Son Goku
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I'd like to see you justify both of those claims.Thrice said:Physics is shortsightedly application driven & math is abstract past meaninglessness.
I'd like to see you justify both of those claims.Thrice said:Physics is shortsightedly application driven & math is abstract past meaninglessness.
Well it was a caricature. I'm just saying I believe the topics allow for many differences & there's no right approach that everyone should converge to. Even in math you'll find discrete vs analysis people or in physics there's theoretical & experimental types.Son Goku said:I'd like to see you justify both of those claims.
Interesting, it's probably due to my limited experience but most of the mathematician's at my university generally learn things from the definitions first, an ability I always found very impressive.mathwonk said:son goku, i also like elementary hands on calculations to begin to understand what a concept means. that's how the subjec=ts began and hoq tgheir discoverers often found them. but after a while one wants to pass to using their proeprties both to understand and to calculTE WITH THEM.
No one learns anything from a definition.Son Goku said:Interesting, it's probably due to my limited experience but most of the mathematician's at my university generally learn things from the definitions first, an ability I always found very impressive.
ObsessiveMathsFreak said:My opinion, for what it's worth, is that differential forms is simply not a mature mathematical topic. Now it's rigourous, complete and solid, but it's not mature. It's like a discovery made by a research scientist that sits, majestic but alone, waiting for another physisist or engineer to turn it into something useful. Differential forms, as a tool, are not ready for general use in their current form.
ObsessiveMathsFreak said:The whole development of forms was likely meant to formalise concepts that were not entirely clear when using vector calculus alone.
ObsessiveMathsFreak said:A one-form must be evaluated along lines, and a two-form must be evaluated over surfaces.
Does this reasoning appear anywhere in any differential form textbook? No.
ObsessiveMathsFreak said:Not even is it mentioned that certain vector fields might be restricted to such evaluations. Once the physics is removed, there is little motivation for forms beyond Stoke's theorem,
ObsessiveMathsFreak said:I don't think differential forms are really going to go places. I see their fate as being that of quaternions. Quaternions were origionally proposed as the foremost method representation in physics, but were eventually superceeded by the more applicable vector calculus. They are still used here and there, but nowhere near as much as vector calculus. Forms are likely to quickly go the same way upon the advent of a more applicable method.
rdt2 said:Several people on this thread have mentioned Flanders’ Differential Forms with Applications to the Physical Sciences (Dover 1989 ISBN 0486661695) and Flanders himself notes that:
'There is generally a time lag of some fifty years between mathematical theories and their applications…(exterior calculus) has greatly contributed to the rebirth of differential geometry…(and) physicists are beginning to realize its usefulness; perhaps it will soon make its way into engineering.'
rdt2 said:However, the formation of engineers is different from that of mathematicians and perhaps even physicists and their aim is usually to get a numerical answer to a _design_ problem as quickly as possible. For example, 'stress' first appears on p.27 of Ashby and Jones’ Engineering Materials, in the context of simple uniaxial structures, but p.617 of Frankel’s Geometry of Physics, in the context of a general continuum. Engineering examples, taken from fluid mechanics and stress analysis rather than relativity or quantum mechanics, usually start with 'Calculate…' rather than 'Prove…'. So many otherwise-excellent books, including Flanders, aren’t suitable for most engineering students. However, what I'm learning here is of great help in trying to put together lecture notes for engineers. So I'd like to add my thanks to those here who've contributed to my limited understanding in this area.
I've read a lot of books on differential forms. Not that one, but still many others. Many of which purport to have applications to physical sciences, but usually just throw down the differential forms version of Maxwell's equations by diktat with little or nothing in the way of semantics. Worked examples are few, probably for the reason that the worked out question would be longer than the route taken by regular vector calculus.Chris Hillman said:Wow! That's quite an impassioned indictment. Did you not read Harley Flanders, Differential Forms, with Applications to the Physical Sciences?
Chris Hillman said:Not really, according to Elie Cartan himself (who introduced the concept of a differential form and was their greatest champion in the first half of the 20th century), the main impetus included considerations like these:
1. the need for a suitable formalism to express his generalized Stokes theorem,
2. the nature desire to express a differential equation (or system of same) in a way which would be naturally diffeomorphism invariant (this is precisely the property which makes them so useful in electromagnetism).
I have decided for myself. I don't approve of differential forms. At least, not as a replacement or improvement for vector calculus. That's just my own opinion, but I would ask others to consider this point of view before imposing forms arbitrarily on undergraduate courses.Calculus on Manifolds said:It is a touchy question whether or not these modern definitions represent a real improvment over classical formalism; this the reader must decide for himself.
That is what is technically referred to as a contextomy. I will simply refer back to the entireity of the original post.Chris Hillman said:This claim seems very contrary to my own reading experience.ObsessiveMathsFreak said:A one-form must be evaluated along lines, and a two-form must be evaluated over surfaces.
Does this reasoning appear anywhere in any differential form textbook? No.
