[HN Gopher] What is the inverse of a vector?
___________________________________________________________________
What is the inverse of a vector?
Author : mkeeter
Score : 295 points
Date : 2021-11-02 12:43 UTC (1 days ago)
(HTM) web link (mattferraro.dev)
(TXT) w3m dump (mattferraro.dev)
| Garlef wrote:
| While the article is written very nicely, It seems that this is
| written out of a perspective of some missing knowledge.
|
| The basic object that the author seems to be interested in is
| that of an "algebra over a field"
| (https://en.wikipedia.org/wiki/Algebra_over_a_field).
|
| Specifically: Invertability of all elements with respect to the
| multiplication leads to the notion of division algebra and these
| have been studied for a long time.
| (https://en.wikipedia.org/wiki/Division_algebra)
|
| When studying math at german universities, alebras are something
| you'll encounter in your 2nd year (latest; but might already show
| up in 1st year analysis albeit with a different focus).
| Implicitly, division algebras show up a lot when students learn
| about field extensions, galois theory and the algebraic closure
| of a field (usually 3rd semester). A more general treatment of
| division algebras is not a common subject, though.
| oikawa_tooru_ wrote:
| Damn. They don't teach these stuff here atleast not in a
| computer science curriculum. What degree did you learn? Iss
| this generally taught in all German engineering courses?
| mam4 wrote:
| German and french engineering school are pretty rough on math
| theory, for the better or the worse. Mostly because a lot of
| theory was born in these two countries.
| [deleted]
| aneutron wrote:
| Can confirm, before I was anywhere close to serious
| computer science, I basically did an undergraduate degree
| in mathematics / physics.
|
| And it goes through some very advanced subjects in both. It
| basically prepares you to be an engineer in whatever field
| you choose, be it a structural engineer, or a computer
| scientist.
| [deleted]
| Garlef wrote:
| > What degree did you learn?
|
| Mathematics (I started a PhD but left academia in favour of
| starting a company). My focus was on category theory /
| algebra.
| conradludgate wrote:
| Speaking from personal experience, not all of us on HN have
| computer science degrees.
|
| I studied mathematics in the UK and can confirm I learnt a
| lot about fields and other forms of linear and abstract
| algebras
| ReleaseCandidat wrote:
| He is talking about math studies. Not much maths in
| engineering/CS in germany either.
|
| But nobody is holding you back from visiting 'real' math
| lectures ;)
| qayxc wrote:
| > Not much maths in engineering/CS in germany either.
|
| That's simply not true. It depends entirely on the
| particular institution and its roots.
|
| There are two origins of CS in German universities:
| electrical engineering and maths. At universities where CS
| originated as a subfield of maths, undergrad CS education
| is very similar to a maths undergrad to the point that most
| of the tests/mid-terms are basically identical between CS
| and maths.
|
| If on the other hand CS came from the electrical
| engineering department, the focus is significantly less on
| maths and the lectures are very different indeed.
|
| So you'd have to look at the history of each university and
| where the CS department originated to find out.
| ReleaseCandidat wrote:
| > There are two origins of CS in German universities:
| electrical engineering and maths.
|
| Ah, I see, thanks for the information.
|
| Which university would be an example for the latter?
| LeanderK wrote:
| Karlsruhe Institute for Technology is an example.
| Analysis 1+2 and Linear Algebra 1+2 is compulsory for
| every CS student (the exams are practically identical,
| just missing a single topic. You can always switch to the
| math lectures and get a few ECTS more). This is a great
| background for more advanced, theoretical lectures
| (CS+Math) but it's lacking electrical engineering
| lectures and especially there's no electrical engineering
| practice.
| pvitz wrote:
| Seriously, which German university (not "FH") doesn't
| teach analysis and linear algebra in the first terms? I
| am really wondering.
| red_trumpet wrote:
| I think also in Germany you have to study pure math to see
| this. Where I study you even can avoid this by never taking
| Algebra 1, which is not obligatory (though imo it should be
| standard. There you learn why polynomials of degree >=5 have
| no closed form solutions)
| matt-noonan wrote:
| No, this is wrong. Geometric algebras aren't division algebras
| in general: they usually have zero divisors. Objects that live
| in a single grade are invertible, but composite objects don't
| always have multiplicative inverses.
|
| As a concrete example, consider the elements 1 + x and 1 - x.
| Their product is 1 + x - x - xx = 1 + x - x - 1 = 0. So
| certainly 1 + x doesn't have an inverse, either.
| yodon wrote:
| Presumably you also meant to say "when X^2 = 1"
| matt-noonan wrote:
| Yes, I meant that equation to be interpreted in the GA used
| in the article. But essentially all geometric algebras also
| have zero divisors, for similar reasons.
| Sharlin wrote:
| It's very intentionally written from the perspective of someone
| who knows linear algebra but maybe not that much, if any,
| abstract algebra, because that's what the target audience is.
| Not university math students or graduates, but highschool
| graduates, CS/physics/engineering students and graduates,
| game/graphics programmers and so on.
| timerol wrote:
| Yeah. It's written to teach people who do not have (what some
| might consider) the standard prerequisite knowledge. I have
| no doubt the author understands algebra over a field. The
| author recommends teaching Geometric Algebra as a pedagogy at
| the end
| ogogmad wrote:
| Important correction: A division algebra is an algebra in which
| _every_ non-zero element has an inverse. The _dual numbers_ for
| instance are a Geometric Algebra which are _not_ a division
| algebra because there are some non-zero dual numbers which don
| 't have inverses. In fact, almost all Geometric Algebras fail
| to be division algebras.*
|
| So your point about division algebras is not particularly
| relevant to the article.
|
| * - Frobenius's theorem classifies all the finite-dimensional
| associative division algebras. They are: The real numbers, the
| complex numbers, and the quaternions. There are no others.
| dan-robertson wrote:
| The object the author is actually interested in is known as a
| geometric algebra. One often sees it discussed as an
| alternative theory for computer graphics or physics as it works
| well for expressing things like rotations.
|
| See here: https://en.m.wikipedia.org/wiki/Geometric_algebra
|
| I think it is probably not so helpful to merely think of it
| like a division algebra, and it is better to stay focused on
| the geometry. Curiously I find it easier to relate "actual"
| linear algebra to geometry than the thing people often call
| "linear algebra" that involves writing columns or rows or grids
| of numbers and manipulating them.
| orangeyjuicey wrote:
| hey, by the way, to prove you right: please check the URL of
| the blog post
| mebassett wrote:
| I think the author is interested in clifford algebras, or more
| specifically geometric algebras, rather than division algebras.
|
| division algebras tend to be quite boring (if they are finite
| then they are just a finite field; if they are finite
| dimensional over an algebraically closed field then they are
| just the field itself. I guess the quaternions are an
| interested example in the non-algebraically closed case. but I
| think if you are over something other than R you're really just
| talking about a field extension)
|
| clifford algebras are a sort of generalization of the exterior
| algebra one would have encountered in differential geometry and
| other spaces.
|
| in fact it could be considered a "quantization" of the exterior
| algebra. as in "quantum groups". which is an entirely different
| part of maths. but that's not what this article is about.
|
| I think using the language of geometric algebras / clifford
| algebras in physics as this article does versus the more
| traditional language is just a matter of taste.
| vanderZwan wrote:
| > _It seems that this is written out of a perspective of some
| missing knowledge._
|
| Well, the author talks about what "we learned in _school_ ",
| not _university_ , so that checks out but only because you two
| have different audiences in mind.
| azalemeth wrote:
| Indeed. I learnt about vector dot and cross products, basic
| linear algebra (including diagonalisation, simple Markov
| chains and similar), partial differentiation, grad div and
| curl, the volume of a parallel piped and "all that Jazz" in
| high school, as a 17-18 year old. I learnt about the
| divergence theorem, Stokes's theorem, multivariate
| integration, integrating factors and higher order ODEs and
| simple PDEs, Fourier transforms and other integral
| transformations and similar in the first year of university
| (studying Physics).
|
| This is not uncommon in the UK - but it is also not _common_
| either, and depends on exactly what A-level modules you did.
| My understanding is that it 's quite rare to do exactly this
| in high school in the united states - but there, I think
| limits are taught much more heavily. I think having a clear,
| short statement of having "assumed knowledge" somewhere
| probably helps avoid these issues.
|
| (I thought the article was excellent, and beautifully
| illustrated!)
| spacedcowboy wrote:
| I'll second this - my experience is pretty much the same
| (though I did some basic Fourier transforms at school),
| studying Physics at college. I did 3 Maths A levels along
| with Physics and Chemistry (and "General Studies" :)
| however, so I too may be atypical.
| [deleted]
| learc83 wrote:
| In the US, it's common to refer to university as school.
| [deleted]
| vanderZwan wrote:
| Really? I was not aware. Do we know if the author is from
| the US though?
| jacobolus wrote:
| The author went to MIT, worked for NASA, and now lives in
| San Francisco https://twitter.com/mferraro89
|
| It is unexceptional (indeed, expected) to get through an
| American undergraduate science or engineering degree
| without ever taking an abstract algebra course (much less
| the 2+ apparently expected of German pure math students).
|
| But in any event, the top post here by Garlef is barking
| up the wrong tree. Division algebras, field extensions,
| and galois theory (per se) are not the tools to use for
| studying arbitrary-dimensional geometry. What you want is
| Clifford algebra (which Clifford himself, and later
| Hestenes, call "geometric algebra") and then geometric
| calculus, which can be used on arbitrary manifolds, in
| non-metrical contexts, etc.
|
| Basic geometric algebra _should_ be taught to advanced
| high school students and all undergraduates studying any
| technical subject.
|
| Math students looking for a math-style introduction to
| geometric algebra should try Chisolm (2012)
| https://arxiv.org/abs/1205.5935
| fouronnes3 wrote:
| As a programmer it seems to me that the number one problem of
| math notation is that it's weakly typed. There's abuse and reuse
| of notation everywhere, which makes learning it needlessly
| difficult. I want a strongly typed fork of math notation. 90% of
| existing math notation would just be laughed at if it had to go
| through code review.
| Grustaf wrote:
| That's a very common criticism, but I don't think mathematics
| would work if you insisted on being 100% explicit all the time.
|
| Clear and short notation, that is just unambiguous enough, is a
| very important factor, without it books wouldn't just be much
| longer, I'm not sure we'd even be able to understand it.
