Introduction to Vector Functions
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1 Introduction to Vector Functions Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Vector Functions Today 1 / 14
2 Introduction Until now, the functions we studied took a real number as input and gave another real number as output. Hence, when defining a function, it was usually enough to simply specify a formula for it. This is no longer the case. In the future, when we define a function we will also need to specify the kind of objects it acts on (numbers, vectors,...) and the kind of output it produces (numbers, vectors,...). The functions we study here will take a real number as input and give a vector as output. Philippe B. Laval (KSU) Vector Functions Today 2 / 14
3 Notation We indicate the kind of input and output as follows: f : A B where A denotes the set of input values and B the set of output values. Remember the name of classical sets: N is the set of natural numbers, Q is the set of rational numbers, R is the set of real numbers, R 2 is the set of pairs of real numbers, R 3 is the set of triplets and so on. All the function you have studied so far took a real number as input and produced another real number as output. For such a function f, we would have written f : R R. Philippe B. Laval (KSU) Vector Functions Today 3 / 14
4 Notation Consider a function f of two variables, say f (x, y) = x 2 + y 2. This functions takes as input a pair of numbers and produces as output a real number. To write its definition, we would write f : R 2 R (x, y) x 2 + y 2 f : R R 2 indicates a function which takes as input a real number and produces a pair or a 2D vector. It is the kind of functions we will study. f : R R 3 indicates a function which takes as input a real number and produces a triplet or a 3D vector. It is the kind of functions we will study. Philippe B. Laval (KSU) Vector Functions Today 4 / 14
5 Definitions Definition A vector-valued function or simply a vector function is a function of the form r : R R 2 or r : R R 3. 1 In the plane, such a function will be of the form r : R R 2 defined by r (t) = f (t), g (t). 2 In space, such a function will be of the form r : R R 3 defined by r (t) = f (t), g (t), h (t). 3 We usually use t to denote the independent variable, often called a parameter. 4 The functions f, g, h are called component functions. They are functions for which both the input and output values are real numbers. In other words, f : R R, g : R R, and h : R R. 5 The domain of a vector function is the set of values of t for which r (t) is defined. That is, it is the set of values of t for which all the component functions are defined. Philippe B. Laval (KSU) Vector Functions Today 5 / 14
6 s Consider the vector function r (t) = sin t, cos t. Find its domain. Consider the vector function r (t) = t 2, ln (3 t), t. Find its domain. Philippe B. Laval (KSU) Vector Functions Today 6 / 14
7 Limits Limit and continuity of vector functions are defined in terms of limits and continuity of their components. Definition If r (t) = f (t), g (t), h (t) then r (t) = lim f (t), lim g (t), lim h (t) t a t a t a lim t a Since the component functions are real-valued functions of one variable, we can use all the tools we learned in calculus I to find their limits. In particular, all the limit rules we learned in calculus I are also true for vector functions. Find lim t 0 r (t) for r (t) = t 2, ln (3 t), t. Philippe B. Laval (KSU) Vector Functions Today 7 / 14
8 Continuity Definition The vector function r (t) is continuous if and only if lim r (t) = r (a) t a From the definition, we see that a vector function is continuous if and only if its component functions are continuous. Since the component functions are real-valued functions of one variable, we can use all the theorems studied in calculus I. Find where r (t) = t 2, ln (3 t), t is continuous. Find where sin t r (t) =, t is continuous. t Philippe B. Laval (KSU) Vector Functions Today 8 / 14
9 Plane and Space Curves Definition Let f, g, h be three continuous real-valued functions. 1 If { r (t) = f (t), g (t) then the set on points (x, y) where x = f (t) and t varies throughout some interval I is called a y = g (t) plane curve. r (t) is the position vector of the point (f (t), g (t)). 2 If r (t) = f (t), g (t), h (t) then the set on points (x, y, z) where x = f (t) y = g (t) and t varies throughout some interval I is called a z = h (t) space curve. r (t) is the position vector of the point (f (t), g (t), h (t)). 3 The above equations are called the parametric equations of the curve. Philippe B. Laval (KSU) Vector Functions Today 9 / 14
10 Plane and Space Curves The curve is being traced by the tip of the position vector as t varies through the interval I as illustrated below. t is called the parameter. A space curve and its position vector at different values of the parameter We will not focus on plotting plane or space curves by hand. However, students should have a basic knowledge of some basic curves. They are illustrated in the examples below. Students should also know how to plot plane and space curves using their favorite computer program. Philippe B. Laval (KSU) Vector Functions Today 10 / 14
11 Plane and Space Curves Recall the parametric equation of the line through (x 1, y 1, z 1 ) with direction vector a, b, c is where t R. x = x 1 + at y = y 1 + bt z = z 1 + ct The parametric equations of the line through two points (x 0, y 0, z 0 ) and (x 1, y 1, z 1 ) are x = (1 t) x 0 + tx 1 y = (1 t) y 0 + ty 1 z = (1 t) z 0 + tz 1 Philippe B. Laval (KSU) Vector Functions Today 11 / 14
12 Plane and Space Curves { x = cos (t) The plane curve given by the parametric equations y = sin (t) t [0, 2π] is a circle of radius 1 centered at the origin. for { x = cos (2πt) The plane curve given by the parametric equations y = sin (2πt) t [0, 1] is also circle of radius 1 centered at the origin. for The last two examples illustrate the fact that different parametric equations can trace the same curve. If we think of t as time, then in the second case, the circle is being traced faster than in the first. Also, when looking at a curve, its shape is important. The direction in which it is being traced is also important. Philippe B. Laval (KSU) Vector Functions Today 12 / 14
13 Plane and Space Curves { If y = f (x), then the corresponding parametric equations are x = t. This shows that it is easy to write the parametric equations y = f (t) of a curve given a function representing the curve. { For example, the x = t parametric equations of a parabola y = x 2 are y = t 2. Another curve we will use often is the helix. Its parametric equations are x = cos t of the form y = sin t for t R. Its trace in the xy-plane is a unit z = t circle of radius 1 centered at the origin. As t increases, the z coordinate of the point corresponding to the position vector will increase. The position vector will actually trace an upward spiral. Philippe B. Laval (KSU) Vector Functions Today 13 / 14
14 Exercises Review the notions of limits and continuity from Calculus I. See the problems at the end of my notes on vector functions: definitions, limits and continuity. Review derivatives and integrals. Philippe B. Laval (KSU) Vector Functions Today 14 / 14
f : R 2 R (x, y) x 2 + y 2
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