The Britannica Guide to Analysis and Calculus

Britannica Educational Publishing, Erik Gregersen

• Publisher: Britannica Educational Publishing
• Year: 2010
• ISBN:
  9781615302208
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

The dynamism of the natural world means that it is constantly changing, sometimes rapidly, sometimes gradually. By mathematically interpreting the continuous change that characterizes so many natural processes, analysis and calculus have become indispensable to bridging the divide between mathematics and the sciences. This comprehensive volume examines the key concepts of calculus, providing students with a robust understanding of integration and differentiation. Biographies of important figures will leave readers with an increased appreciation for the sometimes competing theories that informed the early history of the field.

CHAPTER 1
MEASURING CONTINUOUS CHANGE

Analysis is the branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration. Since the discovery of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz at the end of the 17th century, analysis has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology.

The historical origins of analysis can be found in attempts to calculate spatial quantities such as the length of a curved line or the area enclosed by a curve. These problems can be stated purely as questions of mathematical technique, but they have a far wider importance because they possess a broad variety of interpretations in the physical world. The area inside a curve, for instance, is of direct interest in land measurement: how many acres does an irregularly shaped plot of land contain? But the same technique also determines the mass of a uniform sheet of material bounded by some chosen curve or the quantity of paint needed to cover an irregularly shaped surface. Less obviously, these techniques can be used to find the total distance traveled by a vehicle moving at varying speeds, the depth at which a ship will float when placed in the sea, or the total fuel consumption of a rocket.

Similarly, the mathematical technique for finding a tangent line to a curve at a given point can also be used to calculate the steepness of a curved hill or the angle through which a moving boat must turn to avoid a collision. Less directly, it is related to the extremely important question of the calculation of instantaneous velocity or other instantaneous rates of change, such as the cooling of a warm object in a cold room or the propagation of a disease organism through a human population.

BRIDGING THE GAP BETWEEN ARITHMETIC AND GEOMETRY

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Mathematics divides phenomena into two broad classes, discrete and continuous, historically corresponding to the division between arithmetic and geometry. Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of such infinite decimals lies at the heart of analysis.

The distinction between discrete mathematics and continuous mathematics is a central issue for mathematical modeling, the art of representing features of the natural world in mathematical form. The universe does not contain or consist of actual mathematical objects, but many aspects of the universe closely resemble mathematical concepts. For example, the number 2 does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars. In a similar manner, the real numbers provide satisfactory models for a variety of phenomena, even though no physical quantity can be measured accurately to more than a dozen or so decimal places. It is not the values of infinitely many decimal places that apply to the real world but the deductive structures that they embody and enable.

Analysis came into being because many aspects of the natural world can profitably be considered as being continuous—at least, to an excellent degree of approximation. Again, this is a question of modeling, not of reality. Matter is not truly continuous. If matter is subdivided into sufficiently small pieces, then indivisible components, or atoms, will appear. But atoms are extremely small, and, for most applications, treating matter as though it were a continuum introduces negligible error while greatly simplifying the computations. For example, continuum modeling is standard engineering practice when studying the flow of fluids such as air or water, the bending of elastic materials, the distribution or flow of electric current, and the flow of heat.

The atom is one of the smallest pieces of matter. It is made up of three smaller pieces—the neutron, the proton, and the electron. There are branches of science that study matter on this tiny scale, but calculus takes a larger, more continuous view. Photodisc/Getty Images

DISCOVERY OF THE CALCULUS AND THE SEARCH FOR FOUNDATIONS

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Two major steps led to the creation of analysis. The first was the discovery of the surprising relationship, known as the fundamental theorem of calculus, between spatial problems involving the calculation of some total size or value, such as length, area, or volume (integration), and problems involving rates of change, such as slopes of tangents and velocities (differentiation). Credit for the independent discovery, about 1670, of the fundamental theorem of calculus together with the invention of techniques to apply this theorem goes jointly to Gottfried Wilhelm Leibniz and Isaac Newton.

