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Posts tagged with differential forms

The dual V* of a vector space V  over ℝ matches lists of reals to linear functionals.

What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb quintuple.

image
linear maps as multiplication
linear mappings -- notice they're ALL straight lines through the origin!

pictures of lines through the origin with various slopes

Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is ƒ₅(x)=5•x. That ƒ₅ does things like

  • ƒ₅(■■■)=5+5+5 and
  • ƒ₅(■■■■■■)=5+5+5+5+5+5.

Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like _ / ∕ ...|. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say id, double, triple, quadruple, quintuple, ….

(Why did I jump so suddenly 0,1,… from _ flat to ⁄  45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).
image
To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T
image
hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)

 

These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same kind of thing. One is nouns and one is a verb.

But still, for any real number that I “counted” I could match up a function, just like I did with

  • "5 that I counted" and
  • "function ƒ₅ = quintuple

So these two qualitatively different things are in bijection. (One can hope for insights by viewing things through one lens or the other, noun or verb version.)

This one-dimensional story can be upgraded to a multi-dimensional one where

  • lists of reals (3.1, √2, −2.1852, ..., 6) 

match to

  • many-to-one functions ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].

Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.




a smooth field of 1-vectors in 3-D

a smooth field of 1-vectors in 3-D

(Source: thievess)


hi-res




Discrete differential geometry

Check out page 40 of the source PDF (calcula ex geometrica) — they talk about geometrical computation. Instead of

  1. approximating analog with digital
  2. doing digital arithmetic
  3. smoothing the result to fake an analog again

why not OOP-define computer operations that deal directly with the curves via their defining properties?

Sort of like how MATLAB, Mathematica, Wolfram Alpha, YACAS, SAGE, Maxima, Axiom, Maude, PARI, lie, Singular, GAP — the symbolic calculators — do integrals and derivatives by the same rules that you would on paper rather than numerically approximating a function and then delivering only floating-point answers. The “computer as zillions of arithmetic operations” idea is totally incapable of giving a general answer like ∫cos=sin, or recognise some Green’s-Theorem-type reduction so that a few quintillion computational steps don’t even need to be performed.

The exterior calculus — invention of Élie Cartán — appears to be the right set of principles to object-orientedly program to do the analog-of-symbolic-computation for curves and surfaces.

It makes sense to me. If you have a plane curve with essentially two parameters, why does it need to be represented as an arbitrarily long array of points=pairs? You should be able to get arbitrary precision (just like a symbolic calculator does) with just the two parameters, if the reasoning system you programmed reasons the right way.

Overload dem operators with curve+curve= and surface×curve=, the wedge operator’s leading the way.

(Source: mesh.brown.edu)










Calculus is topology.

The reason is that the matrix of the exterior derivative is equivalent to the transpose of the matrix of the boundary operator. That fact has been known for some time, but its practical consequences have only been understood recently.

[S]uppose you know the boundary of each k-cell in a cell complex in terms of (k−1)-cells, i.e., the boundary operator. Then you also know the exterior derivative of all discrete differential forms (i.e., cochains). So, you know calculus. Smooth or discrete.

Peter Saveliev

(Source: inperc.com)




A beautiful depiction of a 1-form by Robert Ghrist. You never thought understanding a 1→1-dimensional ODE (or a 1-D vector field) would be so easy!

What his drawing makes obvious, is that images of Phase Space wear a totally different meaning than “up”, “down”, “left”, “right”. In this case up = more; down = less; left = before and right = after. So it’s unhelpful to think about derivative = slope.

BTW, the reason that ƒ must have an odd number of fixed points, follows from the “dissipative” assumption (“infinity repels”). If ƒ (−∞)→+, then the red line enters from the top-left. And if ƒ (+∞)→−∞, then the red line exits toward the bottom-right. So no matter how many wiggles, it must cross an odd number of times. (Rolle’s Thm / intermediate value theorem from undergrad calculus / analysis)

Found this via John D Cook.

(Source: math.upenn.edu)