Harley Flanders

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Harley Flanders
Born(1925-09-13)September 13, 1925
DiedJuly 26, 2013(2013-07-26) (aged 87)
Alma materUniversity of Chicago
Known forDifferential forms
Scientific career
InstitutionsUniversity of California, Berkeley
Doctoral advisorOtto Schilling
André Weil
Doctoral studentsTheodore Frankel

Harley 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]


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.[2] He held the Bateman Fellowship at Caltech. He joined the faculty at University of California at Berkeley, then became professor at Purdue University (1960).

Teaching posts he held included the faculty at 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).[3]

Flanders was Editor-in-Chief, American Mathematical Monthly, 1969–1973. He also wrote calculus software MicroCalc, ver 1–7 (1975–).[4][5]

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

Differential forms[edit]

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 indicies. The author [Flanders] gives a mechanism which shows that this is not necessarily so."[6]

In 1954 Flanders considered the converse of the Poincaré lemma.[7]

In 1963 Flanders published Differential Forms with Applications to the Physical Sciences which connected applied mathematics and differential forms.[8] A reviewer affirmed that the book forms such a bridge with differential geometry.[9] The book was republished in 1989 by Dover Books.


  • MAA Lester R. Ford Award 1969[10]
  • NCRIPTAL/EDUCOM Distinguished Software Award 1987
  • NCRIPTAL/EDUCOM Distinguished Software Award 1989
  • Lifetime Senior Member, IEEE 1998


  • Differential forms with applications to the physical sciences (1964)
  • Calculus (Academic Press, 1970). With Justin Jesse Price and Robert R. Korfhage.
  • Elementary functions and analytic geometry (Academic Press, 1973). With Price.
  • First course in calculus with analytic geometry (Academic Press, 1974). With Korfhage and Price.
  • Introductory college mathematics: With linear algebra and finite mathematics (Academic Press, 1974). With Price.
  • C. Loewner's Lectures on Continuous Groups, (MIT Press, 1971). With Murray H. Protter.
  • Second course in calculus (Academic Press, 1974). With Korfhage and Price.
  • Algebra (Academic Press, 1975). With Price
  • Calculus (1985).
  • Single-variable calculus (Academic Press, 1981)
  • Trigonometry (Academic Press, 1975). With Price.
  • Pre-calculus mathematics (Academic Press, 1981). With Price.
  • College algebra (Academic Press, 1982). With Price.
  • Algebra and trigonometry (Academic Press, 1981). With Price
  • Scientific Pascal (1984).[5]
  • Scientific Pascal 2/e (Birkhauser 1996)
  • Calculus: A lab course with MicroCalc (Springer-Verlag, 1996).

Selected papers[edit]

  • 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)
  • Development of an Extended Exterior Differential Calculus, Trans. Amer. Math. Soc. 75 (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)
  • The Schwarzian as a Curvature, J. Diff. Geom. 4 (1970)
  • 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 AXXB = C and AXYB = C, (with H. Wimmer), SIAM Jour. Math. Anal. 32 (1977)
  • Coroutines in Pascal, ACM SIGPLAN Notices 18 (December 1983)
  • Automatic differentiation of composite functions, in Griewank, A. and Corliss, G. F., Automatic Differentiation of Algorithms, SIAM, 1991
  • A minimal code list (with Herbert Fischer), Theoretical Computer Science 215 (1999)
  • Functions not satisfying implicit, polynomial ODE, J Diff Eqs 240:1 (2007)


  1. ^ a b "Harley Flanders Obituary: View Harley Flanders's Obituary by AnnArbor.com". Obits.mlive.com. Retrieved 2013-08-09.
  2. ^ entry from the Mathematics Genealogy Project
  3. ^ memo from uchicago.edu
  4. ^ Dr. Flanders is a unique person Archived 2007-08-08 at the Wayback Machine
  5. ^ a b online googlebook
  6. ^ 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
  7. ^ H. Flanders (1954) An extension theorem for solutions of dω = Ω, Proceedings of the American Mathematical Society 5(3):509, 10
  8. ^ 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.
  9. ^ MR0162198
  10. ^ Flanders, Harley (1968). "A proof of Minkowski's inequality for convex curves". Amer. Math. Monthly. 75: 581–593. doi:10.2307/2313773.