Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1994-1997 by Evans M. Harrell II and James V. Herod. All rights reserved.

This document contains some brief biographical and historical notes as part of the hypertext, Linear Methods of Applied Mathematics. When a name such as Fourier in the text is highlighted, it usually indicates a link to part of this document.

Much more historical information is available from the History of Mathematics Archive or the Museum of the History of Science.

Sir George Biddell Airy, b. 1801, Alnwick, England, d. 1892. British astronomer who studied diffraction (among other topics), in which he used Airy functions, which satisfy a differential equation of the form

u'' = x u.

Actually, these are equivalent to Bessel functions with a fractional index.

Jean le Rond d'Alembert, b. 1717, Paris, d. 1783. D'Alembert was a foundling, who derives his somewhat curious name from the church at which he was left as a baby by his mother, who was a fashionable lady with an intellectual salon in Paris and a distaste for encumbrances. He went on to become one of the scientific leaders of the enlightenment and a close friend of Voltaire. He was the codirector with Diderot of the Encyclopédie until 1758, but abandoned the project because of ecclesiastical attacks; which he returned as best he could in a pamphlet, sur la déstruction des Jésuites (1765).

In our text we encounter his original idea for solving the wave equation, which evolved into the method of characteristics. He also contributed to celestial mechanics, geometry, hydrodynamics, and complex analysis. His most important scientific writing was his Traité de Dynamique (1743), but perhaps his greatest contribution to mankind was his development of the theory of the construction of eyeglasses.

Daniel Bernoulli, b. 1700, Basel, Switzerland, d. 1782. The scion of an illustrious family of mathematicians, he analyzed the physics and mathematics of the vibrating string and pioneered the use of trigonometric series. He also worked on hydrodynamics, thermodynamics, clectial mechanics, and the theory of gases. (His father Johann and his uncle Jakob also contributed to calculus and mechanics, and Jakob Bernoulli was the founder of the theory of probability and statistics.)
Friedrich Wilhelm Bessel, b. 1784, Minden, Germany, d. 1846. Bessel was a German astronomer who used Bessel functions in his analysis of orbital motion, but they were known much earlier. According to G.N. Watson, Theory of Bessel Functions, Cambridge: at the University Press, 1945, Bessel functions were used by Daniel Bernoulli in a memoir of 1738 on oscillations of heavy chains, and Bessel's equation was studied by Euler in 1764, when he investigated the vibrations of a stretched membrane (see chapter X). Bessel is also remembered for his work on double stars, cosmology, and perception of space and time. He was to first to make a good estimate of interstellar distances.
V. Ya. Buniakovskii. Russian discoverer of the inequality also attributed to Cauchy and Schwarz.
Lennart Carleson, Swedish mathematician working at the Royal Institute of Technology in Stockholm. In a 1966 article, he proved that the Fourier series for a square-integrable function converges almost everywhere.
Baron Augustin-Louis Cauchy, b. 1789, Paris, d. 1857, Sceaux. Cauchy made fundamental contributions to mechanics, optics, and astronomy, but is best remembered today as one of the founders of complex analysis. His articles and textbooks on analysis and calculus greatly raised the standards of rigor prevalent at the time, so that it was no longer acceptable to make mathematical arguments based merely on intuition. They became the standard works of their time and had a lasting influence discernible to this day.

Unlike most of the mathematicians of his time, who were revolutionaries, Cauchy was ardently conservative. He supported Catholic and royalist causes throughout the revolutionary period in France, and even went into exile with the Bourbons in 1830.

James Clerk Maxwell, b. 1831, Edinburgh, Scotland, d. 1879, Cambridge, England. Usually referred to as Maxwell, his family name was actually Clerk (pronounced "Clark") Maxwell. His father, James Clerk, appended Maxwell to the family name as part of some legal manoeuvring over the inheritance of the 1500 acre estate on which Clerk Maxwell wrote much of his scientific work. Clerk Maxwell was one of the major figures of nineteenth century physics, contributing to many fields, including the theory of color vision and colorblindness, planetary physics, statistical mechanics, elasticity, and molecular physics. He is most remembered for codifying the theory of electricity and magnetism in four partial differential equations, and for using them to predict the existence of electromagnetic waves with the properties of light.

