Mathematics
Vector Valued Function
A vector-valued function is a mathematical function that maps each input in its domain to a vector in its range. It is commonly used to describe the path of a moving object in space, where the input represents time and the output is a position vector. Vector-valued functions are essential in fields such as physics, engineering, and computer graphics.
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8 Key excerpts on "Vector Valued Function"
eBook - PDFCalculus
Multivariable
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
723 CHAPTER 12 Vector-Valued Functions In this chapter we will consider functions whose values are vectors. Such functions provide a unified way of studying parametric curves in 2-space and 3-space and are a basic tool for analyzing the motion of particles along curved paths. We will begin by developing the calculus of vector-valued functions—we will show how to differentiate and integrate such functions, and we will develop some of the basic properties of these operations. We will then apply these calculus tools to define three fundamental vectors that can be used to describe such basic characteristics of curves as curvature and twisting tendencies. Once this is done, we will develop the concepts of velocity and acceleration for such motion, and we will apply these concepts to explain various physical phenomena. Finally, we will use the calculus of vector-valued functions to develop basic principles of gravitational attraction and to derive Kepler’s laws of planetary motion. Krystian Janczynski/iStockphoto ´ The design of a roller coaster requires an understanding of the mathematical principles governing the motion of objects that move with varying speed and direction. 12.1 Introduction to Vector-Valued Functions In Section 11.5 we discussed parametric equations of lines in 3-space. In this section we will discuss more general parametric curves in 3-space, and we will show how vector notation can be used to express parametric equations in 2-space and 3-space in a more compact form. This will lead us to consider a new kind of function—namely, functions that associate vectors with real numbers. Such functions have many important applications in physics and engineering. Parametric Curves in 3-Space Recall from Section 10.1 that if f and g are well-behaved functions, then the pair of parametric equations x = f (t), y = g(t) (1) generates a curve in 2-space that is traced in a specific direction as the parameter t increases.
eBook - PDF- 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.
eBook - PDF- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
VECTOR-VALUED FUNCTIONS 12 Krystian Janczynski/iStockphoto ´ The design of a roller coaster requires an understanding of the mathematical principles governing the motion of objects that move with varying speed and direction. In this chapter we will consider functions whose values are vectors. Such functions provide a unified way of studying parametric curves in 2-space and 3-space and are a basic tool for analyzing the motion of particles along curved paths. We will begin by developing the calculus of vector-valued functions—we will show how to differentiate and integrate such functions, and we will develop some of the basic properties of these operations. We will then apply these calculus tools to define three fundamental vectors that can be used to describe such basic characteristics of curves as curvature and twisting tendencies. Once this is done, we will develop the concepts of velocity and acceleration for such motion, and we will apply these concepts to explain various physical phenomena. Finally, we will use the calculus of vector-valued functions to develop basic principles of gravitational attraction and to derive Kepler’s laws of planetary motion. 12.1 INTRODUCTION TO VECTOR-VALUED FUNCTIONS In Section 11.5 we discussed parametric equations of lines in 3-space. In this section we will discuss more general parametric curves in 3-space, and we will show how vector notation can be used to express parametric equations in 2-space and 3-space in a more compact form. This will lead us to consider a new kind of function—namely, functions that associate vectors with real numbers. Such functions have many important applications in physics and engineering. Parametric Curves in 3-Space Recall from Section 9.1 that if f and g are well-behaved functions, then the pair of parametric equations x = f (t), y = g (t) (1) generates a curve in 2-space that is traced in a specific direction as the parameter t increases.
eBook - PDFCalculus
Late Transcendentals
- Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
744 12 The design of a roller coaster requires an understanding of the mathematical principles governing the motion of objects that move with varying speed and direction. VECTOR-VALUED FUNCTIONS In this chapter we will consider functions whose values are vectors. Such functions provide a unified way of studying parametric curves in 2-space and 3-space and are a basic tool for analyzing the motion of particles along curved paths. We will begin by developing the calculus of vector-valued functions—we will show how to differentiate and integrate such functions, and we will develop some of the basic properties of these operations. We will then apply these calculus tools to define three fundamental vectors that can be used to describe such basic characteristics of curves as curvature and twisting tendencies. Once this is done, we will develop the concepts of velocity and acceleration for such motion, and we will apply these concepts to explain various physical phenomena. Finally, we will use the calculus of vector-valued functions to develop basic principles of gravitational attraction and to derive Kepler’s laws of planetary motion. 12.1 INTRODUCTION TO VECTOR-VALUED FUNCTIONS In Section 11.5 we discussed parametric equations of lines in 3-space. In this section we will discuss more general parametric curves in 3-space, and we will show how vector notation can be used to express parametric equations in 2-space and 3-space in a more compact form. This will lead us to consider a new kind of function—namely, functions that associate vectors with real numbers. Such functions have many important appli- cations in physics and engineering. PARAMETRIC CURVES IN 3-SPACE Recall from Section 10.1 that if f and g are well-behaved functions, then the pair of para- metric equations x = f (t), y = g(t) (1) generates a curve in 2-space that is traced in a specific direction as the parameter t in- creases.
