
#55
Oct906, 07:02 AM

P: 406

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. 



#56
Dec106, 05:20 PM

Sci Advisor
P: 2,341

Hi, OMF,
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 for my modest attempt to describe a few of the applications I myself use most often. 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). http://www.google.com/advanced_searc...stabrook&hl=en Chris Hillman 



#57
Dec206, 08:24 AM

P: 124

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 wellknown 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 ZhongHeng [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/~mstone5...es/bmaster.pdf '…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 pforms is slowly supplanting traditional vector calculus, much as Willard Gibbs’ vector calculus supplanted the tedious componentbycomponent 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 vectorvalued function v = v(x) .' Hestenes, 1993, Differential Forms in Geometric Calculus. http://modelingnts.la.asu.edu/pdf/DIF_FORM.pdf 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 otherwiseexcellent 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. 



#58
Dec206, 03:19 PM

Sci Advisor
P: 2,341

Hi, Ron,
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 



#59
Dec506, 07:59 AM

P: 406

[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; 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. 



#60
Dec506, 09:46 AM

P: 124

Ron. 



#61
Dec506, 03:19 PM

P: 255

I must say, though, having just recently read The Large Scale Structure of SpaceTime by Ellis and Hawking that a good knowledge of forms (and other elements of differential geometry) are essential to the understanding of GR. 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 windowdressing than anything substantial. But, every so often a big theorem comes into view: I'm thinking of two within my mathematical career: the FermatWiles 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. 



#62
Dec606, 12:04 AM

Sci Advisor
HW Helper
P: 2,589

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



#63
Dec606, 04:02 AM

P: 406





#64
Dec606, 04:54 AM

P: 255





#65
Dec606, 06:25 AM

P: 406





#66
Dec606, 01:01 PM

P: 255





#67
Dec606, 09:30 PM

Sci Advisor
HW Helper
P: 9,421

as my 8th grade teacher used to say about our reaction to the class clown: "you're only encouraging him."




#68
Dec606, 09:46 PM

Sci Advisor
P: 2,341

Oh dear, I wrote a long reply to OMF, then belatedly noticed a crucial remark:
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 CartanAlbert Einstein : letters on absolute parallelism, 19291932. 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, 2529 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, 195362), and IMO remains valuable to this day! 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. 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). 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. 



#69
Dec806, 06:29 AM

Emeritus
Sci Advisor
PF Gold
P: 16,101

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. But that's not the main reason I'm responding... Of course, one of the strengths of the axiomatic method is that it is syntactic, allowing the reader to interpret it in whatever context he desires. I guess, though, that causes a problem for a reader uninterested in forming those interpretations for himself, despite demanding they exist. 



#70
Dec806, 07:22 PM

P: 406

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 understand that in the United States, when people finish their degree, they go on to do six years of coursework to obtain a Phd! I would strongly disagree with this. This is far too much to ask anyone to do. Where I am, the regieme is that Phd's are granted through research. Your research could be, usually, between three and five years. In that time, you truely do learn the skills of your trade, and I can personally say I learned at lot faster, and a lot more by researching than I ever did taking classes. 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. In this regard, even though the concept of an open set is perhaps intuative, I need a precise and clear definition to move on. Most topology books, in fact every topology book I have ever read, fails to meet this criterea. While the definitions are probably precise, they are as far from clear and intuative as it is possible to be. For quite a while, I took a compact set to be a single point or element, as the definition given was; "A set is compact if every open cover has a finite subcover". Seeing this in the context of the support of the delta distibution, I took the definition straightforwardly as decribing a point, as the author had used the strict subset notation when describing a subcover. The author has sacraficed clarity for terseness, unneccesarily in my opinion. Simply stating "A set is compact if every open cover is either finite or has a finite subcover", would be a perfectly clear definition where of course the compactness of paticluar sets could be inferred immediately without invoking subcovers, and where notational laxity would not cause problems later down the road. 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. 



#71
Apr1809, 12:31 PM

P: 64

You are not alone. See THE VECTOR CALCULUS GAP: Mathematics not= Physics
by Tevian Dray and Corinne A. Manogue (24 September 1998) (http://www.math.oregonstate.edu/brid...s/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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