Posts tagged with differential geometry

Double integrals `∫∫ƒ(x)dA` are introduced as a “little teacher’s lie” in calculus. The “real story” requires “geometric algebra”, or “the logic of length-shape-volume relationships”. Keywords

• multilinear algebra
• Grassmann algebra / Grassmanian
• exterior calculus
• Élie Cartán’s differential-forms approach to tensors

These equivalence-classes of blobs explain how

• volumes (ahem—oriented volumes!)
• areas (ahem—oriented areas!)
• arrows (vectors)
• numbers (scalars)

"should" interface with each other. That is, Clifford algebra or Grassman algebra or "exterior algebra" or "geometrical algebra" encodes how physical quantities with these dimensionalities do interface with each other.

(First the volumes are abstracted from their original context—then they can be “attached” to something else.)

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EDIT:user mrfractal points out that Clifford algebras can only have dimensions of 2,4,8,16,… https://en.wikipedia.org/wiki/Clifford_algebra#Basis_and_dimension Yes, that’s right. This post is not totally correct. I let it fly out of the queue without editing it and it may contain other inaccuracies. I was trying to throw out a bunch of relevant keywords that go along with these motivating pictures, and relate it to equivalence-classing, one of my favourite themes within this blog. The text here is disjointed, unedited, and perhaps wrong in other ways. Mostly just wanted to share the pictures; I’ll try to fix up the text some other time. Grazie.

(Source: arxiv.org)

It wasn’t Einstein, but the mathematician Hermann Weyl who first addressed the [distinction] [between gravitational and non-gravitational fields] in 1918 in the course of reconstructing Einstein’s theory on the preferred … basis of a “pure infinitesimal geometry”….

Holding that direct…comparisons of length or duration could be made at near-by points of spacetime, but not … “at a distance”, Weyl discovered additional terms in his expanded geometry that he … formally identified with the potentials of the electromagnetic field. From these, the electromagnetic field strengths can be immediately derived.
Choosing an action integral to obtain both [sorts of] Maxwell equations as well as Einstein’s gravitational theory, Weyl could express electromagnetism as well as gravitation solely within the confines of a spacetime geometry. As no other interactions were definitely known to occur, Weyl proudly declared that the concepts of geometry and physics were the same.

Hence, everything in the physical world was a manifestation of spacetime geometry. (The) distinction between geometry and physics is an error, physics extends not at all beyond geometry: the world is a (`3+1`) dimensional metrical manifold, and all physical phenomena transpiring in it are only modes of expression of the metric field, …. (M)atter itself is dissolved in “metric” and is not something substantial that in addition exists “in” metric space. (1919, 115–16)

Riemannian

Ryckman, Thomas A., "Early Philosophical Interpretations of General Relativity", The Stanford Encyclopedia of Philosophy (Fall 2012 Edition), Edward N. Zalta (ed.), forthcoming URL = <http://plato.stanford.edu/archives/fall2012/entries/genrel-early/>.

via University of David

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)

The Idea of Holonomy by Robert Bryant

from the MAA:

“Can I roll the ball from any point to any other point and have it wind up in a given orientation that we want?” Bryant asked.

If I draw a dot with this marker, can you eventually roll the ball enough times so that the dot would touch down anywhere on the table, anywhere at all? Or is the logic of the situation constrained, so that certain spots on the ball pair with certain spots on the table? The answer, he said, has consequences for fields from robotics to control theory.

To me, the idea of constrained motion sounds more like the fundamental economic dilemma.

• You can’t live in as nice of a house as you want and work as little as you want and have all the other stuff you want.
• Even if you had \$100,000,000, you still couldn’t spend the weekend fishing in Chile and attend the Davos seminar and go to your son’s art exhibition.
• There’s a direct tradeoff between how long you work on building the perfect product (say, a game console) and how soon it will be released. You might be able to achieve a little more of both by investing more money into the project … but that comes at the expense of something else.

The “optimal path" — if such a thing even exists — will never be feasible, because the choice space is fundamentally characterised by tradeoffs.

More drawings by Robert Ghrist. Point-set topology, differential topology, geometric topology, symplectic topology, algebraic topology illustrated. From his (free) notes on applied homology.

GLOSSARY

• Topology = connections between things.
• Manifolds come in a wide variety of shapes, but they’re all tame.
• Vector fields = arrows on a manifold.
• Differential = linear approximation.
• Symplectic = isotropic + antisymmetric + bilinear. (erm, not as complicated as it sounds)
• Phase Space means these things can be conceptual rather than literal.

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)

In gradeschool calculus I learnt that derivative = slope. That was a nice teacher’s lie (like the Bohr atom is a nice teacher’s lie) to get the essential point across. But “derivative = slope” isn’t ultimately helpful because in real life, functions aren’t drawn on a chalkboard. ℝ→ℝ drawings don’t always look like what they feel like (e.g. this parabola).