eBook - PDF- Ken Binmore, Joan Davies(Authors)
- 2002(Publication Date)
- Cambridge University Press(Publisher)
Five Vector functions In this chapter we introduce the idea of a vector function and examine the instances in which it can be visualised. We have seen in Chapter 3 how partial derivatives are used to generalise the idea of the derivative of a real valued function of one real variable to that of a total derivative of a function of several variables. We extend this further to the idea of the total derivative of a vector function. This object is defined to be a matrix, and an understanding of the material covered in this chapter requires a good knowledge of the matrix algebra surveyed in Chapter 1 . (Note especially § 1.10 .) Although it is quite demanding at first to think systematically in terms of matrices and vectors, the payoff is considerable. The theory is elegant, the formulas are easily remembered, and the amount that has to be written down is very much less than would otherwise be necessary. Some effort with this chapter will therefore be amply repaid. 5.1 Vector Valued Functions In this chapter we shall study functions f : R n → R m Such a function assigns to each x ∈ R n a unique vector y ∈ R m . We write y = f ( x ) Suppose that x = ⎛ ⎜ ⎜ ⎜ ⎝ x 1 x 2 . . . x n ⎞ ⎟ ⎟ ⎟ ⎠ and y = ⎛ ⎜ ⎜ ⎜ ⎝ y 1 y 2 . . . y m ⎞ ⎟ ⎟ ⎟ ⎠ Then the vector function f assigns to each set of values of the n independent variables x 1 , x 2 , . . . , x n a unique set of values of the m dependent variables y 1 , y 2 , . . . , y m . The vector equation y = f ( x ) can be written out as a list of m real equations as below. y 1 = f 1 ( x 1 , x 2 , . . . , x n ) y 2 = f 2 ( x 1 , x 2 , . . . , x n ) · · · y m = f m ( x 1 , x 2 , . . . , x n ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ 149 Chapter 5. Vector functions A Vector Valued Function can therefore be studied by looking at the m real valued functions f 1 , f 2 , . . . , f m . We shall say that f 1 , f 2 , . . . , f m are the component functions of f . We shall concentrate mainly on the case f : R 2 → R 2 , which represents the mapping of planes.
eBook - PDF- William Cox(Author)
- 1998(Publication Date)
- Butterworth-Heinemann(Publisher)
8.1 Introduction: what is a vector? You are probably aware from your previous mathematics that one of the most powerful tools for dealing with functions of more than one variable is the use of vector notation. Thus, when describing the motion of a projectile in two dimen-sions we have the option of using (x(t), yet)) coordinates to represent position at any time or of using the position vector ret) = x(t)i + y(t)j with i, j the usual basis vectors. The point about using vectors here is that it effec-tively reduces the number of 'variables' -instead of two coordinates, we have one vector. A natural question therefore is to what extent one can use vectors in the theory of functions of several variables -which is what this book is about. So far, we have managed virtually without them -and indeed it is not always clear that they would be of much use. But in fact they are -the bulk of the rest of this book is essentially about 'vector calculus'. Why should this be so? Why does most of the differen-tiation and integration that we cover occur in a vector context? Is it just the case that vectors are simply a good shorthand notation whenever we have more than one variable? No, there is a deep significance to the idea of a vector which is simply not brought out in the introductory treatments to which you may have been exposed so far. Vectors are of fundamental importance and utility in all physical applications -they are far more than simply a shorthand notation. The reason for this is rather subtle, and requires a leap into abstraction which really has to be deferred until sufficient mathematical foundations have been laid -that is why elementary treatments are usually incomplete. To get an insight into the new view-point, let us stick with our two-dimensional projectile. You may have a number of different definitions, but they are most likely to come from the following list: 1. Any quantity having both a magnitude and a direction, such as velocity as opposed to speed.
