DOCSLIB.ORG

Laplace Transforms SCHOOL of ENGINEERING & BUILT

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Laplace Transforms SCHOOL of ENGINEERING & BUILT SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Laplace Transforms 1. Introduction ……………………………………. 1 2. Definition ………………………………………. 2 3. Properties of the Laplace Transform …………… 4 4. Inverse Laplace Transforms …………………… 6 5. Using Laplace Transforms to Solve ODEs ……. 9 6. The Step Function ……………………………... 11 7. The Dirac Delta Function ……………………… 20 8. Convolution ……………………………………. 25 9. A Note on Control Theory …………………….. 26 Tutorial Exercises ……………………………… 27 Answers to Tutorial Exercises …………………. 33 1 Laplace Transforms 1) Introduction The idea of transforming a “difficult” problem into an “easier” problem is one that is used widely in mathematics. Diagrammatically we have Problem Transform Transformed Problem Difficult Solve Solution Transformed Inverse Solution Transform There are many types of transforms available to mathematicians, engineers and scientists. We are going to examine one such transformation, the Laplace transform, which can be used to solve certain types of differential equations and also has applications in control theory. 2) Definition The Laplace transform operates on functions of t . Given a function f (t ) we define its Laplace transform as ∞ L[ f (t ) ] = f (t ) e − s t dt 0 where s is termed the Laplace variable. We sometimes denote the Laplace transform by F ( s ) or f . We start off with a function of t and end up with a function of s ; s is, in fact, a complex variable, but this need not concern us too much. Because the function of t is often some form of time signal, we often talk about moving from the time domain to the Laplace domain when we perform a Laplace transformation. Note: Laplace transforms are only concerned with functions where t ≥ 0. 2 Examples (1) f (t) = 1: ∞ L[1] = 1.e −st dt 0 t = ∞ = − 1 −st e s t = 0 Strictly speaking, we can’t set t equal to ∞, but we can “take the limit” as t heads towards infinity. Providing s > 0 , thereby ensuring that we have a negative exponential, the limit of the inside of the square brackets as t tends to infinity will be zero. Also, since e 0 = 1, this leaves us with 1 1 L[1] = 0 − − = . s s 1 So f (t) = 1 → L[1] = s (2) f (:t) = t ∞ − s t L[t ] = t .e dt 0 This integral requires integration by parts to complete the process, but with the same assumptions regarding s as before, it can readily be shown that 1 L[t ] = . s 2 1 So f (t) = t → L[t ] = s 2 Very quickly the integrations required to complete the Laplace transformation become difficult and messy. For this reason, we generally work from a table of pre-determined Laplace transforms (see Appendix). 3 Examples Using Table of Laplace Transforms (3) (a) Determine L[t 3 ]. t 3 is not in the table explicitly, but t n is: n ! L[t n ] = s n + 1 For t 3 we require n = 3 : 3! 3×2×1 6 L[t 3 ] = = = s 4 s 4 s 4 (b) Determine L[ e − 2 t ]. From table: −α 1 L[ e t ] = . s + α Set α = 2 : − 1 L[ e 2 t ] = s + 2 (c) Determine L[sin ( 4t ) ]. From table: ω L[sin (ω t ) ] = . s 2 + ω 2 Set ω = 4 : 4 4 L[sin ( 4t ) ] = = s 2 + 4 2 s 2 + 16 4 3) Properties of the Laplace Transform The Laplace transform has several special properties that make it a useful mathematical tool. We consider some of these now. a) Linearity Suppose we have two functions along with their respective Laplace transforms: L[ f (t ) ] = F ( s ) L[ g ( t ) ] = G ( s ) . The property of linearity means that L[ a f (t ) + b g (t ) ] = a F ( s ) + b G ( s ) providing a and b are constants. This makes the transformation of a string of functions straightforward. Example s ω 2 s + 3 ω (4) L[ 2 cos (ω t ) + 3 sin (ω t ) ] = 2 + 3 = s 2 + ω 2 s 2 + ω 2 s 2 + ω 2 Warning: The Laplace transform of a product is NOT EQUAL TO the product of the individual Laplace transforms. We have to invoke other properties of the Laplace transform to deal with such. b) The First Shifting Theorem Suppose a function f (t ) has the Laplace transform F ( s ) . It is easily demonstrated that L[ e −α t f ( t ) ] = F ( s + α ) . Example (5) From tables and Example (3)(a) we have 6 L[t 3 ] = . [1] s 4 By the first shifting property − 6 L[ e 2 t t 3 ] = . [2] ( s + 2) 4 To obtain [2] from [1] we merely replace s by ( s + 2) . 