All very well, but this discussion is in the context of differential forms being a replacement for vector calculus for ordinary physicists and engineers. As per my original point, I believe froms to be unsuited to this task. Whether by design or immaturity, they are not a suitable topic of study for most physicists involved in the study of either electromagnetism and especially fluid dymanics. They may, like other advanced mathematical topics, be of use in describing new theories or methods, but this thread is about their promotion for more basic studies, as per nrqed's initial post.Chris Hillman said:I hardly know where to begin, but perhaps it suffices to mention just one counterexample: the well-known recipe of Wahlquist and Estabrook for attacking nonlinear systems of PDEs is based upon reformulating said system in terms of forms and then applying ideas from differential rings analogous to Gaussian reduction in linear algebra. I can hardly imagine anything more practical than a general approach which has been widely applied with great success upon specific PDEs.
ObsessiveMathsFreak said:I've read a lot of books on differential forms. Not that one, but still many others. Many of which purport to have applications to physical sciences, but usually just throw down the differential forms version of Maxwell's equations by diktat with little or nothing in the way of semantics. Worked examples are few, probably for the reason that the worked out question would be longer than the route taken by regular vector calculus.
I have decided for myself. I don't approve of differential forms. At least, not as a replacement or improvement for vector calculus. That's just my own opinion, but I would ask others to consider this point of view before imposing forms arbitrarily on undergraduate courses.
All very well, but this discussion is in the context of differential forms being a replacement for vector calculus for ordinary physicists and engineers. As per my original point, I believe froms to be unsuited to this task. Whether by design or immaturity, they are not a suitable topic of study for most physicists involved in the study of either electromagnetism and especially fluid dymanics.
ObsessiveMathsFreak said:All very well, but this discussion is in the context of differential forms being a replacement for vector calculus for ordinary physicists and engineers. As per my original point, I believe froms to be unsuited to this task. Whether by design or immaturity, they are not a suitable topic of study for most physicists involved in the study of either electromagnetism and especially fluid dymanics.
ObsessiveMathsFreak said:If I remember correctly, nrqed's inital post was in the context of several other threads on the topic of differential forms and possibly topology, where the supposed benefits of forms were being lauded to nrqed who, quite rightly, simply didn't see the benefit in the frankly massive amount of formalism required to study these topics. He's absolutely right. Topology in paticular is now a disaster area for the newcomer. 100+ years of invesigations, disproofs, counter examples, theorems and revisions have lead to the axioms and definitions of topology being completely unparsable.
A great many topology books offer nothing but syntax with no sematics at all. Differential forms texts fare little better. To a good physicist, sematics is everything, and hence the subject will appear to the great majority of them to be devoid of use. That's actually a problem with a lot of mathematics, and modern mathematics in paticular. Syntax is presented, but sematics is frequently absent.
Why like that? Why not:Hurkyl said:Through axioms! You define d/dx to be an operator that:
(1) is a continuous operator
(2) satisfies (d/dx)(f+g) = df/dx + dg/dx
(3) satisfies (d/dx)(fg) = f dg/dx + df/dx g
(4) satisfies dx/dx = 1
and I think that's all you need.
I was speaking from an andragogical standpoint.Doodle Bob said:I've always considered it very bad manners to criticize someone else's discipline as worthless, and the above seems to me very bad manners.
ObsessiveMathsFreak said:I was speaking from an andragogical standpoint.
I think it's safe to say not many children would be learning differential geometry from textbooks.Doodle Bob said:that may be the case. but i do not see any mention of "adult learners" in this post or any of the others.
ObsessiveMathsFreak said:I think it's safe to say not many children would be learning differential geometry from textbooks.
ObsessiveMathsFreak said:I was speaking from an andragogical standpoint.
ObsessiveMathsFreak said:I've read a lot of books on differential forms. Not that one, but still many others.
ObsessiveMathsFreak said:Many of which purport to have applications to physical sciences, but usually just throw down the differential forms version of Maxwell's equations by diktat with little or nothing in the way of semantics. Worked examples are few, probably for the reason that the worked out question would be longer than the route taken by regular vector calculus.
ObsessiveMathsFreak said:I feel the main impetus for differential forms was to formalise something that was never really valid in the first place, namely concepts like; df or equations like
df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy
instead of the actual equation
\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}
ObsessiveMathsFreak said:I have decided for myself. I don't approve of differential forms. At least, not as a replacement or improvement for vector calculus. That's just my own opinion, but I would ask others to consider this point of view before imposing forms arbitrarily on undergraduate courses.
ObsessiveMathsFreak said:this discussion is in the context of differential forms being a replacement for vector calculus for ordinary physicists and engineers. As per my original point, I believe froms to be unsuited to this task. Whether by design or immaturity, they are not a suitable topic of study for most physicists involved in the study of either electromagnetism and especially fluid dymanics. They may, like other advanced mathematical topics, be of use in describing new theories or methods, but this thread is about their promotion for more basic studies, as per nrqed's initial post.