More about Clerk Maxwell

Edward Salisbury Dana, b. 1849, New Haven, Connecticut, d. 1935. American minerologist, who brought out the sixth edition of the minerology textbook of his father, James Dwight Dana, b. 1813, Utica, New York, d. 1895. He was one of the4 founders of minerology and was the minerologist on the U.S. expedition to Antarctica and the South Seas in 1837-42.
Paul Adrien Marie Dirac, b. 1902, Bristol, d. 1984, Florida. Dirac won the Nobel Prize for predicting the positron. He did other significant work on relativistic quantum theory and wrote an influential textbook. He popularized but did not invent the delta function, which was known much earlier to Kirchhoff, whose treatment of the delta function as a limiting operation is much closer to the modern point of view than is Dirac's.
Pierre Louis Dulong, b. 1785, Rouen, France, d. 1838. He studied specific heat, together with Petit.
Euclid, fl. 3rd c. B.C., Greek mathematics professor working in Alexandria, Egypt, under the reign of Ptolemy I. His Elements are the pattern for millenia of later books on geometry, but it is uncertain how much is due to Euclid himself and how much was collected from other sources.
Leonhard Euler, b. 1707, Basel, Switzerland, d. 1783, St. Petersburg, Russia. Euler was one of the towering geniuses of mathematics and physics, and it is difficult even to list all of his contributions. He was one of the founders of mechanics, introducing among many other things, the notion of action principles, and working out for the first time the motion of rigid bodies. He analyzed the motions of planets and comets. In mathematics he contributed to the foundations of algebra and of probability thory. He was perhaps the most prolific scientist ever, despite his lack of a word processor.

Euler had a large family and although he was early recognized as a genius, he could not find good employment in his native Switzerland. Fortunately, during his early career Catherine the Great founded the Petersburg Academy, the ancestor of the Russian Academy of Sciences, and Euler accepted an invitation to be one of the founding professors of this institution. He became a good friend of Frederick the Great of Prussia - the original "enlightened despot" - and carried on correspondence with him even while Russia and Prussia were at war.

Euler's contributions to our subject include:

Baron Joseph Fourier, b. 1768, Auxerre, France, d. 1830, Paris. While Fourier was not the first to use the series named after him, he did create many things, including the Fourier integral transform. He was one of the chief engineers on Napoleon's expedition to Egypt, where the torrid climate appealed to him; indeed, for the rest of his life he insisted on keeping his living quarters at a temperature that others found intolerably hot. Perhaps not coincidentally, he developed the first correct theory of heat flow and derived the heat equation, which is one of the most basic partial differential equations, describing not only heat flow but also diffusion. In this text we learn how to solve the heat equation with the technique of Fourier series, following his original analysis.

Fourier was also politically talented, becoming at various times the French Commissioner to the Sultan, the Prefect of Isère (comparable to being a governor), an academician, and the chief engineer of many projects, including the construction of a major route between Grenoble and Turin (Torino), still in use today. He managed to remain influential when Napoleon came to power, after Napoleon's defeat, during his return, and after Waterloo. Another interesting accomplishment of Fourier was his discovery and sponsorship of a young linguist named Champollion, who went on to decipher the Rosetta stone.

Fourier's contributions to our subject include:

Among his other contributions to science was his realization, in 1827, that atmospheric gases were responsible for keeping the temperature on earth warm enough for life.

Erik Ivar Fredholm, b. 1866, Stockholm, Sweden, d. 1927, Stockholm. Fredholm is mainly remembered for his theory of integral operators. More information is available at the History of Mathematics Archive.

Carl Friedrich Gauss, also spelled Gauß, b. 1777, Braunschweig (Brunswick), Germany, d. 1855, Göttingen. Perhaps the greatest mathematical genius ever, he contributed in an essential way to algebra, astronomy, celestial mechanics, geometry, surveying, electromagnetism, mechanics, number theory, probability, and statistics.
Josiah Willard Gibbs, b. 1839, New Haven, Connecticut, d. 1903, New Haven. America's first great mathematical physicist, Gibbs studied with Kirchhoff and Helmholtz in Berlin and then returned to teach at Yale. He was one of the founders of vector analysis, potential theory, and statistical mechanics.
George Green, b. 1793, Sneinton, Nottingham, England, d. 1841, Sneinton. Green was a self-educated man, who discovered many important facts about vector calculus as well as the method of Green functions. These are sometimes called Green's functions, by the way, but I have preferred to drop the possessive on the pattern of Bessel functions, Legendre functions, etc. This also avoids the ungrammatical phrase "the Green's function," which is widely used for reasons I cannot fathom. More information is available at the History of Mathematics Archive.