5 c) Transformation of Derivatives As before, denote the Laplace transform (LT) of f (t ) by F ( s ) . Now consider the LT of the derivative of f (t ) , denoted by f ( t ) : ∞ − s t L[ f (t ) ] = f (t ) e dt . 0 Integrate by parts (integrating f ( t ) and differentiating e − s t ): ∞ ∞ = []f ( t ) e − s t − f ( t ) ( − s ) e − s t dt 0 0 ∞ − s t = 0 − f ( 0) + s f ( t ) e dt 0 = − f ( 0 ) + s F ( s ) . So L[ f (t ) ] = s F ( s ) − f ( 0 ) . We have expressed the Laplace transform of a derivative in terms of the Laplace transform of the undifferentiated function. In effect, the Laplace transform has converted the operation of differentiation into the simpler operation of multiplication by s . In a similar fashion, using repeated integration by parts, we can show that L[ f (t ) ] = s 2 F ( s ) − s f ( 0) − f ( 0 ) . This is one of the most important properties of the Laplace transform. The Laplace transform “gets rid of” derivatives; just the thing for solving differential equations! When we come to solve differential equations using Laplace transforms we shall use the following alternative notation: L[x] = x L[ x ] = s x − x(0) 2 L[ x] = s x − s x(0) − x (0) . However, before we can solve differential equations, we need to look at the reverse process of finding functions of t from given Laplace transforms. 6 4) Inverse Laplace Transforms So far, we have looked at how to determine the LT of a function of t , ending up with a function of s . The table of Laplace transforms collects together the results we have considered, and more. When we apply Laplace transforms to solve problems we will have to invoke the inverse transformation. That is, given a Laplace transform F ( s ) we will want to determine the corresponding f (t ) . In general we have γ + ∞ − 1 j L 1 [ F ( s ) ] = F ( s ) e s t ds , 2π j γ − j ∞ where the evaluation of the integral requires a knowledge of complex analysis, which is too difficult to consider here. Instead, we shall rely on the table of Laplace transforms used in reverse to provide inverse Laplace transforms. This will mean manipulating a given Laplace transform until it looks like one or more entries in the right of the table. The inverse is then determined from the left of the table. The following examples illustrate the main algebraic techniques required. These include completing the square, factorisation and the formation of partial fractions. See separate documents for the details of completing the square and partial fractions. Examples 3 (6) Invert the Laplace transform . s 6 n! The closest entry in the table is which is inverted as: sn + 1 n! → t n . sn + 1 Setting n = 5 : 5! 120 → t 5 . [Note: 5! = 5×4×3×2×1 = 120 ], so → t 5 . s 6 s 6 In the given Laplace transform there is a 3 in the numerator but we would like there to be a 120 to match the table entry. We can re-write the transform providing we do not alter its “net value”: 3 1 3 120 1 120 = 3 = = . 6 6 6 6 s s 120 s 40 s The term in the square brackets is now exactly the table entry so we can invert that and simply multiply by the fraction in front: 3 1 → t 5 . s 6 40 7 1 (7) Invert . s 2 + 2 s + 5 This requires the technique of “completing the square” and a little bit of fine tuning to re-write it in a form that can be inverted from tables. Note that the closest entry in the table gives: ω → e −α t sin (ω t ) . ( s + α ) 2 + ω 2 Now complete the square in the denominator: s 2 + 2 s + 5 = ( s + 1) 2 − 1 + 5 = ( s + 1) 2 + 4 . The given transform becomes: 1 1 1 = = . s 2 + 2 s + 5 ( s + 1) 2 + 4 ( s + 1) 2 + 2 2 This is now very close to the table entry with α = 1 and ω = 2 . We would like there to be a 2 on top so “fine tuning” gives 1 = 1 2 2 2 2 2 , ( s + 1) + 2 2 ( s + 1) + 2 where the term in the square brackets is exactly the table entry with α = 1 and ω = 2 . Inverting this and multiplying by the fraction gives 1 2 − → 1 t 2 2 2 e sin ( 2t ) . 2 ( s + 1) + 2 s + 5 (8)(a) Invert . s 2 − 2 s − 3 Factorising the denominator and splitting the result into its partial fractions deals with this one. Note that the details of the partial fraction expansion have been omitted. s + 5 s + 5 2 1 = = − s 2 − 2 s − 3 ( s − 3) ( s + 1) s − 3 s + 1 1 −α Use → e t twice with α = − 3 and α = +1 to give ( s + α ) 2 1 − − → 2e 3t − e t s − 3 s + 1 8 s + 5 (8)(b) Invert by completing the square.
