Engineering Calculus Notes 173

Engineering - 2.3 CALCULUS OF VECTOR-VALUED FUNCTIONS 161(2(1 Suppose xi 1 yi 2 and zi 3 Given > 0 we can nd N1 so that i > N1 guarantees |xi 1 | <

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 161 (2) (1) : Suppose x i 1 , y i 2 , and z i 3 . Given ε > 0, we can fnd N 1 so that i > N 1 guarantees | x i 1 | < ε 3 N 2 so that i > N 2 guarantees | y i 2 | < ε 3 N 3 so that i > N 3 guarantees | z i 3 | < ε 3 . Let L R 3 be the point with rectangular coordinates ( 1 ,ℓ 2 ,ℓ 3 ). Setting N = max( N 1 ,N 2 ,N 3 ), we see that i > N guarantees δ := max( | x i 1 | , | y i 2 | , | z i 3 | ) < ε 3 and hence by Lemma 2.3.1 i > N guarantees dist( −→ p i ,L ) < 3 ε 3 = ε, so −→ p i L . As in R , we say a sequence { −→ p i } oF points is bounded iF there is a fnite upper bound on the distance oF all the points in the sequence From the origin—that is, sup { dist( −→ p i , O ) } < . An easy analogue oF a basic property oF sequences oF numbers is the Following, whose prooF we leave to you (Exercise 5 ): Remark 2.3.6. Every convergent sequence is bounded. A major di±erence between sequences oF numbers and sequences oF points
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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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