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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.

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- Thread starter nrqed
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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.

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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.

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mathwonk

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fundamental groups for instance have a basic proeprty, their homotopy invariance. thi shows immediately that a mobius strp and a circle have the same fundamental group, so there is no reason to calculate it again for a mobius strip.

as for a circle, the ebst calculation is the notice that the exponentil map is a contractible covering covering space. hence the fundamental group of the circle is essentilly the group of lifts of arcs based at a one point of the circle. such lifts are clasified by their endpoint, which must be an integer. hence the fundamental group is the integers.

similarly the fundamental group of a product is the rpoduct of the fundamental groups. so since the torus is a product of two circles, the fundamental group is a prodct of two copies of the integers.

or one could use the contractible covering map showing the torus is the quoteint space of the plane modulo the integr lattice points in the plane, hence that lattice is the fund group.

etc etc

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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. thats 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.

Although as you said, either way of doing it (learning by calculating first and then moving to definition or vice-versa) is just a way of moving on into the interesting stuff.

As mathematician what would you say, in general, seperates the way mathematics is presented in theoretical physics to the way it is presented in maths?

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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.

A mathematical definition is a thing austere and insurmountable. It's form comes only into focus from shelves above it, reached by winding and circuitous paths that loop around its sheer and unforgiving slopes. None can scale its glassy surface, no crack or foothold exists upon it. It is a cliff unmeant for climbing.

Do not accept ropes of rote let down by those on the definitions tip! To understand mathematics, one must muddy one's boots on the longer, less grandiose routes. For if you rely on dangling ropes to ascend this noble peak, then the time will come when your path leads you to a facade as yet unmastered, and no ropes will come. There you will stand awaiting one, surrounded by muddy but fruitful treks to the summit.

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Chris Hillman

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Hi, OMF,

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.

Wow! That's quite an impassioned indictment. Did you not read Harley Flanders, Differential Forms, with Applications to the Physical Sciences?

I am quite confident that you are quite wrong about forms. Not only is the theory of differential forms highly developed as a mathematical theory, it is highly applicable and greatly increases conceptual and computational efficiency in many practical engineering and physics tasks. The elementary aspects of forms and their applications have been taught to undergraduate applied math students at leading universities with great success for many years. (At my undergraduate school, the terminal course for applied math majors was based entirely on differential forms; all engineering students were also required to take this course, as I recall.) I am a big fan of differential forms and feel they are easy to use to great effect in mathematical physics; see for example http://www.math.ucr.edu/home/baez/PUB/joy [Broken] for my modest attempt to describe a few of the applications I myself use most often.

The whole development of forms was likely meant to formalise concepts that were not entirely clear when using vector calculus alone.

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).

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.

This claim seems very contrary to my own reading experience.

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,

Not true at all. 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.

http://www.google.com/advanced_search?q=Wahlquist+Estabrook&hl=en

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.

I am sorry that you have apparently had such a miserable experience trying to learn how to compute with differential forms! I hope you will try again with a fresh outlook, say with a book like the one I cited above.

Chris Hillman

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I've just come back to the forum after almost a year away and found this thread stimulating. The following quotes show why even a mechanical engineer is interested in differential forms:

'The important concept of the Lie derivative occurs throughout elasticity theory in computations such as stress rates. Nowadays such things are well-known to many workers in elasticity but it was not so long ago that the Lie derivative was first recognized to be relevant to elasticity (two early references are Kondo [1955] and Guo Zhong-Heng [1963]). Marsden and Hughes, 1983, Mathematical Foundations of Elasticity.'

'Define the strain tensor to be ½ of the Lie derivative of the metric with respect to the deformation'. Mike Stone, 2003, Illinois. http://w3.physics.uiuc.edu/~m-stone5/mmb/notes/bmaster.pdf [Broken]

'…objective stress rates can be derived in terms of the Lie derivative of the Cauchy stress…' Bonet and Wood, 1997, Nonlinear continuum mechanics for finite element analysis.

'The concept of the Lie time derivatives occurs throughout constitutive theories in computing stress rates.' Holzapfel, 2000, Nonlinear solid mechanics.

'Cartan’s calculus of p-forms is slowly supplanting traditional vector calculus, much as Willard Gibbs’ vector calculus supplanted the tedious component-by-component formulae you find in Maxwell’s Treatise on Electricity and Magnetism' – Mike Stone again.

'The objective of this paper is to present…the benefits of using differential geometry (DG) instead of the classical vector analysis (VA) for the finite element (FE) modelling of a continuous medium (CM).' Henrotte and Hameyer, Leuven.