Load more

Recommended publications
  • Plotting, Derivatives, and Integrals for Teaching Calculus in R
    Plotting, Derivatives, and Integrals for Teaching Calculus in R Daniel Kaplan, Cecylia Bocovich, & Randall Pruim July 3, 2012 The mosaic package provides a command notation in R designed to make it easier to teach and to learn introductory calculus, statistics, and modeling. The principle behind mosaic is that a notation can more effectively support learning when it draws clear connections between related concepts, when it is concise and consistent, and when it suppresses extraneous form. At the same time, the notation needs to mesh clearly with R, facilitating students' moving on from the basics to more advanced or individualized work with R. This document describes the calculus-related features of mosaic. As they have developed histori- cally, and for the main as they are taught today, calculus instruction has little or nothing to do with statistics. Calculus software is generally associated with computer algebra systems (CAS) such as Mathematica, which provide the ability to carry out the operations of differentiation, integration, and solving algebraic expressions. The mosaic package provides functions implementing he core operations of calculus | differen- tiation and integration | as well plotting, modeling, fitting, interpolating, smoothing, solving, etc. The notation is designed to emphasize the roles of different kinds of mathematical objects | vari- ables, functions, parameters, data | without unnecessarily turning one into another. For example, the derivative of a function in mosaic, as in mathematics, is itself a function. The result of fitting a functional form to data is similarly a function, not a set of numbers. Traditionally, the calculus curriculum has emphasized symbolic algorithms and rules (such as xn ! nxn−1 and sin(x) ! cos(x)).
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • 5 Graph Theory
    last edited March 21, 2016 5 Graph Theory Graph theory – the mathematical study of how collections of points can be con- nected – is used today to study problems in economics, physics, chemistry, soci- ology, linguistics, epidemiology, communication, and countless other fields. As complex networks play fundamental roles in financial markets, national security, the spread of disease, and other national and global issues, there is tremendous work being done in this very beautiful and evolving subject. The first several sections covered number systems, sets, cardinality, rational and irrational numbers, prime and composites, and several other topics. We also started to learn about the role that definitions play in mathematics, and we have begun to see how mathematicians prove statements called theorems – we’ve even proven some ourselves. At this point we turn our attention to a beautiful topic in modern mathematics called graph theory. Although this area was first introduced in the 18th century, it did not mature into a field of its own until the last fifty or sixty years. Over that time, it has blossomed into one of the most exciting, and practical, areas of mathematical research. Many readers will first associate the word ‘graph’ with the graph of a func- tion, such as that drawn in Figure 4. Although the word graph is commonly Figure 4: The graph of a function y = f(x). used in mathematics in this sense, it is also has a second, unrelated, meaning. Before providing a rigorous definition that we will use later, we begin with a very rough description and some examples.
    [Show full text]
  • Lecture 11: Graphs of Functions Definition If F Is a Function With
    Lecture 11: Graphs of Functions Definition If f is a function with domain A, then the graph of f is the set of all ordered pairs f(x; f(x))jx 2 Ag; that is, the graph of f is the set of all points (x; y) such that y = f(x). This is the same as the graph of the equation y = f(x), discussed in the lecture on Cartesian co-ordinates. The graph of a function allows us to translate between algebra and pictures or geometry. A function of the form f(x) = mx+b is called a linear function because the graph of the corresponding equation y = mx + b is a line. A function of the form f(x) = c where c is a real number (a constant) is called a constant function since its value does not vary as x varies. Example Draw the graphs of the functions: f(x) = 2; g(x) = 2x + 1: Graphing functions As you progress through calculus, your ability to picture the graph of a function will increase using sophisticated tools such as limits and derivatives. The most basic method of getting a picture of the graph of a function is to use the join-the-dots method. Basically, you pick a few values of x and calculate the corresponding values of y or f(x), plot the resulting points f(x; f(x)g and join the dots. Example Fill in the tables shown below for the functions p f(x) = x2; g(x) = x3; h(x) = x and plot the corresponding points on the Cartesian plane.
