Curl in ND

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Curl in ND

Postby PatrickPowers » Fri Feb 05, 2016 9:16 pm

To be able to use curl in N spatial dimensions the key is to start with bivectors instead of vectors.

Curl is traditionally a vector. Its dot product with another vector is a scalar that measures vorticity. We don't have a dot here, so let | represent the dot of the dot product.

TRAD: curl | n = vorticity.

Both the curl vector and the normal vectors are pseudovectors, each used to define a plane. The curl vector defines a local plane of rotation, while the normal vector defines a locally planar surface. This works only in 3D, as this is the only space where vectors and planes are dual.

In geometric algebra(GA) each of these planes is defined by a bivector. The curl is a bivector and the plane is also a bivector. The dot product of the two bivectors yields the same scalar that measures vorticity. As we shall see, this has the advantage of scaling up to any number of dimensions. But for now we are still in 3D, where

GA 3D: curl | n* = vorticity, where the bivector n* is the dual of n.

We will start out easy. Vector i maps to bivector jk, j to ki, and k to ij. (We use ki instead of ik to get the signs to work out.)

Let

a = dF3/dy - dF2/dz
b = dF1/dz - dF3/dx
c = dF2/dx - dF1/dy

TRAD: [ai, bj, ck] = curl
curl | n = vorticity

GA: ajk + bki + cij = curl
curl | n* = vorticity

We won't be able to use symbols like a, i or dx in N dimensions, so we replace them with indexed symbols.

i=e1, j=e2, k=e3.

a1 = dF3/de2 - dF2/de3
a2 = dF1/de3 - dF3/de1
a3 = dF2/de1 - dF1/de2

Instead of using the proper eiej I'm going to use eij. It's easier to type in. Then ij= e12, ki= e31, jk= e23,

a1e23 + a2e31 + a3e12 = curl

With P any bivector,

curl | P = vorticity

Choose any N other than 3 and P doesn't describe a surface. This can't be helped. Get used to it.

The definition of curl may be found at http://www.ittc.ku.edu/~jstiles/220/handouts/The%20Curl%20of%20a%20Vector%20Field.pdf. We are not able to reproduce it on this page, so if you wish for this to make any sense you better have a good look. This definition isn't extendable. So we need to rewrite it in terms of bivectors. In N D we can't use a normal vector to define a bivector. But in N D we don't have to: just use the bivector directly. Instead of Bi and Ci, we want Bjk and Cjk. Easy. 1 maps to 23, 2 maps to 31, 3 maps to 12. Our new definition of curl is simpler than it was. But all we have done is change names. In 3D everything is functionally the same as it was, it just looks different.

With this change of point of view, coefficient ai no longer explicitly corresponds to normal vector ei.
Coefficient ai now corresponds to bivector ejk. So it makes sense to rename ai to ajk.

a23 = dF3/de2 - dF2/de3
a31 = dF1/de3 - dF3/de1
a12 = dF2/de1 - dF1/de2

Then

a23e23 + a31e31 + a12e12 = curl

Moving on, I know from experience that the e31 basis vector is going to cause us trouble. So change to e13 via e31=-e13.

a23 = dF3/de2 - dF2/de3
a31 = -(dF3/de1 - dF1/de3)
a12 = dF2/de1 - dF1/de2

(a23e23 - a31e13 + a12e12) = curl

This gives us

a23 = dF3/de2 - dF2/de3
a13 = dF3/de1 - dF1/de3
a12 = dF2/de1 - dF1/de2

a23e23 + a13e13 + a12e12 = curl

This superficial change allows a simple rule for the signs. If i < j then dFi/dej is negative, otherwise positive. Now we can add in new terms. Our definition of curl using bivectors does not depend on the number of dimensions. It even works in 1 and 0 dimensions, where the result is 0. Ready to go to 4D!

4D:

a12 = dF2/de1 - dF1/de2
a13 = dF3/de1 - dF1/de3
a14 = dF4/de1 - dF1/de4
a23 = dF3/de2 - dF2/de3
a24 = dF4/de2 - dF2/de4
a34 = dF4/de3 - dF3/de4

a12e12 + a13e13 + a14e14 + a23e23 + a24e24 + a34e34 = curl

The pattern is evident.

aij = dFj/dei - dFi/dej

curl = Sum over { i,j with 1<=i<j<=N } aijeij

vorticity = curl | P

with P any bivector.

As a final step, revert to standard GA notation.
curl = Sum over { i,j with 1<=i<j<=N } aijeiej
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Re: Curl in ND

Postby wendy » Wed Feb 10, 2016 8:21 am

The definition of curl here in N dimensions is largely reproducable in N dimensions, since it is the basis of determining the directions of the 'out-vector' and hence volume.

