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5 - Techniques of Integration
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Summary
Our work on differentiation in the previous chapter has brought us to a fork in the road. We can pursue the implications of the first fundamental theorem to obtain techniques for computing integrals. Alternately, we can use the Fermat and monotonicity theorems to further develop the relationship between functions and their derivatives, leading to new techniques of calculating limits, approximation of functions by polynomials, use of integration to measure arc length, surface area and volume, and error estimates for numerical calculations of integrals.We have chosen to take up integration in this chapter. If you are more interested in the other applications of differentiation you can read Chapter 6 first.
The Second Fundamental Theorem
A function F is called an anti-derivative of f if F = f . Let us make some observations regarding the existence and uniqueness of anti-derivatives:
1. Not every function has an anti-derivative. By Darboux's theorem (Theorem 4.5.12), if f = F then f has the intermediate value property. Thus, a function with a jump discontinuity, like the Heaviside step function or the greatest integer function, cannot have an anti-derivative.
2. On the other hand, the first fundamental theorem shows that every continuous function on an interval has an anti-derivative.
3. A function's anti-derivative is not unique. For example, both sin x and 1 + sin x are anti-derivatives of cos x.
4. On the other hand, two anti-derivatives of the same function over an interval can differ only by a constant. Theorem 4.5.7 states that if F = G on an interval I, then F G is constant. Thus, every anti-derivative of cos x over an interval I has to have the form sin x + C, where C ∊ R.
5. Over non-overlapping intervals, two anti-derivatives of a function need not differ by the same constant. For example, the Heaviside step function and the zero function are anti-derivatives of the zero function over (-∞,0) ∪ (0,∞).
The first fundamental theorem established a connection between integration and differentiation: if we are able to calculate the definite integrals of a continuous function, then the first fundamental theorem gives us its anti-derivative. The next theorem uses that connection to provide an approach for evaluating definite integrals by using anti-derivatives.
1 - Real Numbers and Functions
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- 16 February 2023, pp 1-42
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Calculus can be described as the study of how one quantity is affected by another, focusing on relationships that are smooth rather than erratic. This chapter sets up the basic language for describing quantities and the relationships between them. Quantities are represented by numbers and you would have seen different kinds of numbers: natural numbers, whole numbers, integers, rational numbers, real numbers, perhaps complex numbers. Of all these, real numbers provide the right setting for the techniques of calculus and so we begin by listing their properties and understanding what distinguishes them from other number systems. The key element here is the completeness axiom, without which calculus would lose its power.
The mathematical object that describes relationships is called “function.” We recall the definition of a function and then concentrate on functions that relate real numbers. Such functions are best visualized through their graphs, and this visualization is a key part of calculus. We make a small beginning with simple examples. A more thorough investigation of graphs can only be carried out after calculus has been developed to a certain level. Indeed, the more interesting functions, such as trigonometric functions, logarithms, and exponentials, require calculus for their very definition.
Field and Order Properties
We begin with a review of the set R of real numbers, which is also called the Euclidean line. It is a “review” in that we do not construct the set but just list its key attributes, and use them to derive others. For descriptions of how real numbers can be constructed from scratch, you can consult Hamilton and Landin [11], Mendelson [24], or most books on real analysis. The fundamental ideas underlying these constructions are easy to absorb, but the checking of details can be arduous. You would probably appreciate them more after reading this book.
What is the need for this review? Mainly, it is intended as a warm-up session before we begin calculus proper. Many intricate definitions and proofs lie in wait later, and we need to get ready for them by practising on easier material. If you are in a hurry and confident of your basic skills with numbers and proofs, you may skip ahead to the next section, although a patient reading of these few pages would also help in later encounters with linear algebra and abstract algebra.
6 - Pressure-driven Flow
- V. Kumaran, Indian Institute of Science, Bangalore
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- Fundamentals of Transport Processes with Applications
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- 16 February 2023, pp 271-318
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The momentum flux or the force per unit area on a surface within a fluid can be separated into two components: the pressure and the shear stress. The latter is due to variations in the flow velocity, while the former is present even when there is no flow. Pressure has no analogue in mass and heat transfer, where the fluxes are entirely due to the variations in the concentration/temperature fields. The fluid pressure is the compressive force per unit area exerted on a surface within the fluid in the direction perpendicular to the surface. At a point within the fluid, the pressure is a scalar which is independent of the orientation of the surface; the direction of the force exerted due to the pressure is along the perpendicular to the surface.
