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Collection of coordinate charts.
Bibliography:
- www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds #6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
- www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the Directional derivative of any function along this direction".One thing to remember is that of course, the most convenient way to define a function and to specify a direction, is by using one of the coordinate charts.
- jakobschwichtenberg.com/lie-algebra-able-describe-group/ by Jakob Schwichtenberg
- math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on Stack Exchange
www.youtube.com/watch?v=tq7sb3toTww&list=PLxBAVPVHJPcrNrcEBKbqC_ykiVqfxZgNl&index=19 mentions that it is a bit like a dot product but for a tangent vector to a manifold: it measures how much that vector derives along a given direction.
Shame that the Chinese in the lat 20th early 21st like that bullshit so much. It just weakens everything. Just imaginge those works with more realistic fighting! Would be amazing.
This is true: high budget movies are shit. Just TV Trops can articular it infinitely better than Ciro Santilli can.
Related:
Source code:
- github.com/leanprover/lean4 why a separate repo per version... but it is what it is.
- github.com/leanprover/lean
The way Lean and Coq mix programming and mathematics is a thing of great beauty. This is especially notable in lean as you start to play with with things such as:
partialenv lean functions, and usingterminates_byto prove that certain functions terminate. Lean requires explicitly known if functions terminate or not to be able to use them in proofs.noncomputablefunctions. Lean allows you to define mathematical functions which you can't actually execute, and it tracks that explicitly
They are huge fans of Unicode characters! Check this out from a formal proof of the prime number theorem: github.com/AlexKontorovich/PrimeNumberTheoremAnd/blob/fbdbb5310d036d33b9797b35f3b04b08f2447a6e/PrimeNumberTheoremAnd/ZetaBounds.lean Here's map to Ascii: proofassistants.stackexchange.com/questions/954/does-lean-have-a-standard-ascii-representation/5289#5289
Their dependency graph thingy is just beautiful however: alexkontorovich.github.io/PrimeNumberTheoremAnd/web/dep_graph_document.html
Pinned article: Introduction to the OurBigBook Project
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