| civilized wrote:
| This was what Bourbaki's Elements and Whitehead/Russell's
| Principia Mathematica were about. These books are admired and
| influential but very few people actually read them. As you
| might expect, they're too long. They're for giving a
| different perspective to people who have already achieved the
| highest levels of sophistication in math.
| yobbo wrote:
| Yes, one usually needs to "intuit" the meaning when there are
| mixes of subscripts/superscripts/parameters, assumed
| definitions, mixes of variable naming conventions, the order of
| function arguments randomly adjusted to taste, invented symbols
| and syntaxes.
|
| Since no machine is ever going to read it, it's all up to
| individual taste and prejudice.
|
| How would a math linter work? Don't see it, but this is one of
| the things that makes maths a different world from programming.
| civilized wrote:
| Ironically, this post is an abuse of the concept of "weak
| typing". There's nothing "weakly typed" about, say, the plus
| sign being used to add both numbers and sets, or a dot being
| used for both multiplication of numbers and the dot product of
| vectors. It just means those symbols dispatch based on the
| types of their arguments, which is perfectly consistent with
| strong typing (cf. the Julia language).
|
| The situation that leads to weak typing in computer algorithms
| -- when you get data from a file or another process and don't
| know in advance what type it's going to be -- is basically non-
| existent in blackboard mathematics. Rigorous mathematical
| papers always tell you what set a variable belongs to when it
| is introduced, as well as the domain and co-domain of any
| functions that are defined. This is the blackboard equivalent
| of strong typing.
| fouronnes3 wrote:
| That's 100% fair, but while I could have made my point using
| more rigorous language, I think it's still valid. I'm not
| talking about multiple dispatch based on type which is fine,
| I'm talking about actual abuse like using fractions to mean
| derivatives, omiting non obvious parameters, confusing
| function and value of function at a point, etc. I could go
| on. Physics notation is an even worse offender btw.
|
| Take for example the law of total expectation, usually
| written as E( E(X|Y) ) = E(X). It's totally non obvious (so
| much that it's harmful IMO) that the outer E is a function of
| Y. Hiding the summation parameter of E does nothing but hurt
| math learners here.
| civilized wrote:
| That's interesting. I'm not quite sure what I would call
| those issues, but I agree they can be tricky.
|
| Many of those abuses are for a very good cause. Leibniz
| notation is very powerful, for example, but it's hard to
| master and physicists really go nuts with it.
|
| For E[X|Y], all you have to remember is that conditional
| expectation yields a function f(Y) of the thing you
| conditioned on, and f(Y) is itself a random variable that
| you can take the expectation of. This property is canonical
| and baked into the formal definitions. It's not an abuse.
|
| However, I do fault some machine learning types a bit for
| abusing probability and statistical notation. For example,
| the Elements of Statistical Learning book extensively
| overloads E[], P(), and other symbols and operators in ways
| that it doesn't even bother to define. They randomly throw
| subscripts and decorations onto all sorts of symbols and
| don't even bother to tell you if those decorations mean
| they're marginalizing, conditioning, or something else. The
| book has no glossary of symbols and operators and no
| preliminary chapter setting out notation, which is unusual
| for such an enormous book full of hundreds of equations. It
| would be impossible because the book is a hodge-podge of
| symbols that change from paragraph to paragraph.
| Jensson wrote:
| That is statistics, not pure maths. But yeah, statisticians
| abuse a lot of notations like that, just like physicists. I
| haven't seen pure mathematicians make such unclear
| notation.
|
| (Statistics is as much maths as theoretical physics is,
| both are technically mathematics but in practice the field
| is handled in a very different manner since they are
| applied and intended to solve a specific set of real world
| problems and hence not pure)
| lmm wrote:
| "Abuse of notation" is usually not weak typing but rather
| polymorphism. One of the reasons typed languages don't catch on
| in physics/mathematics is that most of them can't express the
| level of polymorphism that even basic routine mathematics has.
| E.g. take a look at https://yosefk.com/blog/can-your-static-
| type-system-handle-l...
| foxhop wrote:
| Here is something similar I wrote a long while back as notes for
| my future selfs: https://www.foxhop.net/vector-math-for-video-
| games
| aktenlage wrote:
| In what way would that be similar? The OP performs a
| theoretical derivation of geometrical algebra, you wrote a well
| documented python class with the most basic operations on 2d
| vectors.
| haunter wrote:
| Wow the interactive 3D illustrations are awesome. Works perfectly
| on touchscreen, feels very natural.
| mferraro89 wrote:
| thank you! It took a whole month to get the 3D illustrations
| working well.
| anotheraccount9 wrote:
| "It turns out that inverting a vector on its own isn't well
| defined."
| necovek wrote:
| I love the way this was presented (others have pointed out flaws
| with the article already).
|
| I'd appreciate a post from Matt Ferraro on how this is built.
| Bonus points for including nice syntax-highlighted code "widget"
| for a cross between maths/programming.
| deepsun wrote:
| English/American style of explanation fascinates me.
|
| First, they show some algebra formulas and mention dot product
| and cross product. But then they start introducing a definition
| of a vector! With images!
|
| Why, oh why do you need to waste yours and reader's time to
| introduce basic definitions, if any reader of the article
| definitely knows that? If they haven't, they wouldn't be able to
| read the first paragraph at all.
|
| PS: Russian style of explanation is more like: "Here's the
| essence of my idea, maybe with some leading pre-definitions, but
| definitely without basics. If you are here, you probably is as
| curious as I am to already know/heard of all the basics." In
| total, there's more material, because it's easier to write and
| read it, as author didn't need to explain 101s to PhDs.
| j7ake wrote:
| Do you have an example of "Russian style explanation"?
| vitorsr wrote:
| Consider Kolmogorov's entire Foundations of the Theory of
| Probability is 71 pages long [1].
|
| [1] https://archive.org/details/foundationsofthe00kolm
| jjgreen wrote:
| ... and that he revolutionised the understanding of
| turbulence in [1] which is _four pages_ long.
|
| [1] The local structure of turbulence in incompressible
| viscous fluids at very large Reynolds numbers, Dokl. Akad.
| Nauk. SSSR 30, 299-303. Reprinted in Proc. R. Soc. London
| A434, 9-13 (1991).
| IshKebab wrote:
| I definitely agree but there's no way this is "English/American
| style". It's because people have grand plans of making their
| article/book accessible to everyone, and they start off
| explaining e.g. what a vector is, but pretty soon realise the
| don't want to write an entire vector algebra textbook so they
| seamlessly give up and jump straight into Stoke's theorem or
| whatever.
|
| I read a Synopsys simulator manual that explained what double
| clicking was.
| deepsun wrote:
| Probably. But why would so many people want to make their
| article/book accessible to everyone. Let's accept the fact
| that some topics, like vector algebra, are just not that
| interesting to everyone.
| fsloth wrote:
| Vector algebra is at the heart of a fairly large industry -
| games. I don't think there can be enough of accessible and
| understandable content from that point of view.
| mferraro89 wrote:
| Did this article at some point give up and jump straight into
| something too difficult?
| Grustaf wrote:
| There's an urban legend at Swedish universities that American
| text book authors get paid by the word, and that's why their
| books are so incredibly verbose.
| jabl wrote:
| When I studied physics at the university, our undergraduate
| textbooks where relatively thin volumes (e.g. Alonso&Finn
| I-III), whereas the engineering students had these massive
| textbooks (Young&Freedman etc.). When looking into these
| massive tomes, yes, they spend a lot of words, but also they
| apparently don't expect the reader to be able to apply
| calculus. So instead of showing, say, Coulomb's law, and
| assuming the reader is capable of integrating to calculate
| the interaction between a point charge and a line, they have
| a section describing the interaction between two point
| charges. Then an entirely separate section describing the
| interaction between a point charge and a line, with the
| formula as given without actually explaining that, hey, this
| formula, you know, results if we take the fundamental law and
| do this and that. Incredibly infuriating.
| mferraro89 wrote:
| ha, thank you for your honest feedback. My intended audience is
| not PhDs or math majors, it is high school physics teachers,
| practicing engineers, programmers, precocious high school
| students, etc. Many of these people benefit from some
| definitions.
|
| I include 3 sentences defining a scalar so that I could
| introduce the concept of grade.
|
| I include a few sentences defining a vector because just read
| the comments here and you'll see there are many definitions of
| vector and I want to specifically call out the one I care about
| in this post. I am also using a nonstandard, color-based
| notation throughout the article so it is helpful to take a
| concept that people already know just to demonstrate my
| notation. This also lets me introduce the 3D interactive
| illustrations.
|
| Did you read the rest of the article or were these two
| definitions so objectionable that you quit?
| puzzledobserver wrote:
| (I have only briefly skimmed the article, but ...)
|
| Material such as this serves to remind the reader of what they
| already know, and contextualize it in a way that is relevant to
| this article. The article begins by telling the reader, "We all
| know what scalars and vectors are---here they are---but have
| you wondered what if ...", and taking the reader beyond. The
| introduction which you seem to find objectionable is only a
| small part of a much longer article.
|
| In addition, different readers have different, mildly different
| notation styles. These introductory blobs inform the reader of
| the language in the article, and are essentially a friendly
| statement of definitions.
|
| A third purpose is rhetorical: readers sometimes get stuck
| while reading text, and these parts of the article work as
| anchor points where they can loop back and "synchronize" with
| the writer.
| garmaine wrote:
| When you are writing to a general audience you can't assume
| what the reader knows or does not know.
| deepsun wrote:
| I disagree -- whenever you write anything, you always assume
| something about your audience. In other words, there's no
| such thing as "general audience". E.g. in OP's article,
| people are already interested in Math, otherwise they
| wouldn't click to the article, yet alone got through the
| first paragraph.
| da39a3ee wrote:
| > A scalar is a point on a number line.
|
| This is going to confuse readers. A point on a number line is a
| 1D vector; in other words it is a unit vector pointing along that
| number line, multiplied by something which scales its length.
| It's the latter dimensionless and directionless quantity that's
| the scalar.
| adrian_b wrote:
| Nitpicking: a more correct view is that the set of points on a
| straight line is an affine space, so the points are neither
| scalars nor vectors, but elements of an affine space.
|
| The set of translations of the straight line is a vector space
| a.k.a. a linear space.
