Limit of Vector Valued Function

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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  C H A P T E R 7 LIMITS 7.1 INTRODUCTION The concept of a limiting value of a function plays an important role in calculus, because the formal definition of the derivative of a function at a point in its domain can be expressed as the limiting value of a particular expression involving the function. The meaning of “limit” in mathematics is more subtle than that in everyday speech. A speed limit that applies on a highway is a speed that motorists may not exceed. The meaning of “limit” in mathematics is similar to that in the following sentence: “In the minute to win it competition the contestant was pushed to the limit of his abilities.” Thus, a “limit” in mathematics is something (like a number or geometrical figure) that is approached and might or might not be reached. It is in keeping with the historical approach of this book to begin with the method of exhaustion as an example of an occurrence of a limit in mathematics, as this is the method that Archimedes and other Greek mathematicians of his time used to calculate approximate values of certain areas, for example, the area of a disk. In section 7.3, the concept of a limit is explained carefully using number sequences without giving the completely rigorous treatment (involving ε arguments) that are given in more advanced textbooks. Students of this book will probably not benefit from such a theoretical approach at this stage. The notion of the left or right limit of a function, introduced simplistically (by reading from a graph) in section 7.4, leads to the definition of continuity of a function in section 7.5. The property of continuity is important because many theorems about functions, for example, the Intermediate Value Theorem (in section 7.7), apply only to functions that are continuous. Most of the skills that students need to learn in this chapter are introduced in sections 7.6 and 7.8. They are the algebraic skills required for calculating limits.

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  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  47. State and prove a limit law for lim t→a [u(t) · v(t)] 48. State and prove a limit law for lim t→a [u(t) × v(t)] 13.3 Differentiation and Integration of Vector-Valued Functions In Section 13.2, we extended the calculus concepts of limits and continuity to vector-valued functions. In this section, we will see that the definitions, techniques, and properties associated with derivatives and integrals of scalar-valued functions will enable us to define, compute, and interpret derivatives and integrals of vector- valued functions. Derivatives of Vector-Valued Functions The definition of the derivative of a scalar-valued function was motivated by the desire to find tangent lines to curves. For vector-valued functions, we begin with a similar definition and then show that the derivative can be interpreted geometrically as a tangent vector. Figure 13.33 Figure 13.34 Figure 13.35 Definition of the Derivative of a Vector-Valued Function The derivative of a vector-valued function r is r 0 (t) = lim h→0 r(t + h) - r(t) h for all t for which the limit exists. Just as with the derivative of a scalar-valued function, the definition of the derivative of a vector-valued function has an interesting geometric interpretation. Consider the graph of a differentiable vector-valued function r in R 3 such as the one shown in Figure 13.33. In this figure, the point P corresponds to r(t) and Q corresponds to r(t + h). The vector r(t + h) - r(t) is represented by --→ PQ; multiplying by 1 h simply changes the length. Now as h becomes smaller, the point Q moves closer to P , and the vector 1 h [r(t + h) - r(t)] gradually changes direction and length, as suggested by Figure 13.34. The limiting value as h → 0 is the vector r 0 (t), which appears to be tangent to the curve at the point P , as shown in Figure 13.35. For this reason, if r 0 (t) exists and is not the zero vector, we refer to r 0 (t) as a tangent vector to the curve given by r at the point r(t).

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  Calculus, Volume 1

  • Tom M. Apostol(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  14 CALCULUS OF VECTOR-VALUED FUNCTIONS 14.1 Vector-valued functions of a real variable This chapter combines vector algebra with the methods of calculus and describes some appli- cations to the study of curves and to some problems in mechanics. The concept of a vector-valued function is fundamental in this study. definition. A function whose domain is a set of real numbers and whose range is a subset of n-space V n is called a vector-valued function of a real variable. We have encountered such functions in Chapter 13. For example, the line through a point P parallel to a nonzero vector A is the range of the vector-valued function X given by X(t) = P + tA for all real t. Vector-valued functions will be denoted by capital letters such as F, G, X, Y , etc., or by small bold-face italic letters f , g, etc. The value of a function F at t is denoted, as usual, by F(t). In the examples we shall study, the domain of F will be an interval which may contain one or both endpoints or which may be infinite. 14.2 Algebraic operations. Components The usual operations of vector algebra can be applied to combine two vector-valued functions or to combine a vector-valued function with a real-valued function. If F and G are vector-valued functions, and if u is a real-valued function, all having a common domain, we define new functions F + G, uF, and F ⋅ G by the equations (F + G)(t) = F(t) + G(t), (uF)(t) = u(t)F(t), (F . G)(t) = F(t) . G(t). The sum F + G and the product uF are vector valued, whereas the dot product F ⋅ G is real valued. If F(t) and G(t) are in 3-space, we can also define the cross product F × G by the formula (F × G)(t) = F(t) × G(t). 512 Limits, derivatives, and integrals 513 The operation of composition may be applied to combine vector-valued functions with real-valued functions.