ℝ→ℝ drawings’ “slope” feels more like a pulse, a β (observed magnitude), a force, a pay rise, a spike in the price of petrol, a nasty vega wave that chokes out a hedge fund, cruising down the highway (speedometer not odometer), a basic not a derived parameter, a linear operator in the space of all functionals, a blip, a pushforward, an impression, a straight-line projection from data, a deep dive into a function’s infinite profundity, a “bite” in the words of Jan Koenderink.

A derivative “is really” a pulse. And an integral “is really” an accumulation.

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This story, “Bird’s Eye View” by Radiolab (minute 12:00), nicely illustrates a differential-geometry-consistent view of derivative & integral in the pleasantly-unexpected space of rare languages.

English : Derivative :: Pormpuraaw : Integral

In the Pormpuraaw language of Cape York, Australia, people say things like “You have an ant on your south-west leg” and “Move your cup to the north-north-west a bit”. “How ya goin’?” one asks the other. "Headed east-north-east in the middle distance."

• Little kids always know, even indoors, which cardinal direction they’re facing.
• This is very useful when you live in the outback without a GPS.
• American linguistics professor who was exploring there: “After about a week I developed a bird’s-eye view of myself on a map, like a video game, in the upper right corner of my mind’s eye.”

$\large \dpi{200} \bg_white \oint \text{where I am}_t \ \ast\; \text{which way I'm facing}_t \ (\theta_t) \ dt$

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The mental map is like a running integral ∮ xᵗθᵗ dt of moves they make. (Or we could think of it decomposed into two integrals, one that tracks changes in orientation ∮ θᵗ and one that tracks accumulating changes in place ∮x.) In other words, a bird’s-eye view.

left right forward back : derivative :: NSEW : integral

Our English way of thinking is like a differential-geometry-consistent derivative. The time derivative “takes a bite” out of space and so is always relative to the particular moment in time. “Left” and “right” are concepts like this — relative, immediate, and having no length of their own. Just like the differential forms in Élie Cartan’s exterior algebra — tangent to our bodies.

$\large \dpi{200} \bg_white \delta \{ \small{ \; \smallint \!\! _t \ \text{which way I'm facing}_t } \; \}$

There is a way to make this more precise and I think it would make sense to do it on  || with a twistor || spinor. (Help, anyone? David?)

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Our English conception of time & space is like a (time-)derivative of our movements. The Pormpuraawans’ conception of time & space is like an integral of their movements, orientation, and location. When we think of direction it’s an immediate slice of time. When they think of direction they’ve been tracking those relative-direction derivatives and they answer with the sum.

(Source: )

## arXiv

Hola Nerds,

Have you ever found yourself browsing tumblr and thinking to yourself: “Instead of looking at pictures of dyed hair, I should really be thinking about chirped dissipative solitons" ?

Well, now you can. I ported the following arXiv feeds to tumblr:

If you’d like me to add another, tweet @isomorphisms. Or if you don’t tweet, use http://isomorphismes.tumblr.com/ask.

Gracious thanks to codecogs, perl monks, p.t. campbell, tumblr, and of course the arXiv & supporting institutions.

## Clouds

You can see the “edge” of a cloud from far away so it should be obvious what ∂cloud means. But up close (from an airplane) you can see there is no edge. The mist fades gradually into blue sky.

Here’s another job for schwartz functions: to define a “fuzzy boundary”  that looks sharp from far away but blurred up close. In other words, to map each Cartesian 3-point to a fuzzy inclusion % in the set {this cloud}.

Jan Koenderink, in his masterpiece Solid Shape, notes that a typical European cumulus cloud has density 𝓞(100 droplets) per cm³ (times 16 in inch⁻³). Droplets are 3–30 μicrons in diameter. (3–30 hair widths across) Typical clouds have a density of .4g/m³ or 674 pounds of water per cubic football field of cloud.

To lift directly from page 508:

What is actually meant by “density” here? Clearly the answer depends on the inner scale or resolution.

At a resolution of 1 μm the density is either that of liquid water or that of air, depending critically on the position within the cloud. At a resolution of ten miles the density is near zero because the sample in the window is diluted.

Both results are essentially useless. The right scale is about a meter, with maybe an order of magnitude play on both sides.

Rather than having just one sharp boundary, ∂cloud is a sequence of level surfaces that enclose a given density at a given resolution. To avoid having to choose an arbitrary resolution parameter, we can define the fuzzy inclusion with a schwartz function. We get a definite beginning and end (compact support) without going too into particulars (like rate of the % dropoff) and this is true at any sensible resolution.

We can’t say exactly where the boundary is, but we can point to a spot in the sky that’s not cloud and we can point to a spot in the sky that is cloud.

## Manifolds, Metrics, Star Fox, and Self-versus-Other

Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.

Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods" that Newton & Descartes supposed make up the fabric of space.

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved" metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around "0", one’s own perspective — and spread out where the "others" stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

If you stitch together such non-Euclidean flats, you’ve again constructed a manifold.

Think about this: the pixel concept re-presents brush-stroke or natural images by a wall of sequential colored squares.  You could extend it to 3-D, for example representing humans by little blocks—white for the bone, burgundy for the blood, pink for the fingernails, etc.

In a similar fashion, the manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.