# Harley Flanders

**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, electrical networks, scientific computing.[1]

Harley Flanders | |
---|---|

Born | |

Died | July 26, 2013 87) | (aged

Nationality | American |

Alma mater | University of Chicago |

Known for | Differential forms |

Scientific career | |

Fields | Mathematics |

Institutions | University of California, Berkeley |

Doctoral advisor | Otto Schilling André Weil |

Doctoral students | Theodore Frankel |

## 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*[2] by Edouard Goursat, translated by Earle Hedrick, 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 a form now called the Faa di Bruno formula.[3]

Flanders received his bachelors (1946), masters (1947) and PhD (1949) at the University of Chicago on the dissertation *Unification of class field theory* advised by Otto Schilling and André Weil.[4]
He held the Bateman Fellowship at Caltech. He joined the faculty at University of California at Berkeley. 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 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 (1960), Tel Aviv University (1970–77), visiting professor at Georgia Tech (1977–78), visiting scholar at Florida Atlantic University (1978–85), University of Michigan, Ann Arbor (1985–97, 2000–), University of North Florida (1997–2000) and, distinguished mathematician in residence at Jacksonville University (1997–2000).[5]

Flanders was Editor-in-Chief, American Mathematical Monthly, 1969–1973. He also wrote calculus software MicroCalc, ver 1–7 (1975–).[6][7]

In 1991 Flanders was invited to the first SIAM 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 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.[8] 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."[3]

Harley Flanders died July 26, 2013, in Ann Arbor, Michigan.[1]

## Differential forms

Flanders is known for advancing an approach to multivariate calculus that is independent of coordinates through treatment of differential forms. According to Shiing-Shen Chern, "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 [Flanders] gives a mechanism which shows that this is not necessarily so."[9]

In 1954 Flanders considered the converse of the Poincaré lemma.[10]

In 1963 Flanders published *Differential Forms with Applications to the Physical Sciences* which connected applied mathematics and differential forms.[11] A reviewer affirmed that the book forms such a bridge with differential geometry.[12] The book was republished in 1989 by Dover Books.

## Awards

- MAA Lester R. Ford Award 1969[13]
- NCRIPTAL/EDUCOM Distinguished Software Award 1987
- NCRIPTAL/EDUCOM Distinguished Software Award 1989
- Lifetime Senior Member, IEEE 1998

## Mathematics education

In 1970 Flanders published the first of several useful textbooks for topics commonly taught at college level: with Justin Jesse Price and Robert R. Korfhage a text on Calculus was distributed by Academic Press. 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 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: *Scientific Pascal* (1984)[7] for which a second edition was published in 1996 by Birkhäuser. That year he also published *Calculus: A lab course with MicroCalc* (Springer-Verlag).

## Selected papers

- Elementary Divisors of AB and BA, Proc. Amer. Math. Soc. 2 (1951)
- Generalization of a Theorem of Ankeny and Rogers, Ann. of Math. 57 (1953)
- On Certain Functions with Positive Definite Hessian, Ann. of Math. 71 (1960)
- Meaning of the Form Calculus in Classical Ideal Theory, Trans. Amer. Math. Soc. 95 (1960)
- On Spaces of Linear Transformations of Bounded Rank, J. London Math. Soc. 37 (1962)
- Satellites of Half Exact Functors, a correction, Proc. Amer. Math. Soc. 15 (1964)
- Local Theory of Affine Hypersurfaces, J. Anal. Math. 15 (1965)
- The Steiner Point of a Closed Hypersurface, Mathematika 13 (1966)
- Tensor and Exterior Powers, J. Algebra 7 (1967)
- Relations on Minimal Hypersurfaces, Pacific J. Math. 29 (1969)
- 1970: The Schwarzian as a Curvature, Journal of Differential Geometry 4:515–9, via Project Euclid
- Infinite Networks I – Resistive Networks, IEEE Trans. Circuit Theory 18 (1971)
- Natural Frequencies of Cyclic Linear Networks (with P.M. Lin), IEEE Trans. Circuit Theory 18 (1971)
- Differentiation Under the Integral Sign, Amer. Math. Monthly 80 (1973), no. 6, pp. 615–627
- A New Proof of R. Foster's Averaging Theorem, Linear Algebra and its Applications 8 (1974)
- Positive Operators and a Problem in Control Theory, (with Harold Wimmer) Linear Algebra and its Applications 8 (1974)
- An Extremal Problem in the Space of Positive Definite Matrices, linear and Multilinear Algebra 3 (1975)
- On the Maximal Power Transfer Theorem for
*n*-ports, International Journal of Circuit Theory and Applications 4 (1976) - On the matrix equations
*AX*−*XB*=*C*and*AX*−*YB*=*C*, (with H. Wimmer), SIAM Jour. Math. Anal. 32 (1977) - Coroutines in Pascal, ACM SIGPLAN Notices 18 (December 1983)
- A minimal code list (with Herbert Fischer), Theoretical Computer Science 215 (1999)
- Functions not satisfying implicit, polynomial ODE, J Diff Eqs 240:1 (2007)

## References

- "Harley Flanders Obituary: View Harley Flanders's Obituary by AnnArbor.com". Obits.mlive.com. Retrieved 2013-08-09.
- E. Goursat, E.R. Hedrick translator (1904) A Course in Mathematical Analysis via HathiTrust
- H. Flanders (2001) "From Ford to Faa", American Mathematical Monthly 108(6): 558–61 doi:10.2307/2695713
- entry from the Mathematics Genealogy Project
- memo from uchicago.edu
- Dr. Flanders is a unique person Archived 2007-08-08 at the Wayback Machine
- Scientific Pascal via Google Books
- Andreas Griewank & George F. Corliss (editors)(1991)
*Automatic Differentiation of Algorithms: Theory, Implementation, Application*, SIAM ISBN 0-89871-284-X, Flanders' paper: Part III, chapter 10, pages 95–9. - H. Flanders (1953) "Development of an extended exterior differential calculus", Transactions of the American Mathematical Society 75: 311–26, doi:10.1090/S0002-9947-1953-0057005-8MR0057005
- H. Flanders (1954) An extension theorem for solutions of dω = Ω, Proceedings of the American Mathematical Society 5(3):509, 10
- Hermann, Robert (1964). "Review:
*Differential forms with applications to the physical sciences*, by Harley Flanders".*Bulletin of the American Mathematical Society*.**70**(4): 483–487. doi:10.1090/s0002-9904-1964-11159-9. - MR0162198
- Flanders, Harley (1968). "A proof of Minkowski's inequality for convex curves".
*Amer. Math. Monthly*.**75**: 581–593. doi:10.2307/2313773.