There is a distinction between the thermodynamic pressure and the dynamical pressure that drives fluid flow. The thermodynamic pressure is an absolute pressure which is calculated, for example, using the ideal gas equation of state. In contrast, flow is driven by the pressure difference between two locations in an incompressible flow. The velocity field depends on the variations in the dynamical pressure, and the flow field is unchanged if a constant pressure is added everywhere in the domain for an incompressible flow.
A potential flow is a limiting case of a pressure-driven flow where viscous effects are neglected. Some applications of potential flows are first reviewed in Section 6.1. The velocity profile and the friction factor for the laminar flow in a pipe is derived in Section 6.2. As discussed in Chapter 2, there is a transition from a laminar to a turbulent flow when the Reynolds number exceeds a critical value. The salient features of a turbulent flow are discussed in Section 6.3. The oscillatory flow in a pipe due to a sinusoidal pressure variation across the ends is considered in Section 6.4. This flow is used to illustrate the use of complex variables for oscillatory flows, and the approximations and analytical techniques used in the convection-dominated and diffusion-dominated regimes.
Potential Flow: The Bernoulli Equation
At high Reynolds number, viscous effects are neglected in the bulk of the flow, and there is a balance between the pressure, inertial and body forces.
8 - Taylor and Fourier Series
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The decimal expansion of a real number is an instance of an infinite series. For example, 3.14159 … can be viewed as the sum of the series created by its digits: . The convergence of such a series is established by a comparison with the geometric series å¥n=0 1/10n. Isaac Newton realized that by replacing the powers of 1/10 with powers of a variable x, one can create real functions that can be easily manipulated in analogy with the rules of decimal expansions. He developed rules for their differentiation and integration as well as a method for expanding inverse functions in this manner. Today's historians believe that for Newton, calculus consisted of working with these “power series.” (See Stillwell [32, pp 167–70].) In this approach, the main task is to express a given function as a power series, after which it becomes trivial to perform the operations of calculus on it. In the first two sections of this chapter we shall study the general properties of power series, and then the problem of expressing a given function as a power series.
A century after Newton, Joseph Fourier replaced the powers xn with the trigonometric functions sinnx and cosnx to create new ways of describing functions. The “Fourier series” could model much wilder behavior than power series, and forced mathematicians to revisit their notions of what is a function, and especially the definition of integration.We give a brief introduction to this topic in the third section.
In our final section, we introduce sequences and series of complex numbers. These bring further clarity to power and Fourier series, and even unify them through the famed identity of Euler: eix = cos x + i sin x.
Power Series
Let us recall our study of Taylor polynomials in §6.3. Given a function f that can be differentiated n times at x = a, we define the Taylor polynomial
Tn is intended to be an approximation for f near x = a, with the hope that the approximation improves when we increase n. These hopes are not always realized, but in many cases they are. (The remainder theorem gives us a way to assess them.) It is natural to make the jump from polynomials to series, and consider the expression
The questions that arise here are: (a) For which x does this series converge, and (b) When it converges, does it sum to f (x)? To tackle these questions, we initiate a general study of series of this form.
Acknowledgments
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5 - Unidirectional Transport: Curvilinear Co-ordinates
- V. Kumaran, Indian Institute of Science, Bangalore
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- 16 February 2023, pp 231-270
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In the previous chapter, a Cartesian co-ordinate system was used to analyse the transport between surfaces of constant co-ordinate, and the boundary conditions were specified at a fixed value of the co-ordinate z. For configurations with curved boundaries, such as a cylindrical pipe or a spherical particle, the boundaries are not surfaces of constant co-ordinate in a Cartesian system. It is necessary to apply boundary conditions at, for example, x2 + y2 + z2 = R2 for the diffusion around a spherical particle of radius R. It is simpler to use a co-ordinate system where one of the co-ordinates is a constant on the boundary, so that the boundary condition can be applied at a fixed value of the co-ordinate. Such co-ordinate systems, where one or more of the co-ordinates is a constant on a curved surface, are called curvilinear co-ordinate systems.
