Ch13Notes - MATH 214: VECTOR-VALUED FUNCTIONS 1....

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Unformatted text preview: MATH 214: VECTOR-VALUED 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 ( , 10 ) } 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 vector-valued function is a function that takes a number to a vector. Regular functions R R are also known as scalar-valued functions 1.1. Limits and Continuity. Limits of vector-valued 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 > such that | t- t | < , | r ( t )- L | < . We write this as lim t t r ( t ) = L It doesnt 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( t ) i as t is h 1 ,- 1 3 , i . The definition of continuity is also defined the same way for vector-valued functions as it is for scalar-valued functions: Definition. A vector valued function is continuous at a point t if lim t t r ( t ) = r ( t ) . A vector-valued function is continuous if and only if each of the component functions is continuous. 1 2 MATH 214: VECTOR-VALUED FUNCTIONS Example 4. The previous vector-valued function is not continuous at t = 0 because the first two functions are not defined at t = 0 . The vector-valued 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 t r ( t + t )- r ( t ) 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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Ch13Notes - MATH 214: VECTOR-VALUED FUNCTIONS 1....

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