Harley Flanders
   HOME

TheInfoList



OR:

Harley M. Flanders (September 13, 1925 – July 26, 2013) was an American mathematician, known for several textbooks and contributions to his fields: algebra and algebraic number theory,
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
,
electrical network An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sour ...
s,
scientific computing Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...
.


Life

Flanders was a sophomore calculus student of Lester R. Ford at the Illinois Institute of Technology and asked for more challenging reading. Ford recommended ''A Course in Mathematical Analysis'' by Edouard Goursat, translated by
Earle Hedrick Earle Raymond Hedrick (September 27, 1876 – February 3, 1943), was an American mathematician and a vice-president of the University of California. Education and career Hedrick was born in Union City, Indiana. After undergraduate work at ...
, which included challenging exercises. Flanders recalled in 2001 that the final exercise required a proof of a formula for the derivatives of a composite function, generalizing the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
, in a form now called the
Faa di Bruno formula The Federal Aviation Administration (FAA) is the largest transportation agency of the U.S. government and regulates all aspects of civil aviation in the country as well as over surrounding international waters. Its powers include air traffic ...
.H. Flanders (2001) "From Ford to Faa",
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...
108(6): 558–61
Flanders received his bachelors (1946), masters (1947) and PhD (1949) at the
University of Chicago The University of Chicago (UChicago, Chicago, U of C, or UChi) is a private university, private research university in Chicago, Illinois. Its main campus is located in Chicago's Hyde Park, Chicago, Hyde Park neighborhood. The University of Chic ...
on the dissertation ''Unification of class field theory'' advised by Otto Schilling and
André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
. He held the Bateman Fellowship at
Caltech The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...
. He joined the faculty at
University of California at Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant uni ...
. In 1955 Flanders heard Charles Loewner speak there on continuous groups. Notes were taken and the lectures appeared in a limited form with the expectation that Loewner would produce a book on the topic. With his death in 1968 the notes drew the attention of
Murray H. Protter Murray Harold Protter (February 13, 1918 – May 1, 2008) was an American mathematician and educator, known for his contributions to the theory of partial differential equations, as well as his well-selling textbooks in Calculus. Protter earned a ...
and Flanders. They edited Loewner's talks and in 1971 The MIT Press published ''Charles Loewner: Theory of Continuous Groups''. The book was re-issued in 2008. Teaching posts Flanders held included the faculty at
Purdue University Purdue University is a public land-grant research university in West Lafayette, Indiana, and the flagship campus of the Purdue University system. The university was founded in 1869 after Lafayette businessman John Purdue donated land and ...
(1960),
Tel Aviv University Tel Aviv University (TAU) ( he, אוּנִיבֶרְסִיטַת תֵּל אָבִיב, ''Universitat Tel Aviv'') is a public research university in Tel Aviv, Israel. With over 30,000 students, it is the largest university in the country. Locate ...
(1970–77), visiting professor at
Georgia Tech The Georgia Institute of Technology, commonly referred to as Georgia Tech or, in the state of Georgia, as Tech or The Institute, is a public research university and institute of technology in Atlanta, Georgia. Established in 1885, it is part o ...
(1977–78), visiting scholar at
Florida Atlantic University Florida Atlantic University (Florida Atlantic or FAU) is a public research university with its main campus in Boca Raton, Florida, and satellite campuses in Dania Beach, Davie, Fort Lauderdale, Jupiter, and Fort Pierce. FAU belongs to the 1 ...
(1978–85),
University of Michigan, Ann Arbor , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...
(1985–97, 2000–),
University of North Florida The University of North Florida (UNF) is a public research university in Jacksonville, Florida. It is part of the State University System of Florida and is accredited by the Commission on Colleges of the Southern Association of Colleges and Sc ...
(1997–2000) and, distinguished mathematician in residence at
Jacksonville University Jacksonville University (JU) is a private university in Jacksonville, Florida. Located in the city's Arlington district, the school was founded in 1934 as a two-year college and was known as Jacksonville Junior College until September 5, 1956, w ...
(1997–2000). Flanders was Editor-in-Chief,
American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...
, 1969–1973. He also wrote calculus
software Software is a set of computer programs and associated software documentation, documentation and data (computing), data. This is in contrast to Computer hardware, hardware, from which the system is built and which actually performs the work. ...
MicroCalc, ver 1–7 (1975–). In 1991 Flanders was invited to the first
SIAM Thailand ( ), historically known as Siam () and officially the Kingdom of Thailand, is a country in Southeast Asia, located at the centre of the Indochinese Peninsula, spanning , with a population of almost 70 million. The country is bo ...
workshop on automatic differentiation, held in Breckenridge, Colorado. Flanders' chapter in the ''Proceedings'' is titled "Automatic differentiation of composite functions". He presented an algorithm inputting two n-vectors of (higher) derivatives of ''F'' and ''G'' at a point, which used the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
to construct a linear transformation producing the derivative of the composite ''F'' o ''G''. With prompting from editor Griewank, Flanders included application of the algorithm to automatic differentiation of implicit functions. Recalling his early exposure to the formula of Faa di Bruno, Flanders wrote, "I think Faa's formula is quite inefficient for the practical computation of numerical (not symbolic) derivatives." Harley Flanders died July 26, 2013, in Ann Arbor, Michigan.