'The fundamental significance of the vector derivative is revealed by Stokes’ theorem. Incidentally, I think the only virtue of attaching Stokes’ name to the theory is brevity and custom. His only role in originating the theorem was setting it as a problem in a Cambridge exam after learning about it in a letter from Kelvin. He may, however, have been the first person to demonstrate that he did not fully understand the theorem in a published article: where he made the blunder of assuming that the double cross product v ( v) vanishes for any vector-valued function v = v(x) .' Hestenes, 1993, Differential Forms in Geometric Calculus. http://modelingnts.la.asu.edu/pdf/DIF_FORM.pdf [Broken]

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 realise its usefulness; perhaps it will soon make its way into engineering.'

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.

Ron Thomson,

Glasgow.

'The important concept of the Lie derivative occurs throughout elasticity theory in computations such as stress rates. Nowadays such things are well-known to many workers in elasticity but it was not so long ago that the Lie derivative was first recognized to be relevant to elasticity (two early references are Kondo [1955] and Guo Zhong-Heng [1963]). Marsden and Hughes, 1983, Mathematical Foundations of Elasticity.'

'Define the strain tensor to be ½ of the Lie derivative of the metric with respect to the deformation'. Mike Stone, 2003, Illinois. http://w3.physics.uiuc.edu/~m-stone5/mmb/notes/bmaster.pdf [Broken]

'…objective stress rates can be derived in terms of the Lie derivative of the Cauchy stress…' Bonet and Wood, 1997, Nonlinear continuum mechanics for finite element analysis.

'The concept of the Lie time derivatives occurs throughout constitutive theories in computing stress rates.' Holzapfel, 2000, Nonlinear solid mechanics.

'Cartan’s calculus of p-forms is slowly supplanting traditional vector calculus, much as Willard Gibbs’ vector calculus supplanted the tedious component-by-component formulae you find in Maxwell’s Treatise on Electricity and Magnetism' – Mike Stone again.

'The objective of this paper is to present…the benefits of using differential geometry (DG) instead of the classical vector analysis (VA) for the finite element (FE) modelling of a continuous medium (CM).' Henrotte and Hameyer, Leuven.

'The fundamental significance of the vector derivative is revealed by Stokes’ theorem. Incidentally, I think the only virtue of attaching Stokes’ name to the theory is brevity and custom. His only role in originating the theorem was setting it as a problem in a Cambridge exam after learning about it in a letter from Kelvin. He may, however, have been the first person to demonstrate that he did not fully understand the theorem in a published article: where he made the blunder of assuming that the double cross product v ( v) vanishes for any vector-valued function v = v(x) .' Hestenes, 1993, Differential Forms in Geometric Calculus. http://modelingnts.la.asu.edu/pdf/DIF_FORM.pdf [Broken]

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 realise its usefulness; perhaps it will soon make its way into engineering.'

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.

Ron Thomson,

Glasgow.

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- #58

Chris Hillman

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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 realise its usefulness; perhaps it will soon make its way into engineering.'

Which he wrote in the 1960s, right? Referring to Cartan's work during the 1920's and 1930's? Indeed, by the 1980s, leading engineering schools such as Cornell were restructuring their undergraduate curricula to expose their students to differential forms.

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.

Interesting. I entirely agree with you about the need to emphasize computational techniques, adding the need to offer plenty of simple but nontrivial examples. I mentioned Flanders because of the books I've seen (yeah, mostly in math libraries, not engineering libraries!), it comes closest to this spirit. In his introduction, he actually makes the same complaint: most students want to see some interesting applications presented in detail more than they want a lengthy exposition of "dry" theory.

In 1999, about the time I wrote the "Joy of Forms" stuff I linked to above, I actually was briefly involved in trying to teach differential geometry in general and forms in particular to graduate engineering students, so "Joy" is no doubt based in part upon that experience. This project resulted in disaster, in great part (I think) because I was directed to plunge in without having prepared a curriculum in advance and without knowing anything about the background of my students (this is certainly not a procedure which I advocated at the time, nor one which I would ever advise anyone else to adopt under any circumstances!).

Despite this failure, I remain entirely convinced that the world would be a much better place if engineering schools were more successful at teaching their students more sophisticated mathematics, [ITALICS]as tools for practical daily use in their engineering work.[/ITALICS] Certainly exterior calculus and Groebner basis methods would top the list, but I'd also add combinatorics/graph theory, perturbation theory, and symmetry analysis of PDEs/ODEs. So I hope you perservere with your lecture notes.

Chris Hillman

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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.Wow! That's quite an impassioned indictment. Did you not read Harley Flanders, Differential Forms, with Applications to the Physical Sciences?

