Posts tagged with integral

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.)

 

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)










There is a century-old tree at the end of my street. Right before you get to the graveyard with its wrought-iron gates. That tree saw my grandmother play in the street when she was a little girl. It saw her ride the train to the college, carry groceries in a paper sack. The tree—I don’t know its name—it saw my da walk across town—from school to that house on Broad, when they used to live there. It can see my great-grandfather’s grave right now—it’s tall enough. He built this house in 1921. They say he was a drunk. The floor slants a little and the window frames aren’t square. He built the other houses on our block, too. Before he built them, it was just this house and greenhouses. The greenhouses were filled with roses. The whole neighbourhood used to smell like roses. At some point they used to call this Rose City, even though there’s a meatpacking factory only two kilometres away.

 

They also say he could multiply long numbers in his head, without any paper. Now this house is holding a different kind of “family”. I can’t even say it’s a modern one. More like a gathering of moneyless relations. Ambitious failures; I sometimes wonder what the house thinks of us. It’s certainly used to the self-help books: Latin; Linux; teach yourself guitar. The trains in this town used to carry passengers. They took my grandmother to the teacher’s college. My da must have walked past this graveyard a thousand times. No, more—maybe even ten thousand. I walk in the graveyard every day. The tree sees me. My favourite is when it’s snowy. Some of the graves announce strange names. A woman named Ruby. She would be 136 now. A man named Reason. Apparently the brothers who lie beneath the massive Romanesque columns at the highest point in the graveyard invented a transport that was used massively during the War. You can see most of the town standing among those columns. Past the roads there’s a small forest, beyond that farms.

I’m thinking about my path γ(t) versus the tree’s λ(t). Neither of us can be everywhere at once. We’ve stood at or around the same spot often enough. But every time I’ve gone “adventuring”, I haven’t seen what’s happening in λ(t). Is the small-town life “worse” than the jet-setter lifestyle? It depends what functional you convolve against γ(t). I don’t like repetitiveness, but maybe what the tree has seen isn’t so repetitive. Two World Wars. The rise of feminism. A time before plastic, a time before tarmac, a time when white supremacists would parade through the streets. My grandmother recognised someone’s shoes and shouted his surname; her mother covered her mouth. The tree saw her in most stages of life.

On we go, hurtling through spacetime. The speed of γ equals the speed of λ. From a galactic perspective the tree and I are whirling in almost the same place — regardless of whether I whisk from here on the earth to there on the earth by plane. I’m bound to the ground, ultimately. The tree just recognises that. People used to wear hats here. Everybody wore hats. Now it’s practically a ghost town except for pensioners and welfare recipients. The tree’s children can’t have blown too far.

Spinning in the same spot on 360° × [−90°, +90°] = ∂(S²×[0,1]). γ torques and twists about the sphere but its length is exactly the same. Does the tree wish λ had summited a mountain at some point? Perhaps, but it would be blown down up there, and the ground is tough and nutritionless anyway. It’s suited to this life.

It bears the snow. It puts up with the heat.

I go inside after a couple hours out of doors, of course. But the tree spends all night, every night facing the elements. Maybe it likes being strong. Digging. Growing big. Drinking in sunlight like an athlete at a water fountain.

I’m more like a tumbleweed, rootless, quick to change course. Hanging out for a bit and then rolling—without announcing a goodbye. Untethered. Free, yet constrained by the same holonomy constraining the tree. One path, and one path only. The same width as all the others.

γ isn’t so much more interesting than λ. My γ is filled with magazines, airports, computer screens. Parties where people say more or less the same things, always indicating the hope that their gradient’s pointing in the right direction.




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Gauß’ divergence theorem states that, unless matter is created or destroyed, the density within a region of space V can change only by flowing through its boundary ∂V. Therefore

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i.e., you can measure the changes in an entire region by simply measuring what passes in and out of the boundaries of the region.

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"Stuff passing through a boundary " could be:

  • tigers through a conservation zone (2-D)
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  • sodium ions through a biological cell (3-D)
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  • magnetic flux through a toroidal fusion chamber
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  • water through a reservoir (but you’d have to measure evaporation, rain, dew/condensation, and ground seepage in order to get all of ∂V)
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  • in the other direction, you could measure water upstream and downstream in a river (no tributaries in between) and infer the net amount of water that was drunk, evaporated, or seeped
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  • probability mass through a set of possibilities
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  • particulate pollution through "greater Los Angeles"
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  • ¿ ideas through your head ? ¿ electrical impulses through your brain ? ¿ feelings through your soul over time ?
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  • ¿ notes through a symphonic orchestra ?
  • chromium(VI) through a human body
  • smoke or steam through an industrial cooling tower or smokestack
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  • imports and exports through an economy
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  • goods or cash through a limited liability company

Said in words, the observation that you can measure change within an entire region by just measuring all of its boundaries sounds obvious, even trivial. Said symbolically, Gauß’ discovery amounts to a nifty tradeoff between boundaries  and gradients . (The gradient  is the net amount of a flow: flow in direction 1 plus flow in orthogonal direction 2 plus flow in mutually orthogonal direction 3 plus…) It also amounts to a connection between 2-D and 3-D.

 

Because of Cartan-style differential geometry, we know that the connection is much more general: 1-D shapes bound 2-D shapes, 77-D shapes bound 78-D shapes, and so on.

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Nice one, Fred.

(Source: ocw.mit.edu)




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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.

 

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.”

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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.

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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.

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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?)

 

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: )




dt

  • dt is a moment of your life
  • ∫ dt is the moments as they accumulate

  • is the whole thing and what it meant.

Poetic.