|
| So the vectors are the classes of equivalences of the
| differences between 2 points on the straight line (i.e. the
| differences between 2 pairs of points, where the distances are
| the same, are equivalent and they determine the same vector).
|
| While the vectors are classes of equivalence of the differences
| between 2 points, the scalars are classes of equivalence of the
| quotients of 2 (collinear) vectors, i.e. a scalar is the ratio
| between the signed magnitudes of 2 collinear vectors.
|
| If you choose a point on the straight line as the origin, you
| can choose as a representative of each class of equivalence
| that corresponds to a vector, the vector corresponding to the
| origin point together with another point. This gives a
| bijective mapping between vectors and those second points.
|
| If now you also choose a vector as being the unit vector, which
| will correspond with a second point besides the origin point,
| together with the origin point, then you can choose as a
| representative for each class of equivalence corresponding to a
| scalar the ratio between a vector and the unit vector, which
| will correspond to a third point, besides the origin and the
| point corresponding to the unit vector. So you obtain a
| bijective mapping between scalars and those third points.
|
| Because on a straight line there are bijective mappings between
| points, vectors and scalars (after choosing 1 origin point and
| a 2nd point as the extremity of a unit vector), they can be
| used interchangeably in most contexts, but it would be good to
| remember that all 3 are in fact different mathematical
| entities.
| da39a3ee wrote:
| Thanks for that! It was extremely clear.
| SantiagoQ wrote:
| Semantically, the inverse of a vector is something that has no
| magnitude nor direction.
|
| I wonder what would that look like (mathematically), and what
| surfaces or fields would it create?
| Rayhem wrote:
| "A vector is a thing with both magnitude and direction" isn't
| really a good definition. Cars have both -- an SUV is larger
| than a sedan, establishing a magnitude, and they obviously
| point in a direction -- but I don't think anyone would mistake
| them for pointy arrow vectors. If you use the more rigorous
| definition that a vector is an element of a space that obeys
| the vector space axioms it becomes easier to invert a vector
| "semantically" (a thing that doesn't obey the axioms) but quite
| a bit less useful. Cats don't obey the axioms, nor do
| punctuation marks.
| pmarreck wrote:
| Would this potentially enable generalized matrix inversion?
| jefftk wrote:
| _> The similarities are so striking that we might think of them
| as "pseudovpseudovectors". But I won't write them this way
| because I think that obscures their true nature. Written this way
| it looks like a bivector only encapsulates three degrees of
| freedom!_
|
| _> Instead, I will use: ... Because it forces us to remember
| what those coefficients are attached to. Knowing that a bivector
| contains five degrees of freedom, can you figure out what the
| other two describe?_
|
| I'm confused here and don't understand why they keep saying a
| bivector has five degrees of freedom. If you can uniquely
| identify one with three scalar coefficients, doesn't it only have
| three degrees of freedom?
| mferraro89 wrote:
| Author here: I was wrong about the 5 degrees of freedom thing.
| A bivector has 3 degrees of freedom. I'll correct the text
| tonight.
| tgb wrote:
| Yes, only three. As defined, two bivectors are equal if their
| areas are equal and if their oriented planes are equal.
| Therefore two more degrees of freedom are absorbed by taking
| rotations of the two vectors in the plane.Along with the
| rescaling the author noted, we're down to three from six.
| jefftk wrote:
| That makes complete sense to me. But then later on they say
| "The output is a Geometric with a scalar component s and a
| bivector component =c, which has 1 + 5 = 6 degrees of freedom
| so this system is not lossy! It should permit an unambiguous
| inversion operation!" If a bivector only has 3 degrees of
| freedom then the total is 4, which seems like it would be
| lossy?
| tgb wrote:
| I was also wondering this. But note that x^y is always
| perpendicular to x, so really only has two degrees of
| freedom while you need three to recover y (knowing x). Add
| in the dot product part to make up for it.
| kevinwang wrote:
| Interesting, I was confused about the same thing. So the
| author is not correct when they say a bivector has 5 degrees
| of freedom?
| [deleted]
| mferraro89 wrote:
| Yes, a bivector has 3 degrees of freedom. I was mistaken
| when I wrote 5. I'll fix the text tonight.
| debbiedowner wrote:
| Doesn't seem self consistent. He defines the ab multiplication as
| dot product plus "extrusion"/bivector (which seems simpler to
| call convex combinations of 0,a,b,a+b). Then he says aa is a
| scalar, presumably because the "extrusion" is 0, but you can't
| have this identity be 0. Just because it's degenerate does not
| mean it's 0. And the "extrusion" while not a plane is a line in
| his definition.
| portpecos wrote:
| I'm taking a first semester physics course right now, and we're
| learning about Torque and Angular Momentum. I just finished a
| calculus course last semester.
|
| Can someone tell me how I would use t=r[?]F on a physics problem
| for Torque?
| ajkjk wrote:
| You wouldn't really. It's the same concept as `t = r x F`. the
| only difference is that it is useful to think of the 'type' of
| the output as being a bivector instead of a vector -- there's
| no sense in which it points 'out of the plane'; rather, it is a
| single vector in the vector space of (planes), with the same
| magnitude as r x F.
|
| The distinct gets a little more useful when you start dealing
| with covariance under coordinate transformations. There it
| becomes more meaningful, because the _vector_ given by r x F
| doesn't transform the same way as their cross product should.
|
| For an obvious example of why this is true: suppose r=x and
| F=y. Then r x F = z. If you change coordinates by mapping z ->
| 2z, then you would be doubling the torque that you computed ..
| which is wrong; the torque is unchanged. The bivector x^y is
| correctly unchanged by z -> 2z.
|
| Currently in physics courses (usually not until more advanced
| mechanics or relativity) the resolution to this is to wave
| ones' hands and declare that, no, torque is a 'pseudovector'.
| But it is really much easier to think about if you type it as a
| bivector in the first place.
| codeflo wrote:
| The writing is cute and the animations are nice, but none of it
| makes any sense. I stopped reading at
|
| > It is important to remember that bivectors have a certain
| redundancy built into them in the sense that s a [?] b = a [?]
| s b s a [?] b = a [?]s b . We can write them using 6 numbers or
| 3 numbers, but they actually convey 5 degrees of freedom.
|
| Three (real) numbers have three degrees of freedom, by
| definition. (And nothing about complex numbers was mentioned.) Is
| this a parody I don't get? I feel like I have wasted ten minutes
| on nonsense.
| mferraro89 wrote:
| author here. I was mistaken about the 5 degrees of freedom bit.
| Bivectors have three. I'll fix the text tonight. I'm sorry you
| wasted ten minutes on my nonsense.
| Grustaf wrote:
| He's not talking about the triplet or sextet, he means that a
| _bivector_ has 5 degrees of freedom.
|
| That isn't correct either though, the basis consists of three
| unit bivectors, so they have at most 3 degrees of freedom.
| tgb wrote:
| You're right about that being wrong, and the author makes the
| same mistake consistently, but otherwise it looks correct. Some
| steps have details elided where it maybe should have been noted
| that things were being skipped, but with correct results. I
| think it's wonderfully written and a great exposition.
| mferraro89 wrote:
| hey, if you have time to detail those mistakes I'd be happy
| to fix them in the text. Can you email me at
| mattferraro.dev@gmail.com
| codeflo wrote:
| Thanks, that helps. When I notice errors in the stuff I
| already know about, it find it hard to trust the other
| information that's new to me.
| foobarbazbarfoo wrote:
| love your writing, can anyone recommend blogs like these
| shoto_io wrote:
| Curious to understand: What do you exactly love about it?
| mferraro89 wrote:
| thanks! the 3blue1brown website has some great written content
| that you may enjoy: https://www.3blue1brown.com/#lessons
| admin786 wrote:
| THanks you ya
| amatic wrote:
| I would say the title should be "reciprocal" of a vector. Right?
| The inverse of a function (f^-1) has an unfortunate notation
| equality with the reciprocal (x^-1), or multiplicatory inverse.
| Or am I wrong?
| felipeqq2 wrote:
| A rotcev
| wisienkas wrote:
| My thought exactly
| fractal618 wrote:
| The inverse of a any positive or negative vector with an
| amplitude greater than zero points directly into your soul
| relative to it's original amplitude and the how many regrets you
| have.
| gpderetta wrote:
| Great article. I had a similar kind of revelation when I learned
| about generalized linear models after failing to understand all
| the various statistical tests.
| billfruit wrote:
| Is there a book that give comprehensive treatment of euclidean
| geometry, but using vectors?
| pizza wrote:
| David Hestenes's Geometric Algebra for Physicists, maybe?
| ur-whale wrote:
| Geometric algebra (Clifford Algebra) unfortunately came late
| historically.
|
| It's a shame, because the whole theory is a very useful (eg for
| engineering / applied math) superset of linear algebra.
|
| I really wish I had learned this first in my undergrad years,
| would have made a whole bunch of things way clearer from the get
| go: differential forms tensor
| calculus linear algebra etc
|
| From zero to geo is a very good video introduction to the topic:
|
| https://www.youtube.com/watch?v=2hBWCCAiCzQ&list=PLVuwZXwFua...
| an1sotropy wrote:
| Yes, this. Though Zero to Geo is one of the links at the bottom
| of the article.
|
| It is really a shame that article does not clarify that, btw,
| what we've just derived is a re-derivation of a thing that has
| already been expressed and named, by Clifford, and well-
| characterized: https://en.wikipedia.org/wiki/Geometric_algebra
|
| Such a bummer to see very slick but very ahistorical articles.
| mferraro89 wrote:
| Hi. I wrote whole section on the history of GA and what
| happened and why it isn't already the norm, but I chose to
| remove it because the article is already far too long, and I
| don't think that my intended audience (engineers, compsci
| people, university undergrads) would care about the history.
| Apologies that wasn't what you would have preferred.
| adrian_b wrote:
| Its rediscovery was very late.
|
| Its initial discovery was not too late, but both Clifford and
| Maxwell died too young in 1879 and after their too early death
| there was no one left who could finalize the applications of
| this theory to physics.
|
| In their absence, the geometric algebra theory was ignored and
| both the theory of vectors and the theory of electromagnetic
| field were simplified to forms which are good enough for
| restricted contexts, but which are nonetheless inconsistent and
| fail in more general cases (the so-called Maxwell differential
| equations are valid only in much more restricted conditions
| than the original integral equations of Maxwell).