    [Show full text]
  • Maximum and Minimum Definition
    Maximum and Minimum Definition: Definition: Let f be defined in an interval I. For all x, y in the interval I. If f(x) < f(y) whenever x < y then we say f is increasing in the interval I. If f(x) > f(y) whenever x < y then we say f is decreasing in the interval I. Graphically, a function is increasing if its graph goes uphill when x moves from left to right; and if the function is decresing then its graph goes downhill when x moves from left to right. Notice that a function may be increasing in part of its domain while decreasing in some other parts of its domain. For example, consider f(x) = x2. Notice that the graph of f goes downhill before x = 0 and it goes uphill after x = 0. So f(x) = x2 is decreasing on the interval (−∞; 0) and increasing on the interval (0; 1). Consider f(x) = sin x. π π 3π 5π 7π 9π f is increasing on the intervals (− 2 ; 2 ), ( 2 ; 2 ), ( 2 ; 2 )...etc, while it is de- π 3π 5π 7π 9π 11π creasing on the intervals ( 2 ; 2 ), ( 2 ; 2 ), ( 2 ; 2 )...etc. In general, f = sin x is (2n+1)π (2n+3)π increasing on any interval of the form ( 2 ; 2 ), where n is an odd integer. (2m+1)π (2m+3)π f(x) = sin x is decreasing on any interval of the form ( 2 ; 2 ), where m is an even integer. What about a constant function? Is a constant function an increasing function or decreasing function? Well, it is like asking when you walking on a flat road, as you going uphill or downhill? From our definition, a constant function is neither increasing nor decreasing.
    [Show full text]
  • Chapter 3. Linearization and Gradient Equation Fx(X, Y) = Fxx(X, Y)
    Oliver Knill, Harvard Summer School, 2010 An equation for an unknown function f(x, y) which involves partial derivatives with respect to at least two variables is called a partial differential equation. If only the derivative with respect to one variable appears, it is called an ordinary differential equation. Examples of partial differential equations are the wave equation fxx(x, y) = fyy(x, y) and the heat Chapter 3. Linearization and Gradient equation fx(x, y) = fxx(x, y). An other example is the Laplace equation fxx + fyy = 0 or the advection equation ft = fx. Paul Dirac once said: ”A great deal of my work is just playing with equations and see- Section 3.1: Partial derivatives and partial differential equations ing what they give. I don’t suppose that applies so much to other physicists; I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful ∂ mathematical relations If f(x, y) is a function of two variables, then ∂x f(x, y) is defined as the derivative of the function which maybe don’t have any physical meaning at all. Sometimes g(x) = f(x, y), where y is considered a constant. It is called partial derivative of f with they do.” Dirac discovered a PDE describing the electron which is consistent both with quan- respect to x. The partial derivative with respect to y is defined similarly. tum theory and special relativity. This won him the Nobel Prize in 1933. Dirac’s equation could have two solutions, one for an electron with positive energy, and one for an electron with ∂ antiparticle One also uses the short hand notation fx(x, y)= ∂x f(x, y).
    [Show full text]
  • 2.8 the Derivative As a Function
    2 Derivatives Copyright © Cengage Learning. All rights reserved. 2.8 The Derivative as a Function Copyright © Cengage Learning. All rights reserved. The Derivative as a Function We have considered the derivative of a function f at a fixed number a: Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain 3 The Derivative as a Function Given any number x for which this limit exists, we assign to x the number f′(x). So we can regard f′ as a new function, called the derivative of f and defined by Equation 2. We know that the value of f′ at x, f′(x), can be interpreted geometrically as the slope of the tangent line to the graph of f at the point (x, f(x)). The function f′ is called the derivative of f because it has been “derived” from f by the limiting operation in Equation 2. The domain of f′ is the set {x | f′(x) exists} and may be smaller than the domain of f. 4 Example 1 – Derivative of a Function given by a Graph The graph of a function f is given in Figure 1. Use it to sketch the graph of the derivative f′. Figure 1 5 Example 1 – Solution We can estimate the value of the derivative at any value of x by drawing the tangent at the point (x, f(x)) and estimating its slope. For instance, for x = 5 we draw the tangent at P in Figure 2(a) and estimate its slope to be about , so f′(5) ≈ 1.5.