ObsessiveMathsFreak said:If I remember correctly, nrqed's inital post was in the context of several other threads on the topic of differential forms and possibly topology, where the supposed benefits of forms were being lauded to nrqed who, quite rightly, simply didn't see the benefit in the frankly massive amount of formalism required to study these topics. He's absolutely right. Topology in paticular is now a disaster area for the newcomer. 100+ years of invesigations, disproofs, counter examples, theorems and revisions have lead to the axioms and definitions of topology being completely unparsable.
ObsessiveMathsFreak said:A great many topology books offer nothing but syntax with no sematics at all. Differential forms texts fare little better. To a good physicist, sematics is everything, and hence the subject will appear to the great majority of them to be devoid of use. That's actually a problem with a lot of mathematics, and modern mathematics in paticular. Syntax is presented, but sematics is frequently absent.
That's wrong, of course. Without syntax, you are incapable of doing calculations, or communicating with others.ObsessiveMathsFreak said:To a good physicist, sematics is everything
I'm going to have to call you on this one. Most of the terms one would learn in elementary topology can evoke an immediate geometric picture: open set, closed set, compact set, interior, exterior, boundary, compact set, connected set, path, pathwise-connected, sequence, sequentially compact, continuous function... Feynman even tells a story how his (mathematical) colleagues would come to him and describe whatever scenario they had been working, and Feynman would generally build a mental picture of what they're describing, and would was generally quite accurate at guessing the result of their analysis.A great many topology books offer nothing but syntax with no sematics at all.
I was not knocking modern mathematics. I was knocking the way modern mathematics is being taught. It is of course a problem throught mathematics, where we have many people who are very good at it, but very few people who are very good at teaching it.Doodle Bob said:Well, that's a very slippery way of avoiding the essence of my assertion: you spend a great deal of time knocking modern mathematics (and topology in particular) as insignificant technobabble and very little talking about curriculum and undergraduate pedagogy.
Just on this and the previous post. There's no reason to assume from my statement about andragogy that I am not a recent graduate. Nor is it correct to referr to the teaching of undergraduates as pedagogy. Everything from undergraduate up is andragogy. You are teaching people who are adults, and who only learn topics they feel are relavent to them.Chris Hillman said:Sigh... Oh well, here's the longish post I wrote predicated on the (mistaken?) assumption that OMF is a twenty-something recent college graduate:
Spivak's book was by far the best book on forms that I read. By far the best book on calculus for that matter. I consider it as having completed a lot of things left out or paper over in my calculus education so far. Spivak was good because he was what so many other author were not. That is, precise. He fully explained, in the required mathematical detail, what a form was, what it did, etc, etc. He did fall down a bit on tensors, but I think without the physics behind them, tensors remain too up in the air for full conceptual understanding.Chris Hillman said:I take it that one of them was Spivak's book, Calculus on Manifolds? You do realize that the goal of this book was not intended to do what you ask? I will go out on a limb and guess (from your username and the context of this thread) that your undergrad major was math, not physics or engineering. If so, I wonder if you might not have been in the wrong major.
I've taken your recommendation and ordered it. If I see forms working well on a problem, perhaps I'll see them in a new light. But I must mention that I have seen them at work on a good many problems, and I have not yet seen any great advantage in the method.Chris Hillman said:Well, if a worked example was the first thing you wanted, it is certainly too bad that you didn't start with the book by Flanders...
I would describe differential forms as many things. Formal certainly. Interesting there is no doubt. They can even be useful when one moves into higher dimensions. But lovely is not a word I would use for a topic that allows for old ghost of maths class past like dx + dy to rise up and walk the Earth once more.Chris Hillman said:Gosh. You certainly seem to be embittered. That is especially unfortunate since this really is such a lovely subject.
I would strongly think otherwise. Four years is quite a reasonable enough amount of time to spend in any undergraduate degree. Anything more would be far too much.Chris Hillman said:Unfortunately, larger social issues force universities to try to churn out their graduates in four years, rather than the six to ten years which in my view would be more reasonable for most undergraduate students.
I blame the mathematicians. They're not precise enough.Chris Hillman said:I think that if you accept what I said just above, it may be that our positions are not so different after all. Perhaps our real difference is over whether you should blame the math faculty at your school, or the politicians who consistently fail to tackle important long range social issues in the country where you were (mis?)-educated.
I think syntax and sematics should come together. In synergy. One cannot understand one without the other. I a big believer in introducing every new mathematical theory or concept via a problem, because that is invariably where it originated.Hurkyl said:That's wrong, of course. Without syntax, you are incapable of doing calculations, or communicating with others.
And besides, one can define sematics for a formal system in terms of the syntax itself, so you can't say that any formalism is inherently devoid of semantics.
The terms might evoke intuitive ideas and pictures. The definitions certainly do not. Despite any impressions I may be giving off, I still consider myself a mathematician, and preciseness and exactness are important to me.Hurkyl said:I'm going to have to call you on this one. Most of the terms one would learn in elementary topology can evoke an immediate geometric picture: open set, closed set, compact set, interior, exterior, boundary, compact set, connected set, path, pathwise-connected, sequence, sequentially compact, continuous function...