For types I say "dimensions" as in "Dimensional Analysis". A major teaching frustration is that most students can barely handle units, much less dimensions. Then they ask: "What is Electric Flux?", and can't understand the answer that it is a measurable quantity of dimension [F][L]^2 / [Q]. I would like to write out some of my thoughts about how dimensional analysis is a lot like projective geometry, but for now, do not have the time. ( And did Max Planck think about this?) As for co-vectors, vectors, linear functionals etc., If the elements are complex numbers--as they are in quantum-- there will be type errors on mismatched addition, because norming requires complex conjugation. --Brad On Wed, Jan 29, 2020 at 10:11 AM Allan Wechsler <acwacw@gmail.com> wrote:
The flavor of the distinction is very similar to unit analysis from physics, which tags some scalars with "types" which behave coherently under arithmetic. In principle, you are never allowed to add two scalars of differing types.
Presumably, whenever you see a dot product, there is a vector on the right and a covector on the left, and it is a type error to add a vector and a covector. Do these types really behave consistently, or are there exceptions to the strict classification?
On Wed, Jan 29, 2020 at 10:47 AM Tom Knight <tk@mit.edu> wrote:
A huge disappointment from my MIT math and physics education was the lack of explanation early on of the difference between vectors and co-vectors. It wasn’t until I started reading about differential geometry that I realized the pervasiveness of this education-driven ignorance. All of a sudden lots of things that never made sense started clicking. How and why do we allow this fuzziness to continue?
On Jan 28, 2020, at 11:16 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Good luck with convincing the engineers involved! Mind you, it is only getting on for 180 years ago that Hamilton came up with quaternions. So early days ... er, centuries?
But perhaps it might be more constructive to attempt to understand why there is such instinctive resistance among engineers to the whole notion of quaternions, never mind more general geometric (Clifford) algebra.
There's a frustrating pedagogical phenomenon involved in such investigations. While I can well recall my own sense of bewilderment on first encountering Hestenes' early book on these matters, I cannot muster the slightest insight into the cause of those conceptual difficulties. As a result, I now am stranded as far away from offering assistance to the uninitiated as I earlier was from receiving any: I simply cannot understand _why_ they can't understand.
Anyway, here's a couple of possible clues: perhaps others can come up with more suggestions.
(A) It's noteworthy that the first thing Heaviside did was to dissect Hamiton's elegant unity into "scalar" & "vector" parts, which went on to gain pretty much universal acceptance. There seems to be a mental hurdle in human minds obstructing the union of disparate familiar categories under a common umbrella: in this case, familiarity with angles & Cartesian coordinates actively obstructs conceptualisation of quaternions.
(B) It is rarely made explicit that (like vectors) quaternions come in two flavours: polar & axial, depending on application. It seems that Hamilton himself was confused over this, which contributed to early controversy about their validity. There's a informative but slightly muddled paper on this topic somewhere on the internet which proceeds from the quaint premiss that they must exclusively be one or the other, despite the arguments put forward clearly illustrating a dichotomy.
WFL
On 1/28/20, Henry Baker <hbaker1@pipeline.com> wrote:
Perhaps it's time to learn quaternions??
From comp.risks:
Date: Fri, 10 Jan 2020 20:24:07 +0000 From: "Clive D.W. Feather" <clive@davros.org> Subject: Boeing 737s can't land facing west (FAA)
"The FAA received reports earlier this year of three incidents of display electronic unit (DEU) software errors on Model 737 NG airplanes flying into runway PABR in Barrow, Alaska. All six display units (DUs) blanked with a selected instrument approach to a runway with a 270-degree true heading, and all six DUs stayed blank until a different runway was selected. [...] The investigation revealed that the problem occurs when this combination of software is installed and a susceptible runway with a 270-degree true heading is selected for instrument approach. Not all runways with a 270-degree true heading are susceptible; only seven runways worldwide, as identified in this AD, have latitude and longitude values that cause the blanking behavior."
(Note that this is all 6 displays on each plane, not 2 displays on each of three planes.)
The runways in question are:
Runway 26, Pine Bluffs, Wyoming, USA (82V) Runway 28, Wayne County, Ohio, USA (KBJJ) Runway 28, Chippewa County, Michigan, USA (KCIU) Runway 26, Cavern City, New Mexico, USA (KCNM) Runway 25, Barrow, Alaska, USA (PABR) Runway 28, La Mina, La Guajira, Colombia (SKLM) Runway 29, Cheddi Jagan, Georgetown, Guyana (SYCJ)
(The numbers are magnetic bearings, whereas the problem is apparently related to true bearing.)
Original FAA notice: < http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgad.nsf/0/3948342a978cc2...
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