|
| As a child I have also learned the theory of vectors in the
| incorrect way, e.g. including the so-called "vectorial
| product", so I lost time later until understanding that it is
| not a vector, and then I lost more time until understanding
| that the so-called pseudovectors a.k.a. axial vectors and the
| so-called pseudoscalars are not independent entities that come
| from nowhere, but their existence is just the natural
| consequence of the properties of the vectors.
|
| It would have been much more efficient if the complete theory
| of geometric algebra would have been taught from the beginning.
| cannabis_sam wrote:
| I would have loved to implement this in Haskell as an exercise in
| uni!
| jeffwass wrote:
| I stopped reading at this paragraph near the top :
|
| "In this post we will re-invent a form of math that is far
| superior to the one you learned in school. The ideas herein are
| nothing short of revolutionary."
| mferraro89 wrote:
| why? too clickbaity?
| admin786 wrote:
| Nice Article !
| OscarCunningham wrote:
| People might be interested in a similar post I wrote about
| dividing by a vector (https://oscarcunningham.com/4/dividing-by-
| a-vector/) although I came to a different answer as I was
| considering arbitrary vector spaces rather than just 3D space.
| sgt101 wrote:
| I am curious, how could one apply these insights to inference
| over sets of vectors generated by creating embeddings over things
| like photos etc? I understand well the ideas of +/- for things
| like word2vec, but what would multiplication and inverse mean in
| this context?
| unixhero wrote:
| Somehow I got that first math explanation.
| Grustaf wrote:
| Ah, another geometric algebra evangelist? I can't figure out if
| GA actually adds anything substantial, or if it merely lets us
| write some equations in a more succinct fashion. But it certainly
| looks cool.
|
| As to vectors, obviously they have inverses, additive inverses.
| Since vectors don't have multiplication, there is no
| multiplicative inverse, but if you define new operations on them,
| well then that operation can have an inverse but that is not
| really "the inverse of a vector" anymore.
| BlackFly wrote:
| If you are working in ordinary Euclidean space with orthonormal
| bases, it doesn't do much for you. When you start doing
| calculus on embedded surfaces (the beginning of differential
| manifolds) it begins to be more helpful. Since Lie algebras are
| special types of differential manifolds, you can learn quite a
| bit by studying the geometry and this quickly leads into gauge
| theory and modern physics. The Geometry of Physics: An
| Introduction, by Theodore Frankel does a good job illustrating
| a lot of aspects of geometry even offering some geometric
| insight into some classical physics.
|
| It is an absolute necessity for general relativity.
| Grustaf wrote:
| That books seems very interesting, but is it really about
| Geometric Algebra? I'm not talking about geometry in general,
| or differential geometry, which I'm a bit familiar with. And
| not Algebraic Geometry either, for that matter, which is also
| a fascinating subject.
|
| I mean specifically Geometric Algebra.
|
| It sort of seems like a notation, but it has almost a cult
| like following and perhaps it's more than a notation, is it a
| theory, a branch of mathematics?
| yuppiemephisto wrote:
| It really is useful even at the advanced level. Following
| it far enough leads to the Atiyah-Singer Index Theorem and
| Hodge Theory. The advantage over the exterior algebra is
| that you have that and an interior algebra, which leads to
| many formulas in differential geometry becoming very
| natural (like Cartan's magic formula).
| musgravepeter wrote:
| There is Geometric Algebra for Physicists by Doran and
| Lasenby (2003). It recasts mechanics, E&M up to gauge
| theories and GR into geometric algebra. I stalled out at
| mechanics but I've now taken it off the Tsundoku pile and
| may give it another chance.
| Grustaf wrote:
| Thanks for the tip!
| contravariant wrote:
| Is there some canonical metric on Lie algebras that makes
| this possible? Otherwise geometric algebra won't get you far.
| enkimute wrote:
| Every Lie algebra is a bivector algebra (see 'Lie groups as
| Spin groups' http://geocalc.clas.asu.edu/pdf/LGasSG.pdf).
|
| Additionally the GA formalism enables closed form solutions
| for the exponential map for all bivector algebras. (see
| 'Graded symmetry Groups' https://www.researchgate.net/publi
| cation/353116859_Graded_Sy...).
| nocturnial wrote:
| > Ah, another geometric algebra evangelist?
|
| Heh, this made me chuckle and this is reason why I read HN
| comments. I can't help but to picture someone ringing your
| doorbell early on an sunday and saying: "Hi! Have you
| considered inverting vectors?" _slams door_ "Who was it?" "It
| was just another of those damned geometric algebraists"
| Grustaf wrote:
| Haha that reminds me, got to put back that "No beggars or
| geometric albraists" sign!
| LotusFunctor wrote:
| For me, the substantial thing geometric algebra gave me so far
| was a newfound appreciation of the seemingly disparate systems:
| tensors, differential forms, matrix algebra, and also a
| newfound appreciation of stuff like determinants, conjugate
| elements in group theory, lie groups and lie algebras, etc.,
| because it helps clarify the relationship between them, and as
| another user here said, you can get propelled up into some
| pretty advanced stuff later on (said user mentions the Atiyah-
| Singer Index Theorem and Hodge theory, but caveat: I've only
| recently started tacking a crack at the latter. I will say
| that, OTOH, it's pretty nice to be able to see something like
| the wiki on Clifford Analysis and realize its familiar
| territory from geometric calculus).
| thaumasiotes wrote:
| > As vectors, obviously they have inverses, additive inverses.
| Since vectors don't have multiplication, there is no
| multiplicative inverse
|
| A vector is pretty much by definition also a matrix, and there
| is a standard way to multiply matrices. You can define several
| inverses of a vector that way, though you can't define a unique
| inverse.
|
| The standard inner product is of course also an exceptionally
| typical way to multiply vectors, but the concept of an inverse
| there doesn't make much sense.
| Grustaf wrote:
| No, a vector is defined as an object that has certain
| properties, like addition and scalar multiplication. It's a
| very general, and abstract concept.
|
| There are vector spaces of functions, with infinite
| dimension, but there are also vector spaces with a finite
| number of elements.
|
| So only some vectors can even be written as 1xN matrices, if
| that is what you're referring to. But even if you write a
| vector that way, it doesn't mean it IS a matrix or that it
| automatically "has" multiplication.
|
| In mathematics, an object only has an operation if it's part
| of the definition, and as such, vectors don't "have"
| multiplication.
| adrian_b wrote:
| What you say is mostly correct, but only for a certain
| meaning of the word "vector", which has been used with 2
| distinct meanings since its introduction in the first half
| of the 19th century.
|
| The set of elements defined by certain properties of their
| addition and of their multiplication with the elements
| belonging to a set of scalars is named "vector space" by
| some and "linear space" by others.
|
| According to the etymology of the word vector, "linear
| space" would be more appropriate. You have used "vector"
| with the meaning "element of a linear space", and what you
| have said is correct, except that for any "vector" as an
| element of a linear space, considered as a column vector,
| there exists a corresponding row vector, even in the
| infinite-dimensional case.
|
| "Vector" means translation of the space, and this is what
| "vector" meant when the word was introduced by Hamilton.
| While the set of translations is a linear space a.k.a. a
| vector space in the generalized sense, the set of
| translations, i.e. vectors in the strict sense, has
| additional properties due to the multiplication operations
| that must be defined for "vectors" in their strict sense
| (which are needed e.g. to determine the angles between
| translations and the distances).
|
| "Vectors" as elements of linear spaces are a very general
| notion, which appears in many domains, and for all linear
| spaces, including for those infinite-dimensional, you can
| define matrices, i.e. linear functions, and matrix
| multiplication, i.e. composition of linear functions, and
| also the correspondence between a 1xN vector and a Nx1
| vector, more correctly between a vector and an associated
| linear form. The latter also exists for the infinite-
| dimensional case, even if it is less likely to use names
| like row vectors and column vectors (though the names bra
| vectors and ket vectors are still in use for the infinite-
| dimensional case).
|
| For the infinite-dimensional case the vectors and the
| matrices become functions of 1 or of 2 parameters and the
| sums from the formulas of matrix multiplication become
| integrals.
|
| While for most computer applications, "vectors" refer just
| to elements of linear spaces, most "vectors" used in models
| of physical systems are vectors in the original sense of
| the word, where not only the vector addition and the
| product with scalars matter, but the products of vectors
| also have an essential role and their meaning can be best
| understood in the context of the complete geometric algebra
| theory.
| ReleaseCandidat wrote:
| > A vector is pretty much by definition also a matrix, and
| there is a standard way to multiply matrices.
|
| A standard way to multiply a MxN with a NxK matrix, but none
| for a 1xN with a 1xN or a Nx1 with a Nx1 matrix - the two
| possible ways to describe a vector. You have to transpose
| exactly one of the two vectors. And then you have two
| possible results, 1xN multiplied with Nx1 yields a scalar
| (that's actually the 'usual' dot-product/scalar
| product/whatever you call it) and Nx1 multiplied with 1xN,
| where the result is a NxN matrix.
| thaumasiotes wrote:
| Doesn't matter. We already didn't have a unique inverse,
| but it's perfectly possible to find a left pseudoinverse
| and a right pseudoinverse, bearing in mind that they're not
| unique.
|
| Though thinking about it more, it seems like the outer-
| product-inverse of a vector (a) must be unique if it
| exists; and (b) is highly unlikely to exist.
|
| > 1xN multiplied with Nx1 yields a scalar (that's actually
| the 'usual' dot-product/scalar product/whatever you call
| it)
|
| I'm aware of this, but there are two ways we might conceive
| of an "inverse":
|
| - Since a vector is a matrix, the inverse of a vector might
| be defined by matrix multiplication, where A is the inverse
| of B if AB is "the" identity matrix. This is only strictly
| defined for square matrices, but the pseudoinverse concept
| extends it to nonsquare matrices.
|
| - Or, we could go for a more basic sense of "multiplicative
| inverse", where the concept is that if AB = C, then B =
| A-1C. This is what I was thinking of when saying that the
| concept of an inverse doesn't make sense when
| multiplication is the inner product - if I give you a
| vector _v_ , and its inner product with some other vector
| _u_ , there is no way of recovering what _u_ was.