While the utility of calculus in explaining physical phenomena was immediately apparent, its use of infinity in calculations (through the decomposition of curves, geometric bodies, and physical motions into infinitely many small parts) generated widespread unease. In particular, the Anglican bishop George Berkeley published a famous pamphlet, The Analyst; or, A Discourse Addressed to an Infidel Mathematician (1734), pointing out that calculus—at least, as presented by Newton and Leibniz—possessed serious logical flaws. Analysis grew out of the resulting painstakingly close examination of previously loosely defined concepts such as function and limit.

Newton’s and Leibniz’s approach to calculus had been primarily geometric, involving ratios with “almost zero” divisors—Newton’s “fluxions” and Leibniz’s “infinitesimals.” During the 18th century calculus became increasingly algebraic, as mathematicians—most notably the Swiss Leonhard Euler and the Italian French Joseph-Louis Lagrange—began to generalize the concepts of continuity and limits from geometric curves and bodies to more abstract algebraic functions and began to extend these ideas to complex numbers. Although these developments were not entirely satisfactory from a foundational standpoint, they were fundamental to the eventual refinement of a rigorous basis for calculus by the Frenchman Augustin-Louis Cauchy, the Bohemian Bernhard Bolzano, and above all the German Karl Weierstrass in the 19th century.

NUMBERS AND FUNCTIONS

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NUMBER SYSTEMS

There are a variety of number systems—that is, collections of mathematical objects (numbers) that can be operated on by some or all of the standard operations of arithmetic: addition, multiplication, subtraction, and division. These main number systems are:

• The natural numbers N. These numbers are the positive (and zero) whole numbers 0, 1, 2, 3, 4, 5, …. If two such numbers are added or multiplied, the result is again a natural number.

• The integers Z. These numbers are the positive and negative whole numbers …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, … . If two such numbers are added, subtracted, or multiplied, the result is again an integer.

• The rational numbers Q. These numbers are the positive and negative fractions p/q where p and q are integers and q ≠ 0. If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a rational number.

• The real numbers R. These numbers are the positive and negative infinite decimals (including terminating decimals that can be considered as having an infinite sequence of zeros on the end). If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a real number.

• The complex numbers C. These numbers are of the form x + iy where x and y are real numbers and i =

If two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a complex number.

FUNCTIONS

In simple terms, a function f is a mathematical rule that assigns to a number x (in some number system and possibly with certain limitations on its value) another number f(x). For example, the function “square” assigns to each number x its square x². Note that it is the general rule, not specific values, that constitutes the function.

The common functions that arise in analysis are usually definable by formulas, such as f(x) = x². They include the trigonometric functions sin (x), cos (x), tan (x), and so on; the logarithmic function log (x); the exponential function exp (x) or e ^x (where e = 2.71828… is a special constant called the base of natural logarithms); and the square root function

. However, functions need not be defined by single formulas (indeed by any formulas). For example, the absolute value function |x| is defined to be x when x ≥ 0 but -x when x < 0 (where ≥ indicates greater than or equal to and < indicates less than).

THE PROBLEM OF CONTINUITY

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The logical difficulties involved in setting up calculus on a sound basis are all related to one central problem, the notion of continuity. This in turn leads to questions about the meaning of quantities that become infinitely large or infinitely small—concepts riddled with logical pitfalls. For example, a circle of radius r has circumference 2πr and area πr², where π is the famous constant 3.14159…. Establishing these two properties is not entirely straightforward, although an adequate approach was developed by the geometers of ancient Greece, especially Eudoxus and Archimedes. It is harder than one might expect to show that the circumference of a circle is proportional to its radius and that its area is proportional to the square of its radius. The really difficult problem, though, is to show that the constant of proportionality for the circumference is precisely twice the constant of proportionality for the area—that is, to show that the constant now called π really is the same in both formulas. This boils down to proving ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Introduction
• Chapter 1: Measuring Continuous Change
• Chapter 2: Calculus
• Chapter 3: Differential Equations
• Chapter 4: Other Areas of Analysis
• Chapter 5: History of Analysis
• Chapter 6: Great Figures in the History of Analysis
• Chapter 7: Concepts in Analysis and Calculus
• Glossary
• Bibliography
• Index