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There have been many false proofs of the Jacobian Conjecture, and it might be interesting to have a survey of the attempts and why they ultimately failed (i.e., what holes there were in the attempted …

My favorite example is Élie Cartan's original classification of the simple Lie algebras over the real numbers, the so-called real forms. The calculations and case analyses that he does in the five ex …

According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one s …

Perhaps the simplest way to understand this formula is to think about how you would go about deriving it: Try to find the 'best' coordinates you can centered on a given point and see what doesn't cha …

Gaston Darboux' magnum opus Leçons sur la Théorie générale des Surfaces et les Applications géométriques du Calcul infinitésimal (first edition 1890, I think; there is a second edition dating from aro …

The formulation of Laguerre geometry in terms of dual numbers is a decidedly 'synthetic' one, meant to exhibit how this set of transformations can be regarded as a different 'real form' of the well-kn …

You could try the introductory sections of our book, "Exterior Differential Systems and Euler-Lagrange Partial Differential Equations" (authors Bryant, Griffiths, Grossman), which is available at http …

Looking around on the internet, I found an English translation of Lie's 1891 paper Die Grundlagen für die Theorie der unendlichen kontinuierlichen Transformationsgruppen. I. (I.e., The foundations of …

Your quote about Cartan thinking of $B_n$ and $D_n$ as 'projective groups..." is actually Cartan describing the lowest dimensional homogeneous space of these groups (except, of course, for a few excep …

in addition to these excellent examples of non-local curvature quantities and their extensions to the non-smooth setting (which I am not sure the OP was anticipating), I might add the 'original' non-l …