| monktastic1 wrote:
| > The standard inner product is of course also an
| exceptionally typical way to multiply vectors, but the
| concept of an inverse there doesn't make much sense.
|
| Not all vector spaces are equipped with an inner product. The
| point is that you can start with some simple axioms and build
| these more complicated things (inner product spaces, algebras
| over a field, geometric algebras, etc.).
| rsj_hn wrote:
| It's just some of the more advanced theory you'd get from
| studying modules repackaged a bit.
|
| Basically the extra stuff that is usually skipped in first year
| linear algebra courses are the symmetric and asymmetric (often
| called exterior) products. These form algebras, of course. The
| exterior product, or wedge product, has a natural
| interpretation in terms of signed areas (or volumes) and from
| this you get the determinant as a volume form.
|
| These are the natural generalizations of dot products (inner
| products) and wedge products (exterior products).
|
| You can take a vector and associate it with a 1 form
| (asymmetric algebra or exterior algebra), and then multiply two
| vectors to get a 2 form using the standard wedge product, etc.
| In dimension 3, the space of 2 forms is dual to the space of 1
| forms and so you can "multiply" two vectors to get a third
| vector. That is all that's going on here.
|
| Actually a good multi-variable calculus class will cover most
| of this stuff as you need some motivation for Jacobian volume
| forms used to calculate areas and volumes under change of
| basis, and dot/wedge products are useful for generalizations of
| the Gauss divergence theorem and the generalized fundamental
| theorem that says the integral over a function, f, on the n-1
| dimensional boundary of a shape is the differential of the
| integral of the shape.
|
| Moreover any class on Riemannian geometry will give you all the
| linear algebra you need as well.
|
| One thing I would caution students with is that by using
| somewhat non-standard jargon they may not understand how to
| generalize this stuff to n-dimensions, nor will the connections
| between, say, determinants and wedge-forms be clear, or dot
| products and angles be fully understood if only the n=3 cases
| is emphasized. Only in n=3 can you multiply two vectors to get
| a vector. But fun fact: in dimension 3k you can multiply two
| k-forms to get a third k-form (as the space of 2k forms is dual
| to the space of k forms in n=3k). If you think there is this
| new thing called "geometric algebra" other than usual tensor
| products, it may not be obvious how things generalize to n !=
| 3.
| alisonkisk wrote:
| n=3 concern applies ewually well to every formulation, as
| it's a peculiarity if human scale physics that's commonly
| used as an application of linear algebra and anaylsis
| (visualizing linear transformations and rotations, point-
| line-plane geometry in 3D space, curl, Maxwell's equations)
| Grustaf wrote:
| > But no, there is nothing new here beyond marketing.
|
| Yes, that's my sense too. Of course cross products, wedge
| products etc make sense and that's just standard mathematics,
| but the part that I haven't really seen the point of is to
| form the algebra where all these forms live side by side.
|
| It doesn't seem like a useful "fusing", in the way that say
| the complex plane is.
|
| Of course it's very cool that sub-algebras in 2 or 3-space in
| GA are isomorphic to the complex plane or even quaternions,
| but it still feels a bit made up.
|
| For a concrete example, one youtuber showed how Maxwells
| equations simplified to a single equation if you introduce an
| operator that is a combination of div and curl, and also a
| new kind of physical entity that combines the electrical and
| magnetic fields.
|
| This is of course cool, but what I want to know is if this
| new operator makes some physical sense, and if the new multi
| dimensional field has any physical meaning. If they don't, it
| just seems like a parlour trick.
|
| Not saying they actually don't, but I haven't seen any deeper
| explanations of it.
| rsj_hn wrote:
| > For a concrete example, one youtuber showed how Maxwells
| equations simplified to a single equation if you introduce
| an operator that is a combination of div and curl, and also
| a new kind of physical entity that combines the electrical
| and magnetic fields.
|
| Back when I was in university, we covered this in our
| differential geometry class. And yes, you'd use more
| abstract concepts like curvature, hodge dual, and exterior
| product.
|
| Maxwells equations in any dimension can be reduced to: dF =
| 0 and d*F = 0
|
| That's two equations, not one, but you can introduce a new
| D = (d, d*) and then get DF=0 if you want.
|
| The advantage here is the d, and F have all the old
| physical meanings. F is curvature, which is the electro-
| magnetic field E+B, and d is the derivative (exterior
| derivative, but that is the derivative needed in calculus).
|
| Here is a derivation: http://home.lu.lv/~sd20008/papers/ess
| ays/Maxwell's%20equatio...
| Grustaf wrote:
| That looks pretty much equivalent, and they even have a
| version where F = E + B, just like in the youtube video.
| But my question is if this F, or the tensor version for
| that matter, has any physical meaning?
| rsj_hn wrote:
| Yes, F is the electromagnetic field whose laws of motion
| generate the E/M dynamics.
|
| https://en.wikipedia.org/wiki/Electromagnetic_tensor
|
| (This assumes you believe a "Field" has a physical
| meaning.)
| enkimute wrote:
| Not the same concrete example, but one where I do find the
| Geometric Algebra version substantially more insightful, is
| the treatment of rigid body mechanics in the geometric
| algebra of the Euclidean group (R_{n,0,1}).
|
| It has the dual quaternions as even subalgebra (in 3D), and
| unifies all linear and angular aspects. It leads to
| remarkable new insights, as removing the need for force-
| couples (pure angular acceleration is caused by pushing
| along a line at infinity), while pure linear acceleration
| is caused by forces along lines through the center of mass.
|
| These geometric ideas are independent of dimension -
| forces, both angular and linear are always lines. The
| treatment of inertia becomes a duality map, and things like
| Steiners theorem are not needed at all.
|
| On top of this, the separation of the metric that sets GA
| apart means that this formulation of rigid body dynamics
| works not only in flat Euclidean space, but unmodified in
| the Spherical and Hyperbolic geometries. (by a simple
| change of metric of the projective dimension).
|
| For a (graphics/game programmer oriented) tutorial on this
| see https://www.youtube.com/watch?v=pq9YfdPHhIo&ab_channel=
| Bivec...
| rsj_hn wrote:
| Well, I think the point is that in rigid body dynamics,
| the configuration and phase spaces naturally form a
| manifold and then the equations of motion are in terms of
| differential forms on the cotangent bundle of the these
| manifolds. This is commonly expressed in terms of the
| language of exterior algebras, hodge duals, etc. That's
| what is driving all of this, and is usually covered in a
| good class on mathematical physics. Again, there is
| nothing new here except marketing, but marketing plays an
| important and useful role.
|
| I remember for a long time, people coming from the math
| end of things would look down a bit on physicists
| laboriously working everything out in complex tensor
| notation when there are these elegant canonical
| descriptions arising from differential geometry that look
| very simple and beautiful and are completely coordinate-
| invariant.
|
| But then when you want to actually calculate something,
| you end up doing all the painful tensor contractions
| anyway, so the physicists would likewise often lookdown
| on the mathematicians for writing these simple one liners
| that described all of mechanics but not really
| understanding how to calculate stuff.
|
| So if repackaging some of the basic facts of differential
| geometry as "Geometric Algebra" gets physicists to be
| excited about it, then that's a good thing. Just like
| repackaging some of the laborious tensor calculus
| computations into differential geometry has gotten a lot
| of mathematicians excited about physics. It really is
| much more pleasant to work in a coordinate-free manner
| using differential structures associated to the natural
| manifold suggested by the problem, rather than being
| stuck in euclidean space and needing to deal with lots of
| fictional forces and complex change of basis formulas.
|
| For example, look at this text: https://depositonce.tu-
| berlin.de/bitstream/11303/2482/1/Doku...
| BlackFly wrote:
| For those who are interested, this sort of algebra would be known
| as the [Grassman algebra or the exterior
| algebra](https://en.wikipedia.org/wiki/Exterior_algebra). It
| becomes much more interesting if you use non-orthonormal bases
| (or non-euclidean geometry), since then you need to introduce a
| dual basis and distinguish between contravariant vectors and
| covariant vectors. When you add derivatives to the mix you end up
| in differential geometry.
| LotusFunctor wrote:
| Grassmann algebra is a very important part of it, in fact you
| can reconstruct it in geometric algebra. More generally though,
| this algebra would be known as Clifford algebra.
| mathgenius wrote:
| Yes, this is exterior algebra. It's also interesting to figure
| out how this works in ambient dimensions other than three. The
| author has a table of grades: 0 for scalars, 1 for vectors, 2
| for "bivectors", 3 for "trivectors", and they count the number
| of bases for each of these grades as 1 3 3 1. These basis
| counts are the dimensions of the (vector space of) scalars,
| vectors, "bivectors", "trivectors". If you go to two ambient
| dimensions you get 1 2 1, and if you go to four ambient
| dimensions you get 1 4 6 4 1. It's Pascal's triangle.
| LotusFunctor wrote:
| Pascal's triangle, and also with a transparently power-set
| flavor to it :)
| steve76 wrote:
| > What is the Inverse of a Vector?
|
| Manifolds, right? A vector takes a value, adds dimensions, and
| expands the value by spatial definition. A manifold takes a
| value, adds dimensions, and constrains the value by surface
| projection.
|
| Vectors are values. Even though they are made up of a bunch of
| numbers, they are a value like any other input or output. If you
| have a vector function, continuity works with vector sums just
| like continuity between input and output. Maxwell uses this, from
| Gibbs, to describe electromagnetic fields so he could have an
| equation for empty space and get the speed of light. Sometimes
| it's more difficult to find a reason for discontinuity, and are
| forced to assume continuity.
|
| Lie algebra is when you want to describe something with
| symmetries. Whatever you are describing, such as the insides of
| an atom, can't be described with exact values, so you use "what
| things are symmetric" and from there can get a differential
| equation.
| fallingfrog wrote:
| It occurs to me that some of these axioms depend on how many
| dimensions the space has- in 4 dimensions, a vector would have 4
| components, a bivector would have 6, a trivector would have 4,
| and a quadvector would have 1. And so on, in accordance with
| Pascal's triangle.
| spekcular wrote:
| The article begins:
|
| >In this post we will re-invent a form of math that is far
| superior to the one you learned in school. The ideas herein are
| nothing short of revolutionary.
|
| and concludes:
|
| > I firmly believe that in 100 years, Geometric Algebra will be
| the dominant way of introducing students to mathematical physics.
