This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full Document
 Spring '11
 AlexOndrus
 Math, Calculus, Derivative, Vectors, dt

Click to edit the document details