In essence, for a closed ring (hollow N-2 sphere-oid), there the vector surface that spans it is a fixed amount, and the direction of the vector depends on the ringlets that span it.
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Re: Curl in ND

Postby quickfur » Thu Feb 11, 2016 12:40 am

Is it possible for a 4D fluid to exhibit vorticity in two orthogonal planes simultaneously? I.e., the constituents of the fluid in the neighbourhood of some given point have a tendency towards motion along the Clifford parallels (aka Hopf fibre bundle)? If so, I'm not sure how useful a bivector representation for vorticity would be?
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Re: Curl in ND

Postby ICN5D » Thu Feb 11, 2016 1:12 am

Whoa, a double vortex ... wonder what the 3D slices of that would look like. And, beyond that, what about the 4d analogue(s) of a toroidal vortex? Could such a thing exist? A swirling ring of fluid shaped like 4d toratope could have any one of the four distinct possible shapes.
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Re: Curl in ND

Postby quickfur » Thu Feb 11, 2016 1:29 am

I don't know if it's possible for there to be toroidal vortices... wouldn't they also occur in 3D, if they were possible?

And, come to think of it, I'm not so sure anymore whether 4D fluids can have double vortices... perhaps 5D fluids can, on their surface, due to downward motion in the 5th direction? That would be interesting to watch, I'm sure! :lol:
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Re: Curl in ND

Postby ICN5D » Fri Feb 12, 2016 6:21 am

A smoke ring is an example of a toroidal vortex. So are those air bubble rings made by dolphins. There's even such a thing as a half-torus vortex, that links two counter-rotating vortices together.
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Re: Curl in ND

Postby quickfur » Fri Feb 12, 2016 6:43 am

Hmm, you are right!! I've a hard time imagining how a Clifford vortex would form, though. Except maybe in 5D. Or... hmm, now that I think of it, wouldn't a Clifford vortex form as a result of fluid contraction around a point? Perhaps 4D dustclouds would form Clifford vortices as they collapse inwardly due to gravity, as leftover angular momentum, and once they heat up enough to combust, they would turn into a star undergoing Clifford rotation?
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Re: Curl in ND

Postby PatrickPowers » Mon Feb 15, 2016 3:20 am

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Last edited by PatrickPowers on Mon Feb 15, 2016 3:42 am, edited 1 time in total.
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Re: Curl in ND

Postby PatrickPowers » Mon Feb 15, 2016 3:22 am

quickfur wrote:Hmm, you are right!! I've a hard time imagining how a Clifford vortex would form, though. Except maybe in 5D. Or... hmm, now that I think of it, wouldn't a Clifford vortex form as a result of fluid contraction around a point? Perhaps 4D dustclouds would form Clifford vortices as they collapse inwardly due to gravity, as leftover angular momentum, and once they heat up enough to combust, they would turn into a star undergoing Clifford rotation?


I'm venturing into the realm of opinion, but I'm sure in 4+1 D that clifford vortices would be very common. Vortices are very common in our 3+1 world, they just go unnoticed. Just wave your arm and you have made vortices in the air. I feel certain it would be the same way in 4+1. The rule of thumb I use is, if a physical system can do something, then it will. If there is a degree of freedom, the system will exercise it if it possibly can.

The Sun is crammed with vortices larger than the Earth.

I'd like to get something more precise, but it is going to have to wait.
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Re: Curl in ND

Postby PatrickPowers » Mon Feb 15, 2016 3:32 am

quickfur wrote:Is it possible for a 4D fluid to exhibit vorticity in two orthogonal planes simultaneously? I.e., the constituents of the fluid in the neighbourhood of some given point have a tendency towards motion along the Clifford parallels (aka Hopf fibre bundle)? If so, I'm not sure how useful a bivector representation for vorticity would be?


Bivectors are natural because rotation is planar. Essentially 2D. Curl isn't rotation exactly, but it is quite similar.

I've ordered MacDonald's geometric calculus book, so I should understand this better in a few weeks. I don't want to have to figure out and prove all of this stuff myself.
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Re: Curl in ND

Postby PatrickPowers » Mon Feb 15, 2016 3:37 am

ICN5D wrote:A smoke ring is an example of a toroidal vortex. So are those air bubble rings made by dolphins. There's even such a thing as a half-torus vortex, that links two counter-rotating vortices together.


Amazing! I've seen many of those vortices, but had no idea they were linked.

Sunspots are pretty much the same thing. They are linked in pairs by a "flux tube." The spots are where the tube crosses the Sun's surface.
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