| In the same way that Newton's notation for Calculus is no longer
| the dominant one, or that Maxwell's actual equations for
| Electromagnetism have been replaced by Heaviside's, textbooks
| will change because a better system has come along.
|
| These claims are wrong.
|
| There are three standard notation methods in physics: vectors,
| tensors, and differential forms. Geometric algebra is, as the
| article points out, a more powerful version of the usual vector
| notation. But it is deficient in various ways when compared to
| tensor notation (for calculations) and differential forms (e.g.
| if you want to work basis-free). [I'm oversimplifying a bit, but
| a full discussion is too long for a comment here.]
|
| Anyway, geometric algebra is not some esoteric secret. People
| know about it and have decided not to teach it, because the stuff
| that's already taught is better.
|
| [I picked up this specific phrasing from another user here,
| knzhou, which I think is a particularly good way of explaining
| it.]
| sebastos wrote:
| If deficiency to tensors and differential forms is a reason not
| to be taught, then why do we learn about vector notation?
|
| Oh right, because it's a natural language for talking about
| geometry and mechanics, by far the most common and important
| type of reasoning that the average student will need to do. And
| geometric algebra is demonstrably superior for that domain. So
| your comment is pointless.
| matt-noonan wrote:
| > it is deficient in various ways when compared to [...]
| differential forms (e.g. if you want to work basis-free)
|
| There is nothing basis-dependent in Geometric Algebra. This
| presentation started from a basis, but then again so do many
| presentations of differential forms, leading to 2-forms like dx
| \wedge dy and so on.
|
| The _actual_ difference is that Geometric Algebra requires a
| choice of inner product (actually, you can get away with any
| bilinear form), while differential forms do not. However, some
| of the important operations on differential forms in physics
| _do_ require an inner product (e.g. the hodge star operator and
| the codifferential), so you end up back on equal footing with
| GA again.
| spekcular wrote:
| I worded that in an unclear way. My main beef is that GA is
| just way clunkier than differential forms, which are clearly
| the "right" approach if you want to approach the subject from
| a theoretical perspective. I see no advantage over the usual
| treatment, and many disadvantages.
| ogogmad wrote:
| I'm a skeptic too, but I might not be the intended user for
| the GA formalism. Please explain your reasons.
|
| My skepticism of the supposedly superior pedagogy of
| Geometric Algebra is the following:
|
| - 3D vector algebra with the cross product operation and
| the dot product operation is fairly easy and intuitive. Its
| replacement by GA might not be so easy. So maybe GA should
| be introduced after the vector formalism.
|
| - An arbitrary element of a Geometric Algebra might not
| have a geometric meaning. For instance, some elements of a
| GA are vectors, while some are scalars, but there are also
| these exotic mixed quantities which are scalars plus
| vectors. This is pretty hard for me to understand
| intuitively.
|
| - An arbitrary matrix has a geometric meaning. It's
| essentially just a linear transformation. By contrast, I
| don't feel that an arbitrary element of a geometric algebra
| has a geometric meaning.
|
| - Consider those elements of a Geometric Algebra which
| represent rotations -- they are called _rotors_. Observe
| that if "z" is a rotor then the element "-z" is also a
| rotor which stands for the same rotation as "z". So there
| is more to a rotor than whatever rotation it describes.
| This seems very unintuitive and advanced. (I know that this
| behaviour has applications for the study of spin-1/2
| particles in quantum physics).
|
| I also have trouble understanding where the rule for
| multiplying two elements of a geometric algebra _comes
| from_. It 's an operation, introduced from seemingly
| nowhere, which happens to have some applications in some
| areas. But I'm not comfortable with a multiplication rule
| being introduced out of nowhere without being derived out
| of something. The claim that it has a consistent geometric
| meaning from which it can be derived is never justified. My
| criticisms are therefore largely pedagogical.
| spekcular wrote:
| I think your reasons are good and essentially what I
| would give. I also think differential forms are a much
| theoretically cleaner way to express the same concepts.
|
| Further, teaching differential forms prepares my students
| to engage with the (vast majority of the) existing math
| and physics literature. Teaching geometric algebra
| doesn't.
|
| The practical reasons boil down to: I have to teach the
| standard stuff because otherwise they can't read the
| literature. Having done that, what's the marginal benefit
| of teaching GA? Not a lot.
| ogogmad wrote:
| On the other hand, maybe GA _can_ help with differential
| forms. Differential forms involves exterior algebra, and
| I feel like some aspects of exterior algebra are
| elaborated upon in an insightful way by GA. For instance,
| the grade-2 elements of an Exterior Algebra can be
| understood as angular velocities in many circumstances.
| In GA, this is captured by the exponential map that sends
| grade-2 elements to rotors. I don 't know if this can be
| helpful for teaching purposes.
| jacobolus wrote:
| > _People know about it and have decided not to teach it_
|
| Generally teachers don't (can't) individually decide this.
| Decisions about what to teach have incredible historical
| inertia, and are largely decided based on what the teacher
| learned when they themself went to school decades ago, what
| everyone else is teaching, what materials are easily available,
| what notations are used in past literature, etc. Substantial
| transitions in the teaching of existing material take
| generations.
|
| In 2020 our basic math/science curriculum and pedagogy in high
| schools and universities has all been pretty well statically
| fixed for 50+ years (many parts are unchanged in 200+ years),
| except in computer science where some of the basic ideas are
| newer than that, and in graduate-level courses that get closer
| to the cutting edge.
|
| * * *
|
| The place where geometric algebra has seen most rapid adoption
| is in computer programming, where code actually has to work,
| and a more effective formalism makes correct code easier to
| write and reason about, saving a ton of time and effort even
| for basic examples.
|
| Even in physics, where a better formalism leads to improved
| physical intuition and deeper conceptual understanding, a
| transition is an uphill struggle, because symbolic fluency with
| geometric algebra takes years of practice.
|
| (Some) pure mathematicians on their high horses scoff at
| anything that doesn't advance their own obscure abstract
| research, which is unconcerned with conceptual obstacles faced
| by undergraduate students, scientists, or engineers. They can
| hand-wave a better formalism away with "this is isomorphic to X
| and Y other structures, so there's no value in it".
| spekcular wrote:
| > Substantial transitions in the teaching of existing
| material take generations.
|
| > In 2020 our basic math/science curriculum and pedagogy in
| high schools and universities has all been pretty well
| statically fixed for 50+ years (many parts are unchanged in
| 200+ years), except in computer science where some of the
| basic ideas are newer than that, and in graduate-level
| courses that get closer to the cutting edge.
|
| This is wildly incorrect. Even in the past ~20 years we've
| seen a sea change in our understanding of science pedagogy.
| Look up the work of Carl Wieman on active learning, or
| https://www.pnas.org/content/111/23/8410. Inclusive classroom
| practices are another thing that's come into fashion in the
| last ~10 years. The curriculum has also evolved; the most
| obvious thing to point to is the new emphasis on connections
| to data science in math/stats courses.
|
| If you're someone who doesn't stay up-to-date on pedagogy,
| then yes, it takes your retirement to bring about a change.
| But a lot of people, especially those teaching at small
| liberal acts colleges, have continually evolving teaching
| practices. There are entire conferences where people get
| together to talk about college teaching.
|
| > Even in physics, where a better formalism leads to improved
| physical intuition and deeper conceptual understanding, a
| transition is an uphill struggle, because symbolic fluency
| with geometric algebra takes years of practice.
|
| Do you really believe this? To anyone to recognizes that it's
| the standard stuff in (a clunkier) disguise, it shouldn't
| take years.
|
| Appealing to these two frictions does not offer a convincing
| theory of why GA has not been adopted despite being around
| for, what, 50+ years? The fact that it's worse than existing
| notation does.
| jacobolus wrote:
| Have you ever spent a few months trying to solve a wide
| variety problems using GA as a formalism, or tried teaching
| it to e.g. undergraduates? If not, you are speculating
| beyond your experience.
| spekcular wrote:
| I've taught math to plenty of undergraduates - enough to
| know what does and doesn't play well - and I made an
| honest attempt to find problems where GA might have some
| advantage. It is the fact that I sunk several hours of my
| life into this with no reward that explains why I'm a
| little salty in this thread, and motivated to warn other
| people away.
|
| I'd suggest that your comment that math and science
| pedagogy have been static for the last 50+ years reveals
| that you are the one speculating beyond their experience.
| jacobolus wrote:
| There has been continuous research into alternative
| pedagogy, but the typical undergraduate intro
| math/science course looks pretty much unchanged in both
| pedagogy and curriculum. My undergraduate math and
| physics courses circa 2005 were only slightly different
| than similar courses from 1960 (the main differences were
| things like an online discussion board in some courses,
| some courses with power point slides instead of a
| chalkboard, videotaped lectures in some courses, use of
| computers to type up papers instead of
| typewriters/handwriting), and the typical course is still
| not that much different 15 years later.
|
| One of my hobbies is skimming old math textbooks;
| Lacroix's textbook from about 1800 is not essentially
| different in structure or content than a typical 2020
| intro calculus textbook for undergraduates or high school
| students, or almost any book in between. Way less radical
| or era-appropriate than something like
| http://www.math.smith.edu/~callahan/intromine.html
|
| If you hunt you can find teachers trying new ideas (and
| you could also find teachers trying non-mainstream
| pedagogy 20, 40, or 60 years ago), but it takes
| generations for ideas to turn over.
|
| If you are interested in better introductory physics
| pedagogy in particular, David Hestenes (in other work,
| the chief promoter of GA for decades) is a real pioneer
| and a huge influence on e.g. Eric Mazur.
| http://geocalc.clas.asu.edu/html/Modeling.html
| https://mazur.harvard.edu/files/mazur/files/rep_557.pdf
| spekcular wrote:
| > the typical course is still not that much different 15
| years later.
|
| For context, I checked your profile to see where you did
| your undergraduate degree. I am familiar with the way
| calculus is currently taught at that university, and it
| looks quite similar to the "radical [...] era-
| appropriate" textbook that you linked (at least based on
| a quick read of a few chapters). Those courses are also
| taught in a quasi-active learning style (though nothing
| as extreme as a flipped classroom, etc.). Your
| observations may have been accurate 15 years ago, but
| that's thankfully no longer case. There's also pressure
| from the department/admin to make these changes in upper-
| level courses. See e.g.
