We introduce the combinatorial notion of posetted trees and we use it in order to write an explicit expression of the Baker-Campbell-Hausdorff formula.

actually I thought about it again in an elementary example. Drawing x²/2 for a freshman and showing how the centred derivative is more accurate than the right-hand one which is usually used.

Put a point, draw two secant lines (equal ± along the abscissa).

Then the graphically intuitive fact is that, in this 1→1D picture, connecting the two ± points —- which belong to different tangent spaces! —– has “the same” slope as the real tangent. So how to state this in terms of contravariant motion? I.e., motion of the bundles themselves. Because it’s graphically obvious as well that the connected secant-line is not “the” tangent line, since it’s shifted over.

Anyway, also if you work out the average of left- and right- derivatives (limit definition) then you get a middle term which always cancels. And this reminds me of the ∂²=0 dictum, and I think it’s the same thing (as well as topologically amenable).

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I wonder if it’s the same thing Peter Saveliev was saying?

http://isomorphism.es/post/24164852462/calculus-is-topology

actually I thought about it again in an elementary example. Drawing x²/2 for a freshman and showing how the centred derivative is more accurate than the right-hand one which is usually used.

Put a point, draw two secant lines (equal ± along the abscissa).

Then the graphically intuitive fact is that, in this 1→1D picture, connecting the two ± points —- which belong to different tangent spaces! —– has “the same” slope as the real tangent. So how to state this in terms of contravariant motion? I.e., motion of the bundles themselves. Because it’s graphically obvious as well that the connected secant-line is not “the” tangent line, since it’s shifted over.

Anyway, also if you work out the average of left- and right- derivatives (limit definition) then you get a middle term which always cancels. And this reminds me of the ∂²=0 dictum, and I think it’s the same thing (as well as topologically amenable).

Anyway, then I wondered if it connects to this phenomenon: http://isomorphism.es/post/47732865342/product-rule-with-shapes

BTW, currently my analogy for top / diff top / riemannian\curvature is this:

topology = C⁰

diff top = C¹

curvature = C²