Matrix Theory and Linear Algebra (562420), страница 13

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 13)

This article has considered functions of a single independent variable only. Partial derivatives, multiple integrals, and partial differential equations are defined and studied in investigating functions of two or more independent variables.

The English and German mathematicians, respectively, Isaac Newton and Gottfried Wilhelm Leibniz invented calculus in the 17th century, but isolated results about its fundamental problems had been known for thousands of years. For example, the Egyptians discovered the rule for the volume of a pyramid as well as an approximation of the area of a circle. In ancient Greece, Archimedes proved that if c is the circumference and d the diameter of a circle, then 3d<c< 3d. His proof extended the method of inscribed and circumscribed figures developed by the Greek astronomer and mathematician Eudoxus. Archimedes used the same technique for his other results on areas and volumes. Archimedes discovered his results by means of heuristic arguments involving parallel slices of the figures and the law of the lever. Unfortunately, his treatise The Method was only rediscovered in the 19th century, so later mathematicians believed that the Greeks deliberately concealed their secret methods.

During the late middle ages in Europe, mathematicians studied translations of Archimedes’ treatises from Arabic. At the same time, philosophers were studying problems of change and the infinite, such as the addition of infinitely many quantities. Greek thinkers had seen only contradictions there, but medieval thinkers aided mathematics by making the infinite philosophically respectable.

By the early 17th century, mathematicians had developed methods for finding areas and volumes of a great variety of figures. In his Geometry by Indivisibles, the Italian mathematician F. B. Cavalieri, a student of the Italian physicist and astronomer Galileo, expanded on the work of the German astronomer Johannes Kepler on measuring volumes. He used what he called “indivisible magnitudes” to investigate areas under the curves y = xⁿ, n = 1 ...9. Also, his theorem on the volumes of figures contained between parallel planes (now called Cavalieri’s theorem) was known all over Europe. At about the same time, the French mathematician René Descartes’La Géométrie appeared. In this important work, Descartes showed how to use algebra to describe curves and obtain an algebraic analysis of geometric problems. A codiscoverer of this analytic geometry was the French mathematician Pierre de Fermat, who also discovered a method of finding the greatest or least value of some algebraic expressions—a method close to those now used in differential calculus.

About 20 years later, the English mathematician John Wallis published The Arithmetic of Infinites, in which he extrapolated from patterns that held for finite processes to get formulas for infinite processes. His colleague at the University of Cambridge was Newton’s teacher, the English mathematician Isaac Barrow, who published a book that stated geometrically the inverse relationship between problems of finding tangents and areas, a relationship known today as the fundamental theorem of calculus.

Although many other mathematicians of the time came close to discovering calculus, the real founders were Newton and Leibniz. Newton’s discovery (1665-66) combined infinite sums (infinite series), the binomial theorem for fractional exponents, and the algebraic expression of the inverse relation between tangents and areas into methods we know today as calculus. Newton, however, was reluctant to publish, so Leibniz became recognized as a codiscoverer because he published his discovery of differential calculus in 1684 and of integral calculus in 1686. It was Leibniz, also, who replaced Newton’s symbols with those familiar today.

In the following years, one problem that led to new results and concepts was that of describing mathematically the motion of a vibrating string. Leibniz’s students, the Bernoulli family of Swiss mathematicians (see Bernoulli, Daniel), used calculus to solve this and other problems, such as finding the curve of quickest descent connecting two given points in a vertical plane. In the 18th century, the great Swiss-Russian mathematician Leonhard Euler, who had studied with Johann Bernoulli, wrote his Introduction to the Analysis of Infinites, which summarized known results and also contained much new material, such as a strictly analytic treatment of trigonometric and exponential functions.

Despite these advances in technique, calculus remained without logical foundations. Only in 1821 did the French mathematician A. L. Cauchy succeed in giving a secure foundation to the subject by his theory of limits, a purely arithmetic theory that did not depend on geometric intuition or infinitesimals. Cauchy then showed how this could be used to give a logical account of the ideas of continuity, derivatives, integrals, and infinite series. In the next decade, the Russian mathematician N. I. Lobachevsky and German mathematician P. G. L. Dirichlet both gave the definition of a function as a correspondence between two sets of real numbers, and the logical foundations of calculus were completed by the German mathematician J. W. R. Dedekind in his theory of real numbers, in 1872.

Zeno of Elea

Zeno of Elea (flourished 5th century bc), Greek mathematician and philosopher of the Eleatic school, known for his philosophical paradoxes.

Zeno was born in Elea, in southwestern Italy. He became a favorite disciple of the Greek philosopher Parmenides and accompanied him to Athens at the age of about 40. In Athens, Zeno taught philosophy for some years, concentrating on the Eleatic system of metaphysics. The Athenian statesmen Pericles and Callias (flourished 5th century bc) studied under him. Zeno later returned to Elea and, according to traditional accounts, joined a conspiracy to rid his native town of the tyrant Nearchus; the conspiracy failed and Zeno was severely tortured, but he refused to betray his accomplices. Further circumstances of his life are not known.

Only a few fragments of Zeno's works remain, but the writings of Plato and Aristotle provide textual references to Zeno's writings. Philosophically, Zeno accepted Parmenides' belief that the universe, or being, is a single, undifferentiated substance, a oneness, although it may appear diversified to the senses. Zeno's intention was to discredit the senses, which he sought to do through a brilliant series of arguments, or paradoxes, on time and space that have remained complex intellectual puzzles to this day. A typical paradox asserts that a runner cannot reach a goal because, in order to do so, he must traverse a distance; but he cannot traverse that distance without first traversing half of it, and so on, ad infinitum. Because an infinite number of bisections exist in a spatial distance, one cannot travel any distance in finite time, however short the distance or great the speed. This argument, like several others of Zeno, is intended to demonstrate the logical impossibility of motion. In that the senses lead us to believe in the existence of motion, the senses are illusory and therefore no obstacle to accepting the otherwise implausible theories of Parmenides. Zeno is noted not only for his paradoxes, but for inventing the type of philosophical argument they exemplify. Thus Aristotle named him the inventor of dialectical reasoning.