eBook - ePubVector Analysis and Cartesian Tensors
Third Edition
- Donald Edward Bourne(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
Vector functions of a real variable. Differential geometry of curves 3 3.1 VECTOR FUNCTIONS AND THEIR GEOMETRICAL REPRESENTATION The reader should already be familiar with the idea of a real function f (x), say, of a real variable x. In this chapter the properties of vector functions F (t) of a real variable t will be discussed. Suppose that the components of the vector F (t) = (f 1 (t), f 2 (t), f 3 (t)) (3.1) are single-valued functions of a real variable t. Then F (t) is called a vector function of t. In most applications t is a continuous variable and f 1 (t), f 2 (t), f 3 (t) are continuous * over some interval of t. If this is so F (t) is said to be a continuous vector function of t. Examples of such functions are F (t) = (2, t 1 2, sin t) 0 ⩽ t < ∞ and F (t) = { t 3, t, 3 for − ∞ < t ⩽ 2 (2 t 2, 2, 6 t − 1) for 2 < t < ∞. The vector. function F (t) = (1, t, t − 1) − 1 ⩽ t ⩽ 1 is not continuous, because as t increases through zero the z -component t −1 changes value from −∞ to ∞. Geometrical representation of vector functions Let a continuous vector function F (t) be represented by the position vector O P →, where O is the origin and P is the point (f 1 (t), f 2 (t), f 3 (t)) Then, as t varies over its permissible range of values, P describes a continuous curve (in three dimensions, see Fig. 3.1). It is clear that in general both the magnitude and direction of F (t) will vary with t. (A vector is constant only if both its magnitude and direction are fixed.) The equation O P → = r = F (t), (3.2) where r = (x, y, z), is called the parametric equation of the curve described by P. Fig. 3.1 A curve in three dimensions described by a point P whose position is given by an equation of the type (3.2). Fig. 3.2 A curve which intersects itself at X. It should be noted that, although F (t) is taken to be a single-valued function of t, two (or possibly more) values of t may correspond to the same vector F
eBook - PDFCalculus
Resequenced for Students in STEM
- David Dwyer, Mark Gruenwald(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
In the next section, we’ll see that differentiation and integration of vector-valued functions can be con- sidered in a component-wise fashion as well. More importantly, we’ll see that these calculus concepts have some important applications to projectile motion. 13.2. VECTOR-VALUED FUNCTIONS 765 13.2 Exercises Exercises 1–6 Answer true or false. Assume r(t) = f (t) i + g(t) j + h(t) k, where f , g, and h are scalar-valued functions. 1. The range of r consists of real numbers. 2. If a is in the domains of f , g, and h, then a is in the domain of r. 3. The graph of r can be described as a curve in R 3 . 4. The projection of the graph of r onto the xy-plane can be described as a curve in R 2 . 5. If lim t→a r(t) exists, then r is continuous at t = a. 6. If f , g, and h are continuous at a, then lim t→a r(t) = r(a). Exercises 7–16 Find the domain. 7. r(t) = √ t i + 1 t j 8. r(t) = 1 t - 1 i + 1 √ t j 9. r(t) = 1 t 2 - 1 i + t 2 j + √ t + 2 k 10. r(t) = (t 2 - 1) i + ln(t + 1) j + 1 t k 11. r(t) = e t i + cos t j + sin t k 12. r(t) = arcsin t i + 1 t j + 1 t - 1 k 13. r(t) = tan t j + cot t k 14. r(t) = √ t 2 + 1 i + t 2 j + 1 t 2 + 1 k 15. r(t) = ln(2 - t) i + 1 √ t - 1 j + t k 16. r(t) = e -πt j + tan t k Exercises 17–24 Sketch the curve represented by the given func- tion, and indicate the orientation using arrows. 17. r(t) = ht 2 , 2ti 18. r(t) = ht + 1, t 2 i 19. r(t) = h2 cos t, 2 sin ti 20. r(t) = hsin t, - cos ti 21. r(t) = 3 sin t i - 3 cos t j + t k 22. r(t) = 2 cos t i + 2 sin t j + t 2 k 23. r(t) = t i + 3 j + (t 2 + 1) k 24. r(t) = 2 i + t 2 j + 2t k Exercises 25–30 Identify the graph of the given function. Choose from I–VI. I. II. III. IV. V. VI. 25. r(t) = ht, t sin t, t cos ti, t ≥ 0 26. r(t) = ht, 3 sin t, 3 cos ti, t ≥ 0 27. r(t) = hcos t, 3 sin t, 3 cos ti 28. r(t) = ht, t 2 , t 2 i 29. r(t) = ht 2 , t 2 , t 2 i, t ≥ 0 30.
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