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I found some useful information in your post, it was awesome to read, thanks for sharing this great content with my vision, keep sharing.<br />Visit: <a href="https://cabinets.activeboard.com/t68231279/why-learning-spoken-is-so-important/?page=last#lastPostAnchor" rel="nofollow">Spoken English Course in Mumbai </a><br /><a href="https://forum.napiprojekt.pl/viewtopic.php?p=65909#65909" rel="nofollow">Spoken English Classes in Pune </a><br /><a href="https://writeonwall.com/question/why-learning-spoken-is-so-important/" rel="nofollow">Spoken English Classes in Delhi </a><br /><a href="https://forum.napiprojekt.pl/viewtopic.php?p=65909#65909" rel="nofollow">Spoken English Classes in Pune </a><br />richamalhotrahttps://www.blogger.com/profile/13107059294571460938noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-31692217796509710052022-01-31T22:23:51.603-08:002022-01-31T22:23:51.603-08:00This is a very nice one and gives in-depth informa...This is a very nice one and gives in-depth information. 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We're looking /back/ on all this, and the perspective is very different from the one Hamilton would have experienced moving forward into it.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-19241378375658317142020-12-21T21:01:05.577-08:002020-12-21T21:01:05.577-08:00Oh, one other random idea I've had, and I'...Oh, one other random idea I've had, and I'm not sure how useful it might be:<br /><br />Quaternions can be (scalar + vector) OR (tensor * versor).<br /><br />(And we DEFINITELY need a replacement word for Hamilton's 'tensor' now that it belongs to tensor calculus... just like Prolog programmers can't use 'functor' anymore because of category theory.)<br /><br />Could there be a quaternion differentiation operator that ran on tensors * versors rather than scalars * vectors? <br /><br />The reason I like the versor form is that it makes clear that we're dealing with pure rotations separated from pure stretching operations. (With the real part of a 4D pure rotation being something like a phase shift or Lorentz boost, adapted perhaps to the speed of sound in a flowing medium?)<br /><br />This reformulation of nabla might not help, but it might give another angle for visualisation.Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-11868687234265959192020-12-21T20:42:53.566-08:002020-12-21T20:42:53.566-08:00Also this
"Hamilton seems to have first dabb...Also this<br /><br />"Hamilton seems to have first dabbled with it several years before he discovered quaternions, as a sort of "square root" of the Laplacian, at which time naturally he only gave it three components; and when he adapted it to a quaternionic form it still had only three components."<br /><br />Nabla *before* quaternions! Back when Hamilton was still stuck on only using three dimensions! That would maybe explain a lot!<br /><br /><br /><br />Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-1445184039949496832020-12-21T20:31:37.929-08:002020-12-21T20:31:37.929-08:00Oh! I did miss your 2019 'full Nabla' post...Oh! I did miss your 2019 'full Nabla' post, and that's given me a lot to think about! Especially on multiplying vs dividing. You seem to be sharing many of the same questions I have. I'm going to have to go back and reread all the posts I missed. <br /><br />Oh, by the way, on signatures:<br /><br />"the norm of a quaternion is the square root of the sum of the squares of its components, √(t2+x2+y2+z2), whereas in Minkowski spacetime the three spatial elements should be negative, √(t2−x2−y2−z2). "<br /><br />Yes, just (square rooting the) sum of the squares of the *components* of the quaternion seems to be the norm as we currently define it.... but why exactly *should* that be the way we define a quaternion norm? Aren't i,j,k imaginaries? Isn't the square of an imaginary a negative number? If instead of √(t2+x2+y2+z2), we had √(t2+ix2+iy2+iz2).... wouldn't that give us in fact the correct Minkowski signature? And mightn't it maybe even be more mathematically 'pure' to define a norm in such a way, as the sum of the four *actual* components of the quaternion, imaginary multipliers and all? <br /><br />Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-46195431780199927962020-12-21T07:29:30.313-08:002020-12-21T07:29:30.313-08:00I find multiple perspectives on these things can b...I find multiple perspectives on these things can be highly valuable, as different minds may come up with very different sorts of patterns. So saying, have you seen my earlier post on <a href="https://fexpr.blogspot.com/2019/04/nabla_18.html" rel="nofollow">Nabla</a>?<br /><br />I'm in partial sympathy, and at the same time partial disagreement, with your suggestion that "intuition doesn't count for much in physics". Certainly one should tread cautiously with intuitions from ordinary experience of the physical world (though one ought, in any case, to be rather selective in when to tread <i>in</i>cautiously); but modern physics has at least one foot in the mathematical world, and intuition counts for quite a lot in mathematics.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-19695269901246774172020-12-20T17:48:01.446-08:002020-12-20T17:48:01.446-08:00My gut feeling though is that while adding (gradie...My gut feeling though is that while adding (gradient + delta-V) might be okay given a suitable definition of 'density', just adding an acceleration vector to a curl vector is still wrong. <br /><br />I think we would want to *vector divide* the curl by the time-or-density-gradient-induced acceleration somehow, to end up with a curl-plus-negative-divergence.