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'm skeptical. 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

[tex]df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy[/tex]

instead of the actual equation

[tex]\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}[/tex]

This was always a precarious point of view, and in my own view the theory of forms does not legitimise the concept. Even Spikav acknowledges that there is some debate in Calculus on Manifolds at the end of Chapter 2;

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.This claim seems very contrary to my own reading experience.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.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.

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

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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.

My point exactly. A couple of authors try to give examples from mechanics but they always appear very contrived - suggesting that forms may be fundamentally unsuitable in some areas. If you want to read Marsden and Hughes 'Mathematical Foundations of Elasticity' or suchlike, then knowledge of forms is required. The question is, how many engineers and physicists want to read Marsden and Hughes.

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.

I'm less certain. I want to expose students to forms as a complement rather than a replacement for vector calculus. They'll judge in later life whether they're useful or whether, like most of their lecture notes, they can be consigned to the little round filing cabinet.

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.

Oddly enough, fluid dynamics was one of the areas where I thought that differential forms might have most application. I'm less sure about stress analysis, where the tensors are all symmetric.

Ron.

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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.

This may very well be true. This topic might indeed be a bit abstruse for the average physics undergrad. However, one should keep in mind that a good century and a half ago, the very same thing could have been said about the relationship between linear transformations and matrices. There was at the time not much use for them among the physicists until quantum physics came around.

I must say, though, having just recently read The Large Scale Structure of Space-Time by Ellis and Hawking that a good knowledge of forms (and other elements of differential geometry) are essential to the understanding of GR.

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 iseverything, 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.

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.

Modern mathematics is indeed very complex, and a very wild field to start out in. However, formalism and logic is the mortar that keeps it all together. Without proofs and rigorous thinking, math is just magic. Hence, a great deal of research seems to be more fancy window-dressing than anything substantial. But, every so often a big theorem comes into view: I'm thinking of two within my mathematical career: the Fermat-Wiles Theorem and the solving of the Poincare conjecture (and hence of Thurston's Geometrization Conjecture).

These results probably don't mean to you (they are after all worthless to electromagnetism) but they mean a great deal to me and other mathematicians.

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AKG

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Why like that? Why not: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.

[tex]\frac{d}{dx} : f \mapsto \left( x \mapsto \lim _{h \to 0}\frac{f(x+h) - f(x)}{h}\right )[/tex]

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I was speaking from an andragogical standpoint.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.

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I was speaking from an andragogical standpoint.

that may be the case. but i do not see any mention of "adult learners" in this post or any of the others.

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I think it's safe to say not many children would be learning differential geometry from textbooks.that may be the case. but i do not see any mention of "adult learners" in this post or any of the others.

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I think it's safe to say not many children would be learning differential geometry from textbooks.

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.

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mathwonk

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Chris Hillman

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Oh dear, I wrote a long reply to OMF, then belatedly noticed a crucial remark:

I was speaking from an andragogical standpoint.

Sigh... Oh well, here's the longish post I wrote predicated on the (mistaken?) assumption that OMF is a twenty-something recent college graduate:

I've read a lot of books on differential forms. Not that one, but still many others.

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.

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.

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 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

[tex]df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy[/tex]

instead of the actual equation

[tex]\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}[/tex]

1. Well, I guess this depends upon what you mean by "valid". Is a "linear approximation" invalid simply because it is not an identity?

2. Trust me. While historians of mathematics have apparently not yet tackled the career of Elie Cartan (despite his extraordinary influence on the development of modern mathematics), I probably know more about his interests than you do. In particular, I know something about his interests in Lie algebras, differential equations and general relativity, as well as integration.

For Cartan's work on the central problem in Riemannian geometry (in fact a whole class of problems involving differential equations), try Peter J. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995. Notice that this work lies at the heart of the Karlhede algorithm in gtr. For more about Cartan's involvement in the early development of gtr, see Elie Cartan-Albert Einstein : letters on absolute parallelism, 1929-1932. English translation by Jules Leroy and Jim Ritter ; edited by Robert Debever, Princeton University Press, 1979. For more about Cartanian geometry (common generalization of Riemannian and Kleinian geometry), try R. W. Sharpe, Differential geometry : Cartan's generalization of Klein's Erlangen program, Springer, 1997. For "Newtonian spacetime", see the chapter in Misner, Thorne, and Wheeler, Gravitation, Freeman 1973.

It is, or IMO should be, very striking that these sources are almost completely independent of each other. Cartan's work is characterized by a remarkable coherence of purpose and scope, yet adds up to so much that even whole commitees of authors can attempt to explain only bits and pieces.