Evans Harrell, b. 1950, Indianapolis, USA. Harrell was not around early enough to contribute the fundamental ideas of the fairly classical subject of this course, but the material and ideas are used in his research on mathematical physics.
Oliver Heaviside, b. 1850, d. 1925. British applied mathematician and electrical engineer, who developed the "operational calculus" and worked on the telegraph and telephone system. He wrote an important text on electromagnetic theory.

Charles Hermite, b. Dieuze, France, 1822, d. 1901, Paris. Hermite produced a useful set of polynomials which are orthogonal with a weight of the form exp(-x^2). He also worked on approximation theory and is remembered for having proved that e, the base of the natural logarithm, is irrational. More information is available at the History of Mathematics Archive and at Project Hermite.
David Hilbert, b. 1862, Königsberg, Germany (now known as Kaliningrad, Russia), d. Göttingen, 1943. Hilbert invented the infinite dimensional space named after him and contributed enormously to modern mathematics, in functional analysis, algebra, number thory, the foundations of mathematics, and logic. He set the course for much of 20th century mathematics by outlining a list of inspiring problems. While he was elderly when the Nazis came to power, he used his influence to shield many younger mathematicians and help them emigrate. (More.)
Gustav Kirchhoff, b. 1824, Königsberg, Germany (now known as Kaliningrad, Russia), d. 1887. The founder of electrical circuit theory, Kirchhoff introduced the "Dirac" delta function in his lectures on optics, published in the 1880s (he called it zeta, however).
L. Kronecker, b 1823, Liegnitz, Germany (now Legnica, Poland), d. 1891, Berlin. German mathematician working in Berlin, mainly on algebra and the foundations of arithmetic. For philosophical reasons he was opposed to the use of irrational numbers. He is reported to have said, "God made the integers, all the rest is the work of man."
Count Joseph Louis Lagrange, b 1736, Turin, Italy, d. 1813, Paris. His legacy includes the Lagrangian action in classical mechanics, Lagrange multipliers in optimization theory, and the Lagrange points of astronomy, and his textbook, mécanique analytique, published in 1786, is the classic text on the subject. In mathematics he also did important work on group t heory and potential theory. He was the inventor of the notion of generalized coordinates.
Marquis Pierre Simon de Laplace, b. 1749, Beaumont-en-Auge, in the Normandy region of France, d. 1827, Paris. Mathematical physicist, chemist, and astronomer, who first theorized that the solar system condensed from a nebulous body. From humble beginnings he became a protégé of d'Alembert and eventually became a marquis. He worked on the theory of sound, probability theory, and quantitative chemistry. In our text we learn of his equation for potential functions.
Adrien Marie Legendre, b. 1752, Paris, d. 1797. One of the founders of number theory, and a significant contributor to geometry and astronomy

Legendre's contributions to our subject include:

