I'm a pretty low-key annotator but I did write this in the margins of @stevenstrogatz's Infinite Powers.pic.twitter.com/2ze822F1a0
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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I'm a pretty low-key annotator but I did write this in the margins of @stevenstrogatz's Infinite Powers.pic.twitter.com/2ze822F1a0
Interesting how in contemporary expositions of calculus the phenomenon of telescoping cancellation (i.e., that a sequence of differences is easy to sum) is seldom stressed as the essence of the subject's fundamental theorem. Euler of course stressed it as such. We've regressed.
It’s also interesting that Stokes Theorem very often is presented as being about cancellation on common boundaries in the interior, and the 1D fundamental theorem is basically a special case of this, but isn’t usually talked about in the same way.
Indeed, Stokes' theorem is a highly suggestive further elaboration of the same basic phenomenon of telescoping cancellation, which ultimately underpins the vast edifices of homology and cohomology, Simple ideas often reach extremely far -- which is why mathematics is possible.
Same ideas arise in Goursat’s proof of Cauchy’s theorem In complex analysis. I want to write an article called a telescopic look at calculus
It's interesting, too, how intimately Cauchy's theorem is linked to the Sperner lemma, which is a sort of combinatorial-topological counterpart of the algebro-arithmetic phenomenon of telescoping cancellation. Riemann and Klein especially were keenly conscious of these links.
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