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What is the inverse of a vector?

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Re: What is the inverse of a vector?

#161

Post by spekcular »

jacobolus wrote: post #157
spekcular wrote: post #155

Earlier quoted context omitted.

> 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 i…

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.

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.

Re: What is the inverse of a vector?

#162

Post by spekcular »

ogogmad wrote: post #156
spekcular wrote: post #136

Earlier quoted context omitted.

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.

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 a…

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.

Re: What is the inverse of a vector?

#163

Post by chriswarbo »

spacedcowboy wrote: post #116
chriswarbo wrote: post #36

Earlier quoted context omitted.

> 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 t…

> 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 yo…

> 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 königsberg, 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).

Re: What is the inverse of a vector?

#165

Post by jacobolus »

spekcular wrote: post #161
jacobolus wrote: post #157

Earlier quoted context omitted.

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.

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…

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

Re: What is the inverse of a vector?

#166

Post by portpecos »

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 τ=r∧F on a physics problem for Torque?

Re: What is the inverse of a vector?

#167

Post by spekcular »

jacobolus wrote: post #165
spekcular wrote: post #161

Earlier quoted context omitted.

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…

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…

> 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.

Re: What is the inverse of a vector?

#168

Post by Grustaf »

rsj_hn wrote: post #143
Grustaf wrote: post #26

Earlier quoted context omitted.

> 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…

> 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 exter…

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?

Re: What is the inverse of a vector?

#169

Post by spekcular »

captainmuon wrote: post #65
spekcular wrote: post #7

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…

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…

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).

Re: What is the inverse of a vector?

#170

Post by rsj_hn »

Grustaf wrote: post #168
rsj_hn wrote: post #143

Earlier quoted context omitted.

> 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 exter…

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?

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.)

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