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This question may seem peculiar, so let me preface it by saying that it arose while I was trying to understand Legendre transformations better, and in that context it is fairly natural. Anyway, suppos …

Have a look at Sylvanus P. Thompson's "Calculus Made Easy" subtitled, What One Fool Can Do, Another Can. It was written at the end of the 19h Century but takes a pretty much 18th Century approach (and …

If you do not mind a "self-reference" there is "Differential Equations, Mechanics, and Computation", published by AMS and written by me and my son Bob Palais. See the associated website at: http://od …

In Physics, the essential reason boils down to symmetry. One expects the fundamental laws of physics to be independent of where you are or how you are oriented in space, and if they are described by s …

If you go to the Archimedean Spiral page of the Virtual Math Museum: http://virtualmathmuseum.org/Curves/archimedean_spiral/archimedean_spiral.html you will find a lot of material about Archimede's …

Here is a slightly more general take on this question. First notice that your condition $L(1) = 0$ is redundant. That is because if you take $f = g = 1$ in your third condition, you get $L(1) = L(1) + …

Well, I cannot say for certain, but I did know Gleason well (he was my thesis advisor, and we wrote a paper together after that) and I have written an essay about Gleason's work on the Fifth Problem ( …

To say that $n \theta$ is equidistributed means in particular that for any open set $O$ of $(0,1)$ that if $N(n)$ is the the number of $k < n$ such that $k \theta (\mod 1) \in O$, then $N(n) /n $ appr …

It is easy to generalize the proof tha a differentiable real ODE has differentiable solutions (using a fxed point argument) from the real to the complex case, and deduce that a holomorphic ODE has loc …

I know it may seem rather "old", but the notes from the IAS "Seminar on the Atiyah-Singer Index Theorem from back in 1965 (published by Princeton Univ. Press) may be just what you are looking for, sin …

If you don't mind considering a recommendation from one of the co-authors of an ODE textbook, you sound like just the sort of student that we had in mind when we wrote "Differential Equations, Mechani …