Function

Function, in mathematics, term used to indicate the relationship or correspondence between two or more quantities. The term function was first used in 1637 by the French mathematician René Descartes to designate a power xⁿ of a variable x. In 1694 the German mathematician Gottfried Wilhelm Leibniz applied the term to various aspects of a curve, such as its slope. The most widely used meaning until quite recently was defined in 1829 by the German mathematician Peter Dirichlet. Dirichlet conceived of a function as a variable y, called the dependent variable, having its values fixed or determined in some definite manner by the values assigned to the independent variable x, or to several independent variables x₁, x₂, ..., x_k.

The values of both the dependent and independent variables were real or complex numbers. The statement y = f(x), read “ y is a function of x,” indicated the interdependence between the variables x and y; f(x) was usually given as an explicit formula, such as f(x) = x² - 3x + 5, or by a rule stated in words, such as f(x) is the first integer larger than x for all x's that are real numbers (see Number). If a is a number, then f(a) is the value of the function for the value x = a. Thus, in the first example, f(3) = 3² - 3 · 3 + 5 = 5, f(-4) = (-4)² - 3(-4) + 5 = 33; in the second example, f(3) = f(3.1) = f() = 4.

The emergence of set theory first extended and then altered substantially the concept of a function. The function concept in present-day mathematics may be illustrated as follows. Let X and Y be two sets with arbitrary elements; let the variable x represent a member of the set X, and let the variable y represent a member of the set Y. The elements of these two sets may or may not be numbers, and the elements of X are not necessarily of the same type as those of Y. For example, X might be the set of the 50 states of the United States and Y the set of positive integers. Let P be the set of all possible ordered pairs (x, y) and F a subset of P with the property that if (x₁, y₁) and (x₂, y₂) are two elements of F, then y₁≠y₂ implies that x₁≠x₂—that is, F contains no more than one ordered pair with a given x as its first member. (If x₁≠x₂, however, it may happen that y₁ = y₂.) A function is now regarded as the set F of ordered pairs with the stated condition and is written F:X→Y. The set X₁ of x's that occur as first elements in the ordered pairs of F is called the domain of the function F; the set Y₁ of y's that occur as second elements in the ordered pairs is called the range of the function F. Thus, {(New York, 7), (Ohio, 4), (Utah, 4)} is one function that has X = the set of the 50 U.S. states and Y = the set of all positive integers; the domain is the three states named, and the range is 4, 7.

The modern concept of a function is related to the Dirichlet concept. Dirichlet regarded y = x² - 3x + 5 as a function; today, y = x² - 3x + 5 is thought of as the rule that determines the correspondent y for a given x of an ordered pair of the function; thus, the preceding rule determines (3, 5), (-4, 33) as two of the infinite number of elements of the function. Although y = f(x) is still used today, it is better to read it as “y is functionally related to x”.

A function is also called a transformation or mapping in many branches of mathematics. If the range Y₁ is a proper subset of Y (that is, at least one y is in Y but not in Y₁), then F is a function or transformation or mapping of the domain X₁ into Y; if Y₁ = Y,F is a function or transformation or mapping of X₁ onto Y.

Leibniz, Gottfried Wilhelm

Leibniz, Gottfried Wilhelm, also Leibnitz, Baron Gottfried Wilhelm von (1646-1716), German philosopher, mathematician, and statesman, regarded as one of the supreme intellects of the 17th century.

Leibniz was born in Leipzig. He was educated at the universities of Leipzig, Jena, and Altdorf. Beginning in 1666, the year in which he was awarded a doctorate in law, he served Johann Philipp von Schönborn, archbishop elector of Mainz, in a variety of legal, political, and diplomatic capacities. In 1673, when the elector's reign ended, Leibniz went to Paris. He remained there for three years and also visited Amsterdam and London, devoting his time to the study of mathematics, science, and philosophy. In 1676 he was appointed librarian and privy councillor at the court of Hannover. For the 40 years until his death, he served Ernest Augustus, duke of Brunswick-Lüneburg, later elector of Hannover, and George Louis, elector of Hannover, later George I, king of Great Britain and Ireland.

Leibniz was considered a universal genius by his contemporaries. His work encompasses not only mathematics and philosophy but also theology, law, diplomacy, politics, history, philology, and physics.

Leibniz's contribution in mathematics was to discover, in 1675, the fundamental principles of infinitesimal calculus. This discovery was arrived at independently of the discoveries of the English scientist Sir Isaac Newton, whose system of calculus was invented in 1666. Leibniz's system was published in 1684, Newton's in 1687, and the method of notation devised by Leibniz was universally adopted (see Mathematical Symbols). In 1672 he also invented a calculating machine capable of multiplying, dividing, and extracting square roots, and he is considered a pioneer in the development of mathematical logic.

In the philosophy expounded by Leibniz, the universe is composed of countless conscious centers of spiritual force or energy, known as monads. Each monad represents an individual microcosm, mirroring the universe in varying degrees of perfection and developing independently of all other monads. The universe that these monads constitute is the harmonious result of a divine plan. Humans, however, with their limited vision, cannot accept such evils as disease and death as part of a universal harmony. This Leibnizian universe, “the best of all possible worlds,” is satirized as a utopia by the French author Voltaire in his novel Candide (1759).

Характеристики

Список файлов учебной работы