Engineering Calculus Notes 175

Engineering Calculus Notes 175 - t ) . Every function from...

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2.3. CALCULUS OF VECTOR-VALUED FUNCTIONS 163 Since passing to subsequences has not hurt the convergence of the Frst and second coordinates x i ′′ k 1 y i ′′ k 2 we see that −→ p i ′′ k L where L R 3 is the point with rectangular coordinates ( 1 ,ℓ 2 ,ℓ 3 ). In the exercises, you will check a number of features of convergence (and divergence) which carry over from sequences of numbers to sequences of points. Continuity of Vector-Valued Functions Using the notion of convergence formulated in the previous subsection, the notion of continuity for real-valued functions extends naturally to vector-valued functions. Defnition 2.3.8. −→ f : R R 3 is continuous on D R if for every convergent sequence t i t in D the sequence of points −→ f ( t i ) converges to −→ f (
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Unformatted text preview: t ) . Every function from R to R 3 can be expressed as f ( t ) = ( f 1 ( t ) ,f 2 ( t ) ,f 3 ( t )) or f ( t ) = f 1 ( t ) + f 2 ( t ) + f 3 ( t ) k where f 1 ( t ), f 2 ( t ) and f 3 ( t ), the component Functions of f ( t ), are ordinary (real-valued) functions. Using Lemma 2.3.5 , it is easy to connect continuity of f ( t ) with continuity of its components: Remark 2.3.9. A function f : R R 3 is continuous on D R precisely if each of its components f 1 ( t ) , f 2 ( t ) , f 3 ( t ) is continuous on D . A related notion, that of limits, is an equally natural generalization of the single-variable idea:...
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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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