| https://people.math.harvard.edu/~community/inclusive-
| classro... or materials from
| https://bokcenter.harvard.edu/active-learning.
| forgotpwd16 wrote:
| Maybe those are better for calculations but are they better for
| learning?
| alisonkisk wrote:
| Most people need calculations to help them learn...
| LotusFunctor wrote:
| >Geometric algebra is, as the article points out, a more
| powerful version of the usual vector notation
|
| That's not just a gross oversimplification, this is also flat
| out wrong if what you meant was that it only has vectors. It
| has more general objects called multivectors through pretty
| much the same process you get one, two, etc. forms from the
| wedge product.
|
| In fact, both GA and differential forms build from the exterior
| algebra, and you can go from the former to the latter through
| geometric calculus (one key difference e.g. would be the method
| of reciprocal bases to compute inner products with non-
| orthonormal bases, rather than explicitly working out a basis
| and then its dual). So I'm confused about your remark regarding
| its alleged deficiency vs. differential forms if you pretty
| much reconstruct it within the GA/GC system (especially
| regarding working basis-free).
|
| With regards to tensor notation in terms of calculations, if
| you mean all that index gymnastics, well GC still openly
| provides that way of computing things out from what you're used
| to.
|
| What I like about geometric algebra/geometric calculus is
| precisely the way in which it's nothing new: it's putting
| everything people use in one system by clarifying the
| connections between these seemingly disparate systems. Even lie
| groups/lie algebras can be constructed rather efficiently in
| the algebra.
|
| Another appealing feature of GA is its ability to make pretty
| transparent an old theorem from Cartan and Dieudonne that says
| you can view geometric transformations like rotations, and even
| translations (in projective geometry) as compositions of
| reflections.
|
| There's other appealing features like this in terms of
| classifying and relating different geometries together that
| harken back to the Erlangen program, but my point is even in
| terms of concrete calculations, it's not quite right to say
| it's just a "more powerful version of the usual vector
| notation" as it includes more general objects than vectors, and
| still includes a lot of very similar ways of doing calculations
| (almost a kind of "backwards-compatibility?") you're used to
| with tensor index calculations, just with the added bonus of
| making the transition to the tensors used from vector calculus
| seamless, alongside other added relation to other systems made
| more transparent.
| 6gvONxR4sf7o wrote:
| > ... by clarifying the connections between these seemingly
| disparate systems.
|
| This is the big reason I like it. I remember learning bits
| and pieces of linear algebra whose rules seemed so entirely
| random (vectors vs pseudo-vectors, curl, quaternions, some
| spin calculations I can't even remember anymore, etc), that
| turned out to have more unified geometric interpretations
| once I learned GA. It definitely slowed me down as a student
| who hated 'just memorize it' pedagogy.
| spekcular wrote:
| To put it in concrete terms, where does GA really fit into
| the story of undergraduate physics (or mathematics)?
|
| Suppose I want to teach first-semester mechanics. I can get
| through this fine with the usual vector notation. Vectors and
| dot products are intuitive when taught well (the latter just
| being projections), and while cross products are a little
| hairy, they don't play a major role in the course. There's no
| time for GA, and it would confuse more than illuminate in any
| case.
|
| Next, I want to teach E&M. Here, I'd probably lead with the
| usual vector calculus notation (because even if it's ugly,
| it's standard and students should know it), and then follow
| with an explanation in terms of differential forms. [I assume
| this is a more theoretical, or honors, class; I might stick
| with vector calculus if it's more computational.] So now
| students know differential forms, they can do everything in a
| coordinate-free way and on manifolds, and they can access a
| significant amount of standard physics and mathematics
| literature.
|
| Having proceeded in this way, what does introducing GA do
| except suck up a lot of class time? To me, it seems clunky
| and without any distinctive advantages.
|
| Another question to think about: if this notation system is
| so good, why don't working mathematicians or physicists
| actually use it? For example, people thought Feynman diagrams
| were strange at first, but they proved their value and
| consequently caught on.
|
| Again, my argument is that this is not some revolutionary
| esoteric knowledge, it's well-understood stuff that people
| don't teach for good reasons.
| jacobolus wrote:
| > _Suppose I want to teach first-semester mechanics_
|
| If you need to teach undergraduate mechanics, I highly
| recommend you at least read some of Hestenes' _New
| Foundations for Classical Mechanics_
| http://geocalc.clas.asu.edu/html/NFCM.html
|
| > _without any distinctive advantages_
|
| The most basic distinctive advantage is that you can invert
| vectors (which is incredibly useful!!) without needing to
| pretend that vectors are matrices, complex numbers, or some
| other kind of object.
|
| GA takes most of the advantages of complex numbers vs. R2
| for representing plane geometry, but extends them to
| arbitrary dimension, and extends them further (when using
| complex numbers for plane geometry you end up representing
| vector-vector products via the obscure zw product involving
| complex conjugation, and it is easy to get confused about
| the difference between a vector vs. a scalar+bivector).
|
| But there are a wide variety of other powerful (and
| geometrically interpretable) algebraic identities which can
| be applied to vectors, blades, and multivectors, ranging
| from awkward to impossible to express using the language of
| differential forms, Gibbs-style vectors, etc. Physicists
| often end up resorting to tedious coordinate-by-coordinate
| calculations for stuff that would end up being an easy
| vector expression in GA. Learning these identities and how
| to apply them takes years and a lot of practice solving
| problems using GA.
|
| My own experience for the first few years of knowing that
| GA existed but not being too fluent with it was that I
| would work some problem (mostly 2-3 dimensional geometry
| problems) out in coordinates, spending like 2 pages of
| scratch paper for the opaque intermediate calculations,
| with high chance for mistakes, then eventually find that
| most of the ugly bits along the way canceled and yielded a
| nice result. Then I would think a bit more about the
| problem, skim through a list of GA identities, and find I
| could have shortened that 2 pages of work to 3 lines, each
| of which had an obvious geometric interpretation.
| spekcular wrote:
| Can you give an example of a problem that might appear in
| an undergraduate physics or math course, whose solution
| is lengthy and tedious by "usual methods" but
| dramatically simplified by the use of GA?
|
| I have seen examples proposed before and been distinctly
| unimpressed. Any serious simplifications in solutions are
| usually due to some notation-agnostic insight.
| jacobolus wrote:
| The nicest written-up example I know is
| https://www.shapeoperator.com/2016/12/12/sunset-geometry/
|
| * * *
|
| As a relatively recent personal example I spent a few
| months (in bits and pieces) working out a bunch of
| metrical spherical geometry for myself without reference
| to past work, with points represented as displacement
| vectors to stereographically projected points at
| https://observablehq.com/@jrus/planisphere with the
| eventual goal of implementing computational geometry /
| cartography code using that as a canonical
| representation, which I think is superior to
| representations used currently in practical software.
|
| The same spherical relationships are comparable (some
| things slightly slightly easier, some slightly trickier)
| to represent as displacement vectors in an embedded
| sphere. But there again relationships are clearer to
| express in GA terms.
|
| Most of the material there is stated without proof (maybe
| eventually full proofs should be included), but several
| of the identities there I worked out very tediously with
| pages of scratch work in coordinates, then realized
| afterward the same results could be arrived at with only
| a few lines of GA.
|
| Only a bit of the material is truly novel (after doing
| the work myself, I hunted around for sources and found
| some of the same formulas worked out previously using
| classical spherical trigonometry 200+ years ago), and
| e.g. some very similar material where the
| stereographically projected points are represented as
| complex numbers can be found at http://fer3.com/arc/img/1
| 10279.applications%20of%20complex%2...
|
| In theory most of the rest could be also worked out using
| complex numbers or matrices, but (a) some ideas end up
| awkward and unidiomatic there so you would never think to
| do it, so that many identities that are slightly obscure
| in GA are almost unheard of written in other formalisms,
| (b) the algebraic manipulation is at least 2-3x more
| cumbersome.
| csdvrx wrote:
| > Geometric algebra is, as the article points out, a more
| powerful version of the usual vector notation. But it is
| deficient in various ways when compared to tensor notation (for
| calculations) and differential forms (e.g. if you want to work
| basis-free). [I'm oversimplifying a bit, but a full discussion
| is too long for a comment here.]
|
| I have no opinion about the claims, but I loved the article, as
| I quickly saw the gains from this algebra for my very basic
| needs.
|
| After checking out your 2 suggested alternatives, I'm not so
| convinced they are easier to understand.
| captainmuon wrote:
| I wish I would have been introduced to Geometric Algebra or
| calculus of forms or whatever it is called during my physics
| studies. We learned all the conventional things you need for
| classical mechanics and electromagnetism, like div and curl and
| BAC-CAB. But there were a couple of things that we were not
| tought well, which caused problems later. One thing is that at
| first, a vector was just an N-tuple. But in physics, something
| is only a vector or a tensor if it behaves under a very
| specific way during transformations. (The infamous "a tensor is
| an object that transforms like a tensor"...)
|
| The other thing is that I hit a wall when trying to read theory
| papers, because nobody ever explained what a wedge operator or
| a two-form and so on is. I was able to mechanically follow
| calculations, but never got a real good intuition.
|
| We learned math together with the mathematicians in a very
| axiomatic way. Without a geometric intuition, I had a hard time
| understanding why you need dual vectors like one-forms or
| covariant vectors. They seem just like a convoluted way to
| write the scalar product.
| spekcular wrote:
| If you want an "intuitive" approach, you might check out this
| new book: https://www.amazon.com/Visual-Differential-
| Geometry-Forms-Ma....
|
| Disclaimer: I haven't read it (but I have heard good things).
| jacobolus wrote:
| I have read (part of) it, and it is excellent. I highly
| recommend it (alongside Needham's VCA book).
|
| But the lessons learned from there are largely orthogonal
| to the lessons you will learn from working with GA for a
| while.
|
| To the grandparent, I would also recommend the very pretty
| book http://matrixeditions.com/5thUnifiedApproach.html
| [deleted]
| chriswarbo wrote:
| > Geometric algebra is, as the article points out, a more
| powerful version of the usual vector notation
|
| > the stuff that's already taught is better
|
| These two statements seem contradictory.
