the motion at t = t 0 . Integration of Vector-Valued Functions
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Unformatted text preview: Integration also extends to vector-valued 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 (vector-valued) 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.

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