The procedure for deriving balance laws for a Cartesian co-ordinate system can be easily extended to a curvilinear co-ordinate system. First, we identify the differential volume or ‘shell’ between surfaces of constant co-ordinate separated by an infinitesimal distance along the co-ordinate. The balance equation is written for the change in mass/momentum/energy in this differential volume in a small time interval Δt. The balance equation is divided by the volume and Δt to derive the differential equation for the field variable. The balance equations for the cylindrical and spherical co-ordinate system are derived in this chapter, and the solution procedures discussed in Chapter 4 are applied to curvilinear co-ordinate systems.
Cylindrical Co-ordinates
Conservation Equation
A cylindrical surface is characterised by a constant distance from an axis, which is the x axis in Fig. 5.1. It is natural to define one of the co-ordinates r as the distance from the axis, and a second co-ordinate x as the distance along the axis. The third co-ordinate ϕ, which is the angle around the x axis, is considered later in Chapter 7. For unidirectional transport, we consider a variation of concentration, temperature, or velocity only in the r direction and in time, and there is no dependence on ϕ and x.
Frontmatter
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Contents
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Preface
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An anecdote about Prof. P. K. Kelkar, founding director of IIT Kanpur and former director of IIT Bombay, was narrated to me by Prof. M. S. Ananth, my teacher and former director of IIT Madras. A distraught young assistant professor at IIT Kanpur approached the director and complained that ‘the syllabus for the course is too long, and I am will not be able to cover everything’. Prof. Kelkar replied, ‘You do not have to cover everything, you should try to uncover a few things.’ In this book, my objective is to uncover a few things regarding transport processes.
The classic books on transport processes, notably the standard text Transport Phenomena by Bird, Stewart and Lightfoot written about 60 years ago, provided a comprehensive overview of the subject organised into different subject areas. At that time, engineers were required to do design calculations and modeling for different unit operations, and for the sequencing of these operations in process design. This required expertise in laboratory and pilot scale experiments on unit operations and scaling up of these operations using correlations. Proficiency in developing, understanding and using design handbooks and correlations was also needed. In this context, the study of transport processes at the microscopic level, and its implications for design for unit operations, was a pioneering advance that has since become an essential part of the chemical engineering curriculum.
In the last half century, sophisticated computational tools have been developed for detailed flow modeling within unit operations, and for the selection and concatenation of unit operations for achieving the required material transformations. The ease of search for information and data today was inconceivable half a century ago. Routine calculations have been automated, and there is little need for routine tasks such as unit conversion, graphical construction and interpreting engineering tables. There is now a greater need for understanding physical phenomena and processes and their mathematical description.
Using a rigorous understanding of transport processes, an engineer usually contributes to process design in one of two ways. The first is the development and enhancement of models and computational tools for modeling of flows and transformations in unit operations; these result in higher resolution, better representation of the essential physics and inclusion of new phenomena.
Preface
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Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny.
— Richard Dedekind, 1872Calculus is a magical subject. A first encounter in school leads to a radical revision of one's ideas of what is mathematics. We are transported from a rather staid enterprise of counting and measuring to an adventure encompassing change, fluctuation, and a vastly increased ability to understand and predict the workings of the world. At the same time, the student encounters “magic” with both its connotations: awe and wonder on the one hand, mystery and a sense of trickery on the other. Calculus can appear to be a bag of tricks that are immensely useful, provided the apprentice wizard can perfectly remember the spells. As the student pursues mathematics further at university, her instructors may use courses in analysis to persuade her that calculus is a science rather than a mystical art. Alas, all too often the student perceives the new instruction as mere hair-splitting which gives no new powers and may even undermine her previous attainments. The first analysis course is for many an experience that makes them regret taking up higher mathematics.
This book is written to support students in this transition from the expectations of school to those of university. It is intended for students who are pursuing undergraduate studies in mathematics or in disciplines like physics and economics where formal mathematics plays a significant role. Its proper use is in a “calculus with proofs” course taught during the first year of university. The goal is to demonstrate to the student that attention to basic concepts and definitions is an investment that pays off in multiple ways. Old calculations can be done again with a fresh understanding that can not only be stimulating but also protects against error. More importantly, one begins to learn how knowledge can be extended to new domains by first questioning it in familiar terrain. For students majoring in mathematics, this book can serve as bridge to real analysis. For others, it can serve as a base from where they can make expeditions to various applications.