    [Show full text]
  • 3.2 the Graph of a Function
    ✎ ✍3.2 ✌ The graph of a function Introduction A very useful pictorial representation of a function is the graph. On this leaflet we remind you of important conventions when graph plotting. 1. The graph of a function Consider the function f(x)=5x +4. We can choose several values for the input and calculate the corresponding outputs. We have done this for integer values of x between −3 and 3 and the results are shown in the table. x −3 −2 −10123 f(x) −11 −6 −1491419 To plot the graph we first draw a pair of axes - a vertical axis and a horizontal axis. These are drawn at right-angles to each other and intersect at the origin O as shown below. y = f(x) y =5x +4 vertical axis 15 10 The point with coordinates (2, 14) 5 -3 -2 -1 O 123x -5 horizontal axis -10 Each pair of input and output values can be represented on a graph by a single point. The input values are measured along the horizontal axis and the output values along the vertical axis. A uniform scale is drawn on each axis sufficient to accommodate all the required points. The points plotted in this way are then joined together, in this case by a straight line. This is the graph of the function. Each point on the graph can be represented by a pair of coordinates in the form (x, f(x)). Each axis should be labelled to show its variable. 2. Dependent and independent variables The horizontal axis is often called the x axis.
    [Show full text]
  • The Laplace Transform (Intro)
    24 The Laplace Transform (Intro) The Laplace transform is a mathematical tool based on integration that has a number of appli- cations. It particular, it can simplify the solving of many differential equations. We will find it particularly useful when dealing with nonhomogeneous equations in which the forcing func- tions are not continuous. This makes it a valuable tool for engineers and scientists dealing with “real-world” applications. By the way, the Laplace transform is just one of many “integral transforms” in general use. Conceptually and computationally, it is probably the simplest. If you understand the Laplace transform, then you will find it much easier to pick up the other transforms as needed. 24.1 Basic Definition and Examples Definition, Notation and Other Basics Let f be a ‘suitable’ function (more on that later). The Laplace transform of f , denoted by either F or L[ f ] , is the function given by ∞ st F(s) L[ f ] f (t)e− dt . (24.1) = |s = Z0 !◮Example 24.1: For our first example, let us use 1 if t 2 f (t) ≤ . = ( 0 if 2 < t This is the relatively simple discontinuous function graphed in figure 24.1a. To compute the Laplace transform of this function, we need to break the integral into two parts: ∞ st F(s) L[ f ] f (t)e− dt = |s = Z0 2 st ∞ st f (t) e− dt f (t) e− dt = 0 + 2 Z 1 Z 0 2 2 |{zst} ∞ |{z} st e− dt 0 dt e− dt . = + = Z0 Z2 Z0 471 472 The Laplace Transform 2 1 1 0 1 2 T 2 S (a) (b) Figure 24.1: The graph of (a) the discontinuous function f (t) from example 24.1 and (b) its Laplace transform F(s) .
    [Show full text]
  • Integration Not for Sale
    Not for Sale 5 Integration Chapter Preview We are now at a critical point in the calculus story. Many would 5.1 Approximating Areas argue that this chapter is the cornerstone of calculus because it explains the relationship under Curves between the two processes of calculus: differentiation and integration. We begin by ex- 5.2 Definite Integrals plaining why finding the area of regions bounded by the graphs of functions is such an important problem in calculus. Then you will see how antiderivatives lead to definite in- 5.3 Fundamental Theorem tegrals, which are used to solve the area problem. But there is more to the story. You will of Calculus also see the remarkable connection between derivatives and integrals, which is expressed 5.4 Working with Integrals in the Fundamental Theorem of Calculus. In this chapter, we develop key properties of 5.5 Substitution Rule definite integrals, investigate a few of their many applications, and present the first of sev- eral powerful techniques for evaluating definite integrals. 5.1 Approximating Areas under Curves The derivative of a function is associated with rates of change and slopes of tangent lines. We also know that antiderivatives (or indefinite integrals) reverse the derivative operation. Figure 5.1 summarizes our current understanding and raises the question: What is the geo- metric meaning of the integral? The following example reveals a clue. f(x) f Ј(x) ͵ f (x)dx Slopes of ??? tangent lines Figure 5.1 Area under a Velocity Curve Consider an object moving along a line with a known position function.
