VISUAL DIFFERENTIAL GEOMETRY AND FORMS, by Tristan Needham
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Differential Geometry contains the word Geometry.

If A and B depend on a small ε, and their ratio → 1 as ε → 0:
	A is ultimately equal to B
	A ≍ B

Symbol: ≍
	Unicode code point: U+224D
	Unicode name: “EQUIVALENCE ≈” (also called “equivalence sign”)
	Needham calls it "ultimate equality"
	Block: Mathematical Operators

This equality holds transitive and other properties.
It can also be applied to objects, like triangles.

Act I   : The Nature of Space
Act II  : The Metric
Act III : Curvature
Act IV  : Parallel Transport
Act V   : Forms

Act III : Curvature ---> Climax ---> Global Gauss-Bonnet Theorem 
	local geometry <---> global topology

Manifold: generalisation to n dimension of 2d surfaces
	The Riemann tensor measures its curvature.

Act V:
	Exterior calculus ---> differential forms, or just forms
	- Forms have immense power, reminiscent of complex numbers.
	- They give back more than they have been put in, a sign they are Platonic Forms.
	- They unify and clarify all vector calculus.
	- Green's, Gauss's and Stokes's theorems are != manifestations of a single th.
	- Undergraduates leaving college w/o knowing this ---> a scandal.
	- Forms are simple and beatiful: they, and Cartan, deserve more credit.

The focus is on intrinsic, not extrinsic geometry.

Notation: (equation), [figure]

If ugly terms are cancelled, they should never have been there in the first place.