|
| > But it is deficient in various ways when compared to tensor
| notation (for calculations) and differential forms (e.g. if you
| want to work basis-free)
|
| The author made no claims about tensor notation or differential
| forms; perhaps those might displace all vector-like notation
| (GA or otherwise) in 100 years, in which case the author's
| claim can be weakened to "in 100 years, GA will be the dominant
| form of vector notation".
|
| As for "the stuff that's already taught is better", notice that
| such stuff includes:
|
| - Complex numbers
|
| - Pseudovectors
|
| - Cross-products
|
| - Matrix algebra
|
| - Dirac notation
|
| You seem to agree that GA is "more powerful" than the Gibbs-
| style vector algebra normally taught. The article is arguing
| that GA is also a simpler and more consistent approach, which I
| agree with (complex numbers are certainly simpler on their own,
| but are a little redundant if we're using GA for the rest).
|
| From my own experience in formal education (UK high school and
| undergraduate physics), I never encountered tensors or
| differential forms. I've since learned a little about tensors
| (for general relativity), but that's been due to my own
| curiosity; I've learned a little about GA for the same reason.
| I never used quaternions (hence why I left them out of the
| above list), although I'm aware of them and that they're used
| e.g. in computer graphics. I used vectors, pseudovectors, and
| matrices _a lot_ , and I'm certain those topics would have been
| easier to learn and comprehend if they'd used GA instead.
| spacedcowboy wrote:
| > I never encountered tensors or differential forms.
|
| Your UK undergraduate physics must have been a bit different
| to mine. About a third of my physics course was taught by the
| maths dept, and tensors/algebras were very much a part of
| that.
|
| I recall, after freshers week, the dean getting everyone
| together. He said two things:
|
| - Hopefully you all had a great fresher's week, now it's down
| to business, and
|
| - Make sure you have fun at college.
|
| He also had a projection on the overhead saying "If you can't
| blind them with science, baffle them with bullshit". I'm
| reasonably certain the second statement above was the latter,
| because...
|
| He then casually mentioned a "maths refresher" 2 week course
| that all freshers had to take before "the real stuff"
| started. That "maths refresher" was the entire Further Maths
| 'A' level syllabus. In two weeks. Those of us who had done
| Further Maths at school were fine. Those that hadn't were
| shell-shocked.
| tsimionescu wrote:
| > That "maths refresher" was the entire Further Maths 'A'
| level syllabus. In two weeks. Those of us who had done
| Further Maths at school were fine. Those that hadn't were
| shell-shocked.
|
| Heh, that reminds me of my first physics course in an
| under-graduate computing degree (in Romania). The
| curriculum was so well designed overall that this Physics
| course needed linear algebra concepts that would be taught
| halfway-through the semester in Algebra, integration along
| a surface and similar that would be taught at a similar in
| the Calculus course, and some Statistics I don't remember
| that would be only be taught in the second semester.
|
| The prof's solution? He taught a 3-hour course covering all
| of the above, and considered that good enough for all
| future courses. This particular Physics course later went
| on to cover analytical mechanics (generalized coordinates,
| Lagrangians, Hamiltonians), electricity, general
| relativity, statistical thermodynamics, and quantum
| mechanics, all in a single semester.
|
| Needless to say, 99% were happy they passed and couldn't
| tell you a single thing about any of these subjects a few
| minutes after the final exam.
| chriswarbo wrote:
| > That "maths refresher" was the entire Further Maths 'A'
| level syllabus. In two weeks. Those of us who had done
| Further Maths at school were fine. Those that hadn't were
| shell-shocked.
|
| Heh, I recall managing to coast for a short time thanks to
| having done AS Further Maths.
|
| The Further Maths syllabus was quite modular, and the
| modules our teachers picked had some discrete math (sorting
| algorithms, Dijkstra's algorithm, bridges of konigsberg,
| etc. which was useful for comp. sci.), and some which
| complemented the regular maths course (complex numbers and
| more calculus, which was certainly useful for physics).
| spacedcowboy wrote:
| Our Further Maths was a lot less modular. Preparation for
| it started in the 2nd year (so 12/13 years old), when we
| were streamed for maths - if you were in set-1, you
| studied to take 'O' level (showing my age here) in 4th
| year (so a year earlier than most) on an accelerated
| schedule.
|
| That meant you could take AO (a halfway house between O
| and A) when everyone else was taking their normal O
| levels. The thing is that the extra stuff in AO was all
| Pure Maths, and formed a fair amount of the easier "P1"
| maths syllabus for the normal A level maths exam, which
| had P1 and Me1 (Maths with mechanics 1, basically
| statics).
|
| Because you'd done that work already prior to the A level
| years, you could take "A level maths" after only 1 year
| (which looked really good on UCCA applications :), so
| you've now done an exam consisting of the two 'P1' and
| 'Me1' papers in the first year of your A levels.
|
| Which meant that in the second year of your 'A' levels,
| you could do 'Pure Maths' (P1, P2) and 'Further Maths'
| (Me1, Me2) for a total of 3 maths A levels.
|
| On top of that, you had your other two subjects (mine
| were Physics and Chemistry), and because it was the JMB
| board, everyone got to do "General Studies".
|
| Getting all of them gave you 6 A levels, even though some
| of the work was duplicated in the maths arena (over
| different years of course :)
|
| S levels were a bonus on top - there was no fudging for
| those, though, you just took what you thought would be
| useful to study. They gave me maths and physics because
| I'd said I was going to do physics at college... :)
| Cybiote wrote:
| spekcular is correct. One could draw the following graph very
| roughly capturing how these concepts developed:
|
| Grassman's exterior algebra -> Differential forms
|
| Cartan's Exterior Derivatives -> Differential forms
|
| Quaternions -> Clifford Algebra
|
| exterior algebra -> Clifford Algebra -> geometric algebra
|
| Differential forms are a particular kind of tensor and
| tensors can be defined in terms of multilinear maps. As
| spekcular says, the standard curriculum covers differential
| forms, tensors and vectors. This entails becoming familiar
| with multivectors, the wedge product and multilinear algebra,
| making geometric algebra a relatively small delta to pick up.
|
| On the other hand, the standard course will also prepare you
| for mathematical topics like lie derivatives, differential
| geometry and de Rham cohomology.
|
| Other than physics, the standard approach equips you with the
| mathematical machinery underlying many topics in machine
| learning and statistics like Hamiltonian monte carlo,
| automatic differentiation, information geometry and geometric
| deep learning.
|
| The central advantage of geometric algebra over the standard
| approach isn't that it's better or more general, it's that
| pedagogical material for it is generally leagues and
| magnitudes better than those for the standard course.
| chriswarbo wrote:
| > As spekcular says, the standard curriculum covers
| differential forms, tensors and vectors... making geometric
| algebra a relatively small delta to pick up.
|
| Could you be a bit more specific about which "standard
| curriculum"/"standard approach" you're talking about?
|
| For example, in my formal education (high school; masters
| with physics major, comp. sci. minor; 4 years of a comp.
| sci. PhD (abandoned)), I did not encounter differential
| forms, tensors, multivectors, the wedge product or
| multilinear algebra (or quaternions, lie derivatives,
| differential geometry, (co)homology, etc.).
|
| Maybe you're talking about a "standard approach" for a pure
| mathematics curriculum, or perhaps physics/math grad
| school?
|
| All I can say is that high school and undergraduate
| _physics_ (in the UK, circa the late naughties) (a) does
| not standardise on those topics, (b) is filled with tricky
| operations which are easy to mix up or perform the wrong
| way around (e.g. cross products, matrix multiplication,
| pseudovectors), and (c) many of those annoyances would
| simplify-away under GA.
|
| It's a cliche that physicists (certainly when teaching)
| cherry-pick the parts of mathematics they find useful. All
| of those concepts would certainly be useful in a physics
| course, but would perhaps be too much to fit in; yet
| there's certainly enough scope to cherry-pick GA (since we
| can drop Gibbs-style vector algebra[0] to make room).
| Perhaps something else, like differential forms, might be
| _even better_ ; I honestly don't know (maybe I'll do some
| reading about it).
|
| [0] By "Gibbs-style" I mean the 'cross product and dot
| product ought to be enough for anyone' approach that
| permeated my undergraduate learning.
| Cybiote wrote:
| By standard approach I mean the typical material covered
| for someone studying vector calculus properly. This will
| be stuff like differential forms and the basics of
| tensors, manifolds and multilinear maps at the undergrad
| level. Differential geometry and cohomology are examples
| of courses which build on them.
|
| I agree with you that pseudovectors, cross products and
| vector calculus are a terribly adhoc way to teach this
| stuff but a course covering linear algebra with
| differential forms elegantly unifies, corrects and
| generalizes them. Standard is also in contrast to the
| geometric algebra/calculus alternate path.
| jacobolus wrote:
| > _for someone studying vector calculus properly_
|
| If you can't invert vectors, you aren't studying vector
| calculus properly. ;-)
|
| Differential forms are a half-baked formalism.
|
| Unfortunately I don't know of any great undergraduate
| level geometric calculus textbooks. Ideally there would
| be something like Hubbard & Hubbard's book
| (http://matrixeditions.com/5thUnifiedApproach.html)
| written using GA as a formalism.
|
| Hestenes & Sobczyk's book
| (http://geocalc.clas.asu.edu/html/CA_to_GC.html) is a
| hard slog, and not appropriate for an undergraduate
| audience.
| spekcular wrote:
| > Differential forms are a half-baked formalism.
|
| I can't emphasize enough how wrong this is. It's the
| standard formalism in research-level physics and math for
| good reasons.
| [deleted]
| LotusFunctor wrote:
| From my view, it goes both ways: geometric algebra/calculus
| is a more transparent version of the standard approach and
| the translation back to it is also a relatively small delta
| to pick up.
|
| Either way of going about what is in essence the same
| material entails becoming familiar with multivectors, the
| wedge product, and multilinear algebra, whether you do it
| through geometric algebra or the standard approach.
| Cybiote wrote:
| That makes sense but my argument is since further
| material (some examples which I listed) assumes and
| builds upon the standard approach, you'll likely be
| better off taking that path.
| topaz0 wrote:
| This may be true in some places, but my undergraduate physics
| education spent a lot of time on standard Gibbs-style vector
| calculus. Taylor Classical Mechanics and Griffiths
| Electrodynamics especially depend on them. Maybe there is a
| case to be made that first years should start with differential
| forms, but until that happens I think geometric algebra could
| be a big improvement.
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