Differential forms

Flanders is known for advancing an approach to multivariate calculus that is independent of coordinates through treatment of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. According to
Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geom ...
, "an affine connection on a differentiable manifold gives rise to covariant differentiations of tensor fields. The classical approach makes use of the natural frames relative to local coordinates and works with components of tensor fields, thus giving the impression that this branch of differential geometry is a venture through a maze of indices. The author landersgives a mechanism which shows that this is not necessarily so." In 1954, Flanders considered the converse of the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
. In 1963, Flanders published ''Differential Forms with Applications to the Physical Sciences'' which connected
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemat ...
and differential forms. A reviewer affirmed that the book forms such a bridge with differential geometry. The book was republished in 1989 by
Dover Books Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...
.


Awards

*MAA Lester R. Ford Award 1969 *NCRIPTAL/EDUCOM Distinguished Software Award 1987 *NCRIPTAL/EDUCOM Distinguished Software Award 1989 *Lifetime Senior Member,
IEEE The Institute of Electrical and Electronics Engineers (IEEE) is a 501(c)(3) professional association for electronic engineering and electrical engineering (and associated disciplines) with its corporate office in New York City and its operati ...
1998


Mathematics education

In 1970, Flanders published the first of several useful
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textboo ...
s for topics commonly taught at college level: with Justin Jesse Price and Robert R. Korfhage a text on
Calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
was distributed by
Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes refere ...
. With J. J. Price, Flanders also wrote ''Elementary Functions and Analytic Geometry'' (1973) and ''Introductory College Mathematics: with Linear Algebra and Finite Mathematics'' (1974). With both J.J. Price and R.R. Korfhage, Flanders wrote ''First Course in Calculus with Analytic Geometry'' (1974) and ''Second Course in Calculus'' (1974). To support the recruitment of students with capacity to follow these courses, some works on
precalculus In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometr ...
mathematics were written with J.J. Price: ''Algebra'' (1975), ''Trigonometry'' (1975), ''Algebra and Trigonometry'' (1981), ''Precalculus Mathematics'' (1981), and ''College Algebra'' (1982). Flanders continued with ''Single-Variable Calculus'' (1981) and another ''Calculus'' in 1985 In 1984, Flanders published his textbook on
Pascal language Pascal is an Imperative programming, imperative and Procedural programming, procedural programming language, designed by Niklaus Wirth as a small, efficient language intended to encourage good programming practices using structured programming an ...
: ''Scientific Pascal'' (1984)Scientific Pascal
via
Google Books Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical ...
for which a second edition was published in 1996 by
Birkhäuser Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particu ...
. That year he also published ''Calculus: A lab course with MicroCalc'' (Springer-Verlag).


Selected papers

* * * * * * * * * * * * *with P. M. Lin: * * *with Harald K. Wimmer: * * *with Harald K. Wimmer: * *with Herbert Fischer: *


References

{{DEFAULTSORT:Flanders, Harley 20th-century American mathematicians 21st-century American mathematicians University of Chicago alumni University of California, Berkeley faculty Purdue University faculty Tel Aviv University faculty 2013 deaths 1925 births Jacksonville University faculty University of Michigan staff American textbook writers The American Mathematical Monthly editors