For an attempted overview of Cartan's influence on modern mathematics, Francophones can try Elie Cartan et les mathématiques d'aujourd'hui, Lyon, 25-29 juin 1984 : the mathematical heritage of Elie Cartan, Société mathématique de France, 1985. For anglophones, an important textbook on mathematical physics, which is contemporary with Cartan's career, which emphasizes the utility of differential forms, and which might provide a few hints about why these techniques should be mastered by any serious student of mathematics, is Courant and Hilbert, Methoden der mathematischen Physik. This book went through various German language editions beginning in 1924. It has been translated into English (Interscience Publishers, 1953-62), and IMO remains valuable to this day!

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.

Gosh. You certainly seem to be embittered. That is especially unfortunate since this really is such a lovely subject.

About your experience in school, I'd just comment that I think it is very unfair to assume that faculty make arbitrary decisions when designing curricula. I have spent enough time as a math student (and teacher) that I think I can confidently assure you that decisions of this kind, while never easy, are not made lightly.

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.

Well, I happened to know two of the mathematicians (John Hubbard and Beverly West) who redesigned the undergraduate curriculum at Cornell two decades ago, and I know that they did not take this responsibility lightly! And decades later, I see that the math courses have been redesigned again (good, these decisions should be revisited at least five times per century), but differential forms remain firmly at the heart of the applied mathematics background for the engineering major. See http://www.engineering.cornell.edu/programs/undergraduate-education/minors/applied-mathematics.cfm: [Broken]

and note these two courses:

MATH 321 Manifolds and Differential Forms II

MATH 420 Differential Equations and Dynamical Systems

Quite frankly, I feel that this demanding curriculum is one reason why the Cornell Engineering School is one of the best: it ensures that graduates have mastered the techniques they will need to work as engineers (or to go on to graduate work in engineering).

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.

For those whose minds are not made up, I would offer an alternative take on the question of why math courses are so demanding. New math builds on old math. New ideas which rest upon old ideas are not neccessarily any harder to learn, as long as the student masters the older context first. A mathematically trained intuition is a very different thing from what a random process (natural selection) has equipped most humans with. Humans are adapted to learn, and do so very well, and many humans are probably quite capable of retraining their intuition to the point of being able to apply powerful theories like topology and the theory of manifolds in applications in physics, engineering, and other areas. But this retraining takes time.

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. This is really a problem too big for the universities, but I feel that it would be more intelligent to adjust upwards both the standard age when an educated youngish person is expected to enter the workforce, and the standard age when an oldish person is expected to retire.

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 iseverything, 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.

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.

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Hurkyl

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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

And besides, one can define sematics for a formal system

But that's not the main reason I'm responding...

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... Feynmann even tells a story how his (mathematical) colleagues would come to him and describe whatever scenario they had been working, and Feynmann 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.

Of course, one of the strengths of the axiomatic method is that it

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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.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.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.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.

My undergraduate degree was in applied mathematics, and I consider myself an applied mathematician. I see mathematics as a disipline to be learned, studied and indeed advanced in the context of problems, be they from physics, chemistry, statistics, or even philosophical questions. In retrospect I see my degree choice as being a very good one over physics, engineering or even theoretical physics.

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.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 [tex]dx + dy[/tex] to rise up and walk the earth once more.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.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 understand that in the United States, when people finish their degree, they go on to do

Most topics would only really require a good solid week in a workshop anyway. Differential forms for example. I spent about a month dipping in and out of it. To be honest I don't think a huge amount more is required in most fields, especially if you may not end up using the topic much. Not just differential forms, any topic. I don't agree with spending ten years in classes. I think you learn more out of them than in, on your own initiative of course.

I blame the mathematicians. They're not precise enough.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.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 systemin terms of the syntax itself, so you can't say that any formalism is inherently devoid of semantics.

TheI'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...

In this regard, even though the concept of an open set is perhaps intuative, I need a precise and

For quite a while, I took a

This is only one of the many examples where topology books resemble more a house of mirrors than what they should resemble, which is "Calculus on Manifolds". Definitions, examples and exercises. Explanations wouldn't go amiss either.

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by Tevian Dray and Corinne A. Manogue (24 September 1998)

(http://www.math.oregonstate.edu/bridge/papers/calculus.pdf)

BRIDGING THE VECTOR (CALCULUS) GAP by TEVIAN DRAY and CORINNE A. MANOGUE

(http://www.physics.orst.edu/bridge/papers/pathways.pdf)

Bridging the Vector Calculus Gap Workshop (http://www.math.oregonstate.edu/bridge/)

Physicists and Mathematicians often use mathematics in different ways and have different viewpoints on what some mathematical symbol means and how it is to be interpreted. E.g. Mathematicians think of vectors in terms of tuples of numbers in a linear space while physicists think of something with a magnitude and a direction. Sometimes physicists create their own mathematical objects to better understand something and occasionally they muddy the water by splicing two different formalisms together such as in the use of the Pauli matrices in quantum mechanics.

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