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mthsc206-fall-2010-notes-14.1

# mthsc206-fall-2010-notes-14.1 - 14 14.1 Vector Functions...

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14 Vector Functions 14.1 Vector Functions and Space Curves Definition 14.1.1 Let r : R R 3 (or R 2 ) be given by r ( t ) = < f ( t ) , g ( t ) , h ( t ) > = f ( t ) i + g ( t ) j + h ( t ) k , where f , g , and h are real-valued functions. r ( t ) is called a vector-valued function. Definition 14.1.2 Let r ( t ) be given by r ( t ) = < f ( t ) , g ( t ) , h ( t ) > , where f , g and h are real-valued func- tions of t . The domain of r is the largest set of real numbers t so that r ( t ) is defined. Definition 14.1.3 Let r ( t ) be given by r ( t ) = < f ( t ) , g ( t ) , h ( t ) > , where f , g and h are real-valued func- tions of t . We say that the limit of r as t approaches a is the vector v if for any > 0 , there is a number δ > 0 so that whenever 0 < | t - a | < δ then || r t - v || < . Theorem: Let r ( t ) = < f ( t ) , g ( t ) , h ( t ) > be a vector-valued function with real-valued component functions f , g , and h . If the component functions all have limits as t approaches a , i.e., lim t a f ( t ) = L 1 , lim t a g ( t ) = L 2 , lim t a h ( t ) = L 3 , then lim t a r ( t ) exists and lim t a r ( t ) = lim t a < f ( t ) , g ( t ) , h ( t ) > = D lim t a f ( t ) , lim t a g ( t ) , lim t a h ( t ) E = < L 1 , L 2 , L 3 > . Definition 14.1.4 We say that the vector-valued function r ( t ) is continuous at a if (1) r ( a ) is defined, i.e., a is in the domain of r , (2) lim t a r ( t ) exists, and (3) lim t a r ( t ) = r ( a ) . More simply put, r ( t ) is continuous at a if lim t a r ( t ) = r ( a ) . (Why?) Example 14.1.1 Let r
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