Introduction
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A calculus text written at this point of time can make its plea for existence on the novelty of its exposition and choice of content, rather than any originality in its mathematics. Let me state the case for this book. These explanations are primarily aimed at teachers, but students may also gain some perspective from them if they peruse them after going through at least the first two chapters.
The book aims to give a complete logical framework for calculus, with the proofs reaching the same levels of rigour as a text on real analysis. At the same time it eschews those aspects of analysis that are not essential to a presentation of calculus techniques. So it has the completeness axiom for real numbers, but not Cauchy sequences or the theorems of Heine–Borel and Bolzano–Weierstrass. At the other end of the spectrum, it omits the case for the importance of calculus through its applications to the natural sciences or to economics and finance.
The first chapter provides a description of real numbers and their properties, followed by functions and their graphs. For the most part, the material of this chapter would be known to students, but not in such an organized way. I typically use the first class of the course to ask students to share their thoughts on various issues. What are rational numbers? What are numbers? Which properties of numbers are theorems and which are axioms? What is the definition of a point or a line? What do we mean by a tangent line to a curve? These flow from one to another and from students’ responses. By the end of the hour, with many students firm in their beliefs but finding them opposed just as firmly by others, I have an opportunity to propose that we must carefully put down our axioms and ways of reasoning so that future discussions may be fruitful. One may still ask whether the abstract approach is overdone; is it necessary to introduce general concepts like field and ordered field? The reason for doing so is that it provides a context within which simple questions can be posed and the student can practice creating and writing small proofs as a warm-up to harder tasks that wait ahead.
3 - Limits and Continuity
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Integration can be seen as accumulation, the summing up of local changes to get a global result. In the previous chapter, we set up the general process for achieving this. We also saw how a specific kind of information—monotonicity—could be used to obtain global results. With its help we were able to formally define the natural logarithm and exponential functions, which are usually taken for granted in school mathematics.
Further progress requires a closer look at the local behavior of functions. The more we know of the local behavior, the better our chances of extracting global information. These considerations underlie our development of the notions of limit and continuity in this chapter. As applications, we will rigorously develop angles and their radian measures, followed by the trigonometric functions and their properties.
Limits
You have seen in school, the notation lim x!p f (x)= L, which is read as “the limit of f (x) at p is L” and is interpreted as “the values of f (x) approach L as the values of x approach p.” We need a clear definition of what we mean by “approaches.”
Example 3.1.1
Consider f (x) = 2x + 5. What happens if we take values of x that approach 0? Here are some calculations:
We see that as x gets closer to 0, f (x) appears to be getting closer to 5. Can we control this? Can we get the output f (x) close to 5 within any required accuracy level, simply by making the input x appropriately close to 0?
7 - Conservation Equations
- V. Kumaran, Indian Institute of Science, Bangalore
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The mass/energy conservation laws are derived for two commonly used co-ordinate systems—the Cartesian co-ordinate system in Section 7.1 and the spherical co-ordinate system in Section 7.2. For unidirectional transport, we have seen that the conservation equation has different forms in different co-ordinate systems. Here, conservation equations are first derived using shell balance in three dimensions for the Cartesian and spherical co-ordinate systems. The conservation equations have a common form when expressed in terms of vector differential operators, the gradient, divergence, and Laplacian operators; the expressions for these operators are different in different co-ordinate systems. The conservation equation derived using shell balance is used to identify the differential operators in the the Cartesian and spherical co-ordinate system, and the procedure for deriving these in a general orthogonal co-ordinate system is explained.
Since the conservation equation is universal when expressed using vector differential operators, it is not necessary to go through the shell balance procedure for each individual problem; it is sufficient to substitute the appropriate vector differential operators in the conservation equation expressed in vector form. It is important to note that the derivation here is restricted to orthogonal co-ordinate systems, where the three co-ordinate directions are perpendicular to each other at all locations.
