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Unformatted text preview: MATH 214: VECTORVALUED FUNCTIONS 1. Introduction In this section we consider the following Definition. A curve is a collection of points of the form { ( x,y,z )  x = f ( t ) ,y = g ( t ) ,z = h ( t ) } for some functions f,g,h : R → R . The functions f,g,h are called the coordinate functions of the curve. Given a point P ( f ( t ) ,g ( t ) ,h ( t )) on the curve, the vector→ OP is called the position vector of P . We usually denote r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i for the function that takes a value t to the position vector for P ( f ( t ) ,g ( t ) ,h ( t )) . Example 1. Lines, considered in the last section, are the simplest example of curves. The function r ( t ) =→ OP + tv gives the position vector for a point on the line. Example 2. Consider the set of points: { (cos( t ) , sin( t ) , 1 t )  t ∈ ( π, 10 π ) } this is a curve. For a curve, the function r ( t ) takes a number t to a vector r ( t ) (the position vector associated to the point on a curve). This is an example of a Definition. A vectorvalued function is a function that takes a number to a vector. Regular functions R → R are also known as scalarvalued functions 1.1. Limits and Continuity. Limits of vectorvalued functions are defined the same way as limits of regular functions: Definition. Say that r ( t ) approaches the limit L ∈ R 3 as t approaches t if, for every > , there exists a δ > such that  t t  < δ ,  r ( t ) L  < . We write this as lim t → t r ( t ) = L It doesn’t take much work (although it is not directly from the definition) to see that Proposition 1.1. If r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i , then lim t → t r ( t ) = h ‘ 1 ,‘ 2 ,‘ 3 i if and only if lim t → t f ( t ) = ‘ 1 , lim t → t g ( t ) = ‘ 2 , and lim t → t h ( t ) = ‘ 3 Example 3. The limit of r ( t ) = h 1 t sin( t ) , tan 1 ( t ) t t 3 , sin( π · t ) i as t → is h 1 , 1 3 , i . The definition of continuity is also defined the same way for vectorvalued functions as it is for scalarvalued functions: Definition. A vector valued function is continuous at a point t if lim t → t r ( t ) = r ( t ) . A vectorvalued function is continuous if and only if each of the component functions is continuous. 1 2 MATH 214: VECTORVALUED FUNCTIONS Example 4. The previous vectorvalued function is not continuous at t = 0 because the first two functions are not defined at t = 0 . The vectorvalued function r ( t ) = h ln( t ) , cos( t ) , t 2 t 1 i is continuous for all t > except t = 1 . 1.2. Derivatives. Definition. If r ( t ) = h f ( t ) ,g ( t ) ,h ( t ) i and f,g,h are differentiable then r ( t ) is differentiable and the derivative of r ( t ) is defined as r ( t ) = lim Δ t → r ( t + Δ t ) r ( t ) Δ t = h f ( t ) ,g ( t ) ,h ( t ) i r ( t ) is smooth if it is differentiable and r ( t ) 6 = h , , i for any t . A curve is piecewise smooth if it can be constructed by joining smooth curves together, end to end....
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This note was uploaded on 07/31/2011 for the course MATH 214 taught by Professor Alexondrus during the Spring '11 term at University of Alberta.
 Spring '11
 AlexOndrus
 Math, Calculus

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