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  The Fundamentals of Mathematical Analysis

  • G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  If we say that the variable ranges over some sequence of values, then the reader might imagine that the variable takes its value in consecutive instances of time; in fact, however, it has nothing to do with time. Only for clarity sometimes the fol-lowing expressions are used: remote values of the variable, starting from some place or from some instant of variation, etc. 28. Definition of the limit of a sequence. The ordering of the values of the variable x n , according to increasing numbers which led us to consider the sequence (2) of these values, simplifies the concept of the process of the variable x n approaching its limit a— as n increases to infinity. The number a is called the limit of the function x n if the latter differs from a by an arbitrarily small amount, beginning from a certain place, i.e. for all sufficiently large numbers n. This statement clearly expresses the essence of the matter, but what arbitrarily small or sufficiently great means has to be explained. We now present a longer but comprehensive and precise definition of limit. The number a is said to be the limit of the variable x n if for any positive number , no matter how small it is, a number N exists such that all values of x n the numbers of which n > N satisfy the inequality χ η -α<ε. (3) The fact that a is the limit of variable x n is written as follows: lim x n = a (lim is an abbreviation of the Latin word limes, meaning limit). Sometimes it is said that the variable tends to a and we then write x n -+ a. § 1. LIMIT OF A FUNCTION 55 Finally, the number a is also called the limit of sequence (2) and we may say that this sequence converges to a. The inequality (3) where ε is arbitrary is the precise statement of the fact that x n differs from a by an arbitrarily small amount, and the number N indicates the place beginning from which this fact occurs, so that all numbers n >N are sufficiently large.

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  CounterExamples

  From Elementary Calculus to the Beginnings of Analysis

  • Andrei Bourchtein, Ludmila Bourchtein(Authors)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 7 Limits and continuity General Remark . Part II (chapters 7-9) contains the topics of the func-tions of two variables, which are quite representative for the most topics of the multivariable functions. The counterexamples presented in this part of the work are separated into two groups. The first one includes those examples that have intimate connections with counterexamples of one-variable functions considered in Part I. They show how the ideas applied in the one-dimensional case can be generalized/extended to many variables. Accordingly, the exam-ples of this group are placed in the sections titled “one-dimensional links” in each of chapters 7-9. The examples of the second group have a weak or no connection with one-dimensional case, highlighting a specificity of concepts and results for multivariable functions. Some of them illustrate the situations that are feasible for two-variable functions but cannot happen in the case of one-variable functions. In each chapter, all the examples of the second group are collected in sections titled “multidimensional essentials”. 7.1 Elements of theory Limits. Concepts Limit (general limit) . Let f ( x, y ) be defined on X and ( a, b ) be a limit point of X . We say that the limit of f ( x, y ), as ( x, y ) approaches ( a, b ), exists and equals A if for every ε > 0 there exists δ > 0 such that for all ( x, y ) ∈ X such that 0 < √ ( x − a ) 2 + ( y − b ) 2 < δ it follows that | f ( x, y ) − A | < ε . The usual notations are lim ( x,y ) → ( a,b ) f ( x, y ) = A and f ( x, y ) → ( x,y ) → ( a,b ) A . Remark . In calculus, a non-essential simplification that f ( x, y ) is defined in some deleted neighborhood of ( a, b ) is frequently used. 207 208 Counterexamples: From Calculus to the Beginnings of Analysis Partial limit . Let f ( x, y ) be defined on X , S be a subset of X and ( a, b ) be a limit point of S .

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  Calculus

  One and Several Variables

  • Saturnino L. Salas, Garret J. Etgen, Einar Hille(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  694 ■ CHAPTER 14 VECTOR CALCULUS As in the examples given, and under conditions to be spelled out later, as t ranges over the interval I, the radius vector f (t ) = f 1 (t ) i + f 2 (t ) j + f 3 (t ) k traces out a curve C. We say that f parametrizes C. The equations x = f 1 (t ), y = f 2 (t ), z = f 3 (t ) formed from the components of f serve as parametric equations for C. If one of the components is identically 0 on I, for example, if f has the form f (t ) = f 1 (t ) i + f 2 (t ) j, then C is a plane curve; otherwise C is a space curve. In this chapter we study the geometry of curves and then apply this geometry to obtain a useful description of curvilinear motion. But first we have to extend the limit process to vector functions, establish the notion of vector-function continuity, define vector derivative, and spell out the rules of differentiation. ■ 14.1 LIMIT, CONTINUITY, VECTOR DERIVATIVE The Limit Process The limit process for vector functions does not require that f(t ) be attached to any particular point. It can be applied to any vector-valued function. DEFINITION 14.1.1 LIMIT OF A VECTOR FUNCTION lim t →t 0 f (t ) = L provided that lim t →t 0 f (t ) − L = 0. Note that for each t in the domain of f, f(t ) − L is a real number, and therefore the limit on the right is the limit of a real-valued function. Thus we are still in familiar territory. The first thing we show is that (14.1.2) if lim t →t 0 f (t ) = L, then lim t →t 0 f (t ) = L. PROOF We know that 0 ≤   f (t ) − L   ≤ f (t ) − L. (Exercise 24, Section 13.2) It follows from the pinching theorem that if lim t →t 0 f (t ) − L = 0, then lim t →t 0   f (t ) − L   = 0. Remark The converse of (14.1.2) is false. You can see this by setting f (t ) = k and taking L = −k. ❏ We can indicate that lim t →t 0 f (t ) = L by writing as t → t 0 , f (t ) → L. We will state the limit rules in this form. As you will see below, there are no surprises.

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