Gustav Lejeune-Dirichlet, b. 1805, Düren, Germany, d. 1859, Göttingen. The family name is double, although most people refer to him simply as Dirichlet. He was a German mathematician of French extraction (the family had fled France after the St. Barthélemy massacre of the Huguenot religious minority in the sixteenth century), and he studied in Paris, circumstances which account for some of the variation in the way his name is pronounced. In Paris he became a protégé of Fourier, and on returning to Germany he introduced many French educational and mathematical ideas. He is known for his work on boundary value problems, integral forms, and number theory, into which subject he introduced the use of Fourier series.
Franz E. Neumann, b. 1798, in the Prussian forest in Uckermark, d. 1895, Königsberg. A German physicist who used Neumann boundary conditions in his investigations of electromagnetic fields. He also made notable contributions to crystallography and optics.
John von Neumann, b. 1903, Budapest, d. 1957, Princeton. Brilliant twentieth century mathematician who, among many accomplishments, created game theory and mathematical economics; worked out the mathematical foundations of the theory of quantum mechanics and proved the spectral theorem which ensures that it works; and spearheaded the program to build the computer called MANIAC, which was used for the calculations for the design of the hydrogen bomb. Von Neumann was able to perform complicated calculations in his head and could recite long passages from books he had read only once.
Sir Isaac Newton, b 1642, Woolsthorpe, Lincoln, England, d. 1727, Kensington, Middlesex (buried in Westminster Abbey). Coinventor of the calculus and of the science of mechanics as we now know it. He also did important work on gravitation, heat, and optics; for instance, it was Newton who discovered that white light is composed of colors of the entire spectrum. After his burst of scientific creativity in his early 20's, he turned to other interests, and after 1666 his writings were predominantly on theology, history, and alchemy. He became a political man, and was at various times a representative to Parliament, warden of the mint, and president of the Royal Society.
Alfred Nobel, b. 1833, Stockholm, d. 1896. The inventor of dynamite, which made him rich enough to be a philanthropist.
Parseval, one of the many nineteenth century French mathematicians working on Fourier series. It is not clear that the Parseval formula was new; it may have been known to Dulong.
Denis Poisson, b. 1781, Pithiviers, France, d. 1840, Paris. One of the major academicians, known in mechanics, electromagnetism, vibration theory, heat transfer, probability, and elasticity.
Pythagoras of Samos, ca. 560-ca. 480 BC. Ancient Greek mathematician. Unfortunately, none of Pythagoras's own writings have survived, but we know that he founded a religion based on the worship of numbers and a religious colony at Crotona in southern Italy. Today only a few thousand adherents of the Pythagorean religion can be found, scattered around the world except for brief meetings at rituals called mathematics conferences. Pythagoras also preached radical notions such as the transmigration of souls and the equality of the sexes. It appears to be true that he is responsible for the theorem which bears his name; less widely appreciated is his discovery of the numerical relationships between the lengths of strings and their musical tones, which enabled him to codify the musical scale on which most of Western music is based.
Frigyes Riesz, b. 1880, d. 1956, Hungarian mathematician working in the area of functional analysis, especially Hilbert space.

Hermann Amandus Schwarz, b. 1843, Hermsdorf, Silesia, which at that time was Austrian; after World War I, Silesia was partitioned, and Hermsdorf is probably now a village with a different name in Poland or the Czech Republic. He settled in Berlin, and was a noted geometer and analyst. He is responsible for the Schwarz reflection principle of complex analysis as well as for the inequality he discovered, in common with Cauchy and Buniakovskii.

Brook Taylor, b. 1685, Edmunton, England, d. 1731. A younger contemporary of Newton, who is best remembered for power-series approximations of functions, but who also worked on machanics and acoustics. For some reason the special case when the point of expansion is 0 is often attributed to the Scottish mathematician Colin Maclaurin (1698-1746), although Taylor published the more general result in 1715, while Maclaurin published it only in 1742. On the other hand, Taylor has been suspected of plagiarizing his series from earlier work by Johann Bernoulli (1667-1748).

Vito Volterra, b. 186, Ancona, Papal States, d. 1940, Rome. Volterra is mainly remembered for his studies of integral equations. More information is available at the History of Mathematics Archive.

Bibliography

These historical notes are culled from many sources, but the ones I have used somewhat systematically are:

Solomon Bochner, The Role of Mathematics in the Rise of Science, Princeton: Princeton Univ. Press, 1981.

William Bridgewater and Seymour Kurtz, editors, The Columbia Encyclopedia, Third edition. New York: Columbia University Press, 1963.

W.F. Bynum, E.J. Browne, and Roy Porter, eds., Dictionary of the History of Science, Princeton: Princeton University Press, 1981.

Charles Coulston Gillispie, editor-in-chief, Dictionary of Scientific Biography, New York: Scribner, 1970-.

Felix Klein, Vorlesungen über die Entwicklung der mathematik im 19. Jahrhundert, New York, Chelsea, 1967 (reprint of two volumes published in Berlin, 1926-1927).

Nouveau Petit Larousse, Paris: Librairie Larousse, 1971.

Since writing most of these biographical sketches, I have discovered a very nice site with biographies of many mathematicians and scientists. From it there is a link to a historical discussion of the concept of abstract vector spaces

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Evans M. Harrell II (correct my scholarship!)