    [Show full text]
  • Infinitesimal Methods in Mathematical Economics
    Infinitesimal Methods in Mathematical Economics Robert M. Anderson1 Department of Economics and Department of Mathematics University of California at Berkeley Berkeley, CA 94720, U.S.A. and Department of Economics Johns Hopkins University Baltimore, MD 21218, U.S.A. January 20, 2008 1The author is grateful to Marc Bettz¨uge, Don Brown, Hung- Wen Chang, G´erard Debreu, Eddie Dekel-Tabak, Greg Engl, Dmitri Ivanov, Jerry Keisler, Peter Loeb, Paul MacMillan, Mike Magill, Andreu Mas-Colell, Max Stinchcombe, Cathy Wein- berger and Bill Zame for their helpful comments. Financial sup- port from Deutsche Forschungsgemeinschaft, Gottfried-Wilhelm- Leibniz-F¨orderpreis is gratefully acknowledged. Contents 0Preface v 1 Nonstandard Analysis Lite 1 1.1 When is Nonstandard Analysis Useful? . 1 1.1.1 Large Economies . 2 1.1.2 Continuum of Random Variables . 4 1.1.3 Searching For Elementary Proofs . 4 1.2IdealElements................. 5 1.3Ultraproducts.................. 6 1.4InternalandExternalSets........... 9 1.5NotationalConventions............. 11 1.6StandardModels................ 12 1.7 Superstructure Embeddings . 14 1.8AFormalLanguage............... 16 1.9TransferPrinciple................ 16 1.10Saturation.................... 18 1.11InternalDefinitionPrinciple.......... 19 1.12 Nonstandard Extensions, or Enough Already with the Ultraproducts . 20 1.13HyperfiniteSets................. 21 1.14 Nonstandard Theorems Have Standard Proofs 22 2 Nonstandard Analysis Regular 23 2.1 Warning: Do Not Read this Chapter . 23 i ii CONTENTS 2.2AFormalLanguage............... 23 2.3 Extensions of L ................. 25 2.4 Assigning Truth Value to Formulas . 28 2.5 Interpreting Formulas in Superstructure Em- beddings..................... 32 2.6TransferPrinciple................ 33 2.7InternalDefinitionPrinciple.......... 34 2.8NonstandardExtensions............ 35 2.9TheExternalLanguage............. 36 2.10TheConservationPrinciple.......... 37 3RealAnalysis 39 3.1Monads..................... 39 3.2OpenandClosedSets............. 44 3.3Compactness.................
    [Show full text]
  • 2.1 Functions: Definition, Notation a Function Is a Rule (Correspondence)
    Chapter 2: 2.1 Functions: definition, notation A function is a rule (correspondence) that assigns to each element x of one set , say X, one and only one element y of another set, Y. The set X is called the domain of the function and the set of all elements of the set Y that are associated with some element of the set X is called the range of the function. For a function described by above diagram, the domain is the set X= {1,2,3,4} and the range is the set {a,b,c}. The elements of a set X are often called inputs and the elements of the set Y are called outputs and a function can be visualized as a machine, that takes x as an input and returns y as an output. The domain of a function is the set of all inputs x that return an output. The range of a function is the set of all outputs y. We will deal with functions for which both domain and the range are the set (or subset) of real numbers Most often functions in mathematics are defined by a formula or by a graph. Very often we define a function by giving a formula or an equation which specifies how the y is obtained when x is given. Example: The equation y = 3x +2 defines a function since we can easily find the output y when the input, x, is given. To obtain the y that correspond to a given x, multiply x by 3 and add 2 to the result.
    [Show full text]