[1] 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 Édouard Goursat, translated by Earle Hedrick, which included challenging exercises.
[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.
Notes were taken and the lectures appeared in a limited form with the expectation that Loewner would produce a book on the topic.
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.
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.
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.