The discussion in Section 7.1 and 7.2 is restricted to mass/energy transfer. The constitutive relation (Newton's law) for momentum transfer for general three-dimensional flows is more complicated than that for mass/heat transfer. Mass and heat are scalars, and the flux of mass/heat is a vector along the direction of decreasing concentration/temperature. Since momentum is a vector, the flux of momentum has two directions associated with it: the direction of the momentum vector and the direction in which the momentum is transported. Due to this, the stress or momentum flux is a ‘second order tensor’ with two physical directions—the direction of momentum and the orientation of the perpendicular to the surface across which momentum is transported.
2 - Dimensionless Groups and Correlations
- V. Kumaran, Indian Institute of Science, Bangalore
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The number of independent parameters in a problem is reduced when the dependent and independent parameters are expressed in dimensionless form. In the problem of the settling sphere in Section 1.6.1 and the flow through a pipe in Section 1.6.3, the original problem contained one dependent and four independent dimensional quantities. Using dimensional analysis, this was reduced to one independent and one dependent dimensionless groups. The mass transfer problem in Section 1.6.2 contained one dependent and six independent quantities. The problem was reduced to a relationship between one dependent and two independent dimensionless groups, using dimensional analysis and the assumption that the solute mass and total mass can be considered as different dimensions. In the heat transfer problem in Section 1.6.3, there were one dependent and eight independent dimensional quantities. This was reduced to a relationship between one dependent and three independent dimensionless groups, using dimensional analysis and the assumption that the thermal and mechanical energy can be considered as different dimensions. Thus, dimensional analysis has significantly reduced the number of parameters in the problem.
It is not possible to further simplify the problem using dimensional analysis. In order to progress further, experiments can be carried out to obtain empirical correlations between the dimensionless groups. Another option, pursued in this text, is to do analytical calculations based on a mathematical description of transport processes. Before proceeding to develop the methodology for the analytical calculations, a physical interpretation of the different dimensionless groups is provided in this chapter.
In dimensional analysis, there is ambiguity in the selection of the dimensional parameters for forming the dimensionless groups. This ambiguity is reduced by a physical understanding of the dimensionless groups as the ratio of different types of forces. Here, a broad framework is established for understanding the different dimensionless groups and the relations between them. The forms of the correlations depend on several factors, such as the flow regime, flow patterns and the boundary conditions.
It is important to note that the correlations listed here are indicative, but not exhaustive. Some commonly used correlations are presented to obtain a physical understanding of the terms in the correlation, and to illustrate their application. More accurate correlations applicable in specific domains can be found in specialised handbooks/technical reports.
References
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Bibliography
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- 16 February 2023, pp 491-492
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Contents
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Index
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4 - Differentiation
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In this chapter, we take a closer look at the idea that local information about functions should help us resolve integration problems.We already saw that continuity guarantees integrability. Another application of continuity combined with integration was in enabling the definition of the trigonometric functions. On the other hand, continuity did not give us new tools to calculate integrals. In this chapter we shall study the stronger property of differentiability, which will eventually give us techniques for calculating integrals. It also has its own significance, independent of integration. We shall use it for a better understanding of the shapes of graphs of functions.
Among the continuous functions, the ones that are easiest to integrate are the “piecewise linear” ones. Their graphs consist of line segments, such as in the example below:
This suggests that we try to locally approximate functions by straight line segments. If we can get good approximations of this type, we can use them to assess the integral. We shall give the name “differentiable” to functions that can be locally approximated by straight lines. Most of this chapter is devoted to identifying these functions and to calculating the corresponding straight lines. Then we make the first connection between the processes of differentiation and integration, the so-called first fundamental theorem of calculus. Finally, we see that differentiation has a life of its own, and we use it to explore the problems of finding the extreme values and sketching the graph of a function.
Derivative of a Function
Let us consider what happens if we zoom in for a closer look at the graph of a function such as y = x2, near a point such as (1,1).
We see that the graph of the function y = x2 looks more like the line y = x2 - 1 as we zoom in towards (1,1), and at some stage becomes indistinguishable from it.
This can happen even for functions with rapid oscillations. Let us look at the function defined by y = x2 cos(1/x) if x ≠ 0, and y = 0 if x = 0, near the origin.
No matter how much we zoom in, the function has infinitely many oscillations. Nevertheless, their amplitudes decrease and in that sense the function becomes closer to the line y = 0 as we zoom in.
Index
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