<br /><br /><br />Eg, if we've got a flow field like a river - let's say an undersea river, so it's embedded in a 3D flow field - that has a curl pointing 'upwards'... and then we see that whole flow field also accelerating in the upward direction... we wouldn't expect the curl to increase or decrease, as it would if we just added the acceleration vector to the curl vector. We'd expect to see the curl *rotating* along the axis needed to rotate the sideways flow of the river into the upwards flow it's suddenly acquired. At least I think would.<br /><br />And it feels like that rotation would be a multiplication, and I don't know if it would be mathematically consistent with the good old quaternion multiplication equation we know and love.<br /><br />But almost nobody in the world (except Sweetser) seems to be thinking about quaternion nabla, and those who are, aren't thinking in terms of extremely naive and dumb physical analogies like fluid flow-and-density-fields over time (and with good reason, because quaternions are really the algebra of rotations of 4D spheres), so this might just be a big weird dead end.Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-31703528274586058072020-12-20T17:32:28.474-08:002020-12-20T17:32:28.474-08:00I suppose the question I have, given (delta + nabl...I suppose the question I have, given (delta + nabla) , eg as Sweetser uses here: https://dougsweetser.github.io/Q/Math/multiplying/ <br /><br />is, can this be given *any* classical physical interpretation?<br /><br />If we had say a field of fluid flow (ignoring for the moment any associated scalar field - which *might* perhaps represent the density, or something like density), then nabla gives us the curl of it. If that flow then were to change over time, eg, to accelerate, then we might be bringing in the quaternionic elements of nabla - such as delta-V. The naive thing to do, as represented by Sweetser's equation here, would just be to add the delta-V to the curl. Which would represent the axis of rotation being skewed - rotating forward - in the direction of the local acceleration of the fluid flow. Is that a physically sensible interpretation, I wonder? <br /><br />If 'curl skewed in the direction of acceleration' was a sensible thing to imagine, then we could also perhaps imagine the scalar field as a density field, and therefore a variation of density (ie the gradient) as being something a little like an acceleration of the flow in that direction.<br /><br />And then we could perhaps put our components of the full quaternionic nabla together as:<br /><br />* delta s = local density increase over time<br />* -div v = local density decrease caused by flow away from this point<br /><br />* delta v = skew of curl of flow caused by local acceleration of flow<br />* grad s = skew of curl of flow caused by density increase in a direction causing acceleration of flow<br />* curl s = the unskewed curl of the flow assuming no change of parameters in time<br /><br />I don't know if this is correct, but it's a naive interpretation of what just banging the parameters of (delta + nabla) together with a somewhat physically intuitive interpretation of an (s + v) flow field might mean. (phi + A) probably works nothing like that actually.Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-12803721446199833162020-12-20T14:37:14.646-08:002020-12-20T14:37:14.646-08:00Also it's A + Phi, I think, not A + B. And the...Also it's A + Phi, I think, not A + B. And there's a whole big weird-physics whisper-network rabbit hole there - which Barrett touches on - about gauge freedom and the Lorenz Gauge being part of the mechanism by which mainstream science accidentally lopped off part of the full quaternion field.Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-8497773019306837102020-12-20T14:11:12.771-08:002020-12-20T14:11:12.771-08:00Another possibility that crosses my mind, though, ...Another possibility that crosses my mind, though, is:<br /><br />Could there be a 'quaternion nabla' which is NOT (delta + nabla)? Is part of the problem that we've been conceptualising the operator wrong? <br /><br />For example, if we extended curl into four dimensions (ignoring relativity for the moment, and assuming that the fourth dimension is time) then curl might be defined over a difference of four vectors, not three:<br /><br />* the change of vector from 'left'<br />* the change of vector from 'north'<br />* the change of vector from 'above'<br />* the change of vector from 'before'<br /><br />and then we might see that a rapidly changing vector field might gain an extra bonus to its curl. Could this extra component tell us something helpful about highly dynamic vector fields, maybe rapidly expanding or contracting magnetic fields, that we normally ignore?<br /><br />(This would actually be curl AND divergence, because they're linked together, so it would be an operator that took four 3D linear-translation vectors and gave us one 3D axis-of-rotation vector plus one scalar quantity representing divergence. Or 'convergence', negative divergence, as Hamilton's maths actually defines it; Heaviside inverted the sign.)<br /><br />I'm in very deep waters here, though, and I don't have the maths to understand what I'm doing. Physically it feels intuitive, but intuition doesn't count for much in physics.Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-63482881395309232612020-12-20T13:43:02.612-08:002020-12-20T13:43:02.612-08:00This is Doug Sweetser: https://dougsweetser.github...This is Doug Sweetser: https://dougsweetser.github.io/Q/<br /><br />To be clear, both Sweetser and myself have been thinking of 'quaternion nabla' as (delta + nabla). Which comes out (for a quaternion field defined as scalar field s + vector field v) as:<br /><br />delta s + delta v<br />+ nabla s + nabla v<br /><br />==<br /><br />delta s + delta v<br />+ grad s + curl v - div v<br /><br />==<br /><br />scalar = delta s - div v<br />vector = delta v + grad s + curl v<br /><br />From the standpoint of modern physics, and modern vector/geometric algebra, this equation feels extremely wrong, because its vector part is a sum of both vector quantities (gradient and delta-v) and pseudovectors (curl) which are considered two entirely separate mathematical objects.<br /><br />But this seems to be what the underlying quaternion maths just does.... and quaternions are where we got our idea of 'vector cross multipication' as well as 'div, grad and curl' from. And quaternions are just pure mathematical objects, and they're also a very nice, simple, closed algebra, and closure feels like it ought to be an important thing. Most physicists assume that it's just an accident that you an cross-multiply two vectors and get a pseudovector; it's a kind of trick or pun, that only works in three dimensions, not really a real thing, and anyway a vector (v) just isn't a pseudovector (curl v). One's a linear translation and the other's an axis of rotation; physically, it does seem obvious that they're very different. So what's going on here? How could a very simple, fundamental equation be 'doing maths wrong' by mixing two different kinds of quantities?<br /><br />I assume it's because quaternions are about rotation, not translation, in which case the question still is: why are chunks of quaternion nabla so physically useful, but the whole quaternion nabla isn't?<br /><br />Nate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-8578971538330181662020-12-20T13:26:22.973-08:002020-12-20T13:26:22.973-08:00This is really fascinating because I've had an...This is really fascinating because I've had an interest in off-mainstream physics for a long time - in fact since encountering Tom Bearden's writings in the 1980s, where he began ranting extensively about quaternions and the idea that the scalar field was dropped by Heaviside et al. I've been trying to understand that claim ever since.<br /><br />On quaternions particularly, I do feel there's something very curious missing there. I'm particularly fascinated by the origins of the vector 'del' operator (grad, curl, div) as Hamilton's 'nabla' - which I think Hamilton himself only defined on the vector part of a quaternion field - and wondering if there could ever be a fully integrated 'quaternion nabla', operating on a coupled vector and scalar field. One might think that the canonical example of such a field might be the electromagnetic scalar+vector potentials (A+B). But I struggle to understand what it would mean, physically, to add a curl to a gradient (or both to a delta-V). There aren't many writers on this subject at all. A modern writer who you might want to check out if you haven't is Doug Sweetser.<br /><br />Another, more mainstream, scientific writer who also touches on the quaternion mystery is Terence Barrett, and his 'Topological Foundations of Electromagnetism' (2008), who repeats again this legend of the quaternion field being cut down to vectors, perhaps missing the very important contribution of the scalar field and what it could mean, and advocates for a 'SU(2) Electromagnetism', by which he means an electromagnetism with full quaternion symmetry, exhibiting beyond-Maxwell yet non-quantum physical effects. (And names several known effects that he thinks fall into that category already).<br /><br />https://books.google.co.nz/books?id=e0-QdLqT-pICNate Cullhttps://www.blogger.com/profile/08219250912232851452noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-29029668626102866812020-12-08T13:55:31.128-08:002020-12-08T13:55:31.128-08:00I too am rather fond of the pairwise thing. The f...I too am rather fond of the pairwise thing. The first speaker about Weber at the Prague conference, iirc, was quite taken with it too. As you mention time, though, that seems to me to be the biggest hurdle the pairwise approach needs to clear. Forces-at-a-distance between particles aren't instantaneous; and even if they were, relativity would introduce disagreements between observers about simultaneity. So how to handle propagation time of a Weber-style force, and how to integrate it with relativity, are imho quite fascinating topics for speculation.John Shutthttps://www.blogger.com/profile/00041398073010099077noreply@blogger.comtag:blogger.com,1999:blog-7068528325708136131.post-46199193713282489382020-12-08T10:19:53.219-08:002020-12-08T10:19:53.219-08:00I am fond of the idea of pairwise interactions bet...I am fond of the idea of pairwise interactions between particles instead of fields because it is nicely symmetrical in time. However I know nothing about quantum mechanics and so I would need to study in order to be anything more than fond.<br /><br />Also I wanted to understand how magnitism could be explained as moving charges plus relativity. It turns out in every popular science mention of relativity I'd encountered, I'd completely skipped over "now" being a gradient. patchworkZombiehttps://www.blogger.com/profile/18279408952877283952noreply@blogger.com