First Approach to Differential Forms

[Math Amateur]

Can anyone suggest a good text or a good online set of notes from which to make a first approach to the topic of differential forms ... ?

Similarly a first approach to to tensors ... ?

The thought is to use these notions in order to gain an understanding of differential geometry and ... later ... differential topology ...

Any suggestions ... ?

Peter

 

[Fantini]

While not a book on differential forms, I recommend J. J. Duistermaat and J. A. C. Kolk's two-volume set Multidimensional Real Analysis. It aims to teach you analysis in $\mathbb{R}^n$ but it is just as much a beginning differential geometry book. Differential forms will be touched upon in the eighth chapter, in the second volume, when the authors discuss Stokes's theorem.

 

[Math Amateur]

While not a book on differential forms, I recommend J. J. Duistermaat and J. A. C. Kolk's two-volume set Multidimensional Real Analysis. It aims to teach you analysis in $\mathbb{R}^n$ but it is just as much a beginning differential geometry book. Differential forms will be touched upon in the eighth chapter, in the second volume, when the authors discuss Stokes's theorem.

Thanks Fantini ... on the basis of your advice I will purchase a copy of the set ...

Peter

 

[Math Amateur]

While not a book on differential forms, I recommend J. J. Duistermaat and J. A. C. Kolk's two-volume set Multidimensional Real Analysis. It aims to teach you analysis in $\mathbb{R}^n$ but it is just as much a beginning differential geometry book. Differential forms will be touched upon in the eighth chapter, in the second volume, when the authors discuss Stokes's theorem.

Hi Fantini,

I was about to purchase your suggested set of books ... but then I read the (reasonably complimentary) review ... ... as follows:

" ... ... hink of this two volume series as the Mother of All Multivariable Calculus books. It's NOT an intro to multivariable calculus for someone who has finished a couple semesters of calculus; you'll need a good stiff course (see my review of Derivatives and Integrals of Multivariable Functions by Guzman) in m.v. calculus, a dose of linear algebra, and mathematical maturity at the junior/senior undergrad level to tackle it. But if you want to go deeper - much deeper - than a first course in multivariable calculus, this is a great book.

You can look inside the book and see the contents for yourself, so I'll limit myself to general comments. The book is refreshingly free of errors (a few trivial typos are about the extent of them, at least as far as I've gotten) and well translated from the Dutch. There are hundreds of problems after the main text; although solutions aren't given in general, the material is well enough explained so that the reader should be able to solve them. (one of the authors has a website giving corrections to the text plus some solutions - see [...] ).

All theorems are proved in full detail, but be aware of two things: first, the proofs are "mathematicians proofs"; short and slick methods are favored over pedagogically softer ones. (example: one of the main theorems of m.v. calculus is the chain rule. Most undergrad texts would simply prove it head on from the definitions of derivative and composition of functions, but here the authors rely on a slick piece of machinery in the form of something called Hadamard's theorem.) Second, the reader WILL have to take out her pencil and paper and fill in some details, but the good news is that the text gives enough information to make this possible in all cases.

It's a beautiful book, really - so why only 4 stars? It has to do with the old tension between the reader and writer of math texts: how explicit should the writing be? I'm dubious about the educational value of having to fill in the algebraic details, and in any case I think the text should be as explicit as possible, leaving the reader to develop his mathematical muscles in the exercises. There are a number of points where 5% more explanation by the authors would have saved the reader 50% of the effort in understanding a theorem - so I'm witholding one star as a protest. Mathematical authors beware! ... ... "I like maths books that give detailed and easy to understand proofs ... yet the review above states:

" ... ... All theorems are proved in full detail, but be aware of two things: first, the proofs are "mathematicians proofs"; short and slick methods are favored over pedagogically softer ones. (example: one of the main theorems of m.v. calculus is the chain rule. Most undergrad texts would simply prove it head on from the definitions of derivative and composition of functions, but here the authors rely on a slick piece of machinery in the form of something called Hadamard's theorem.) Second, the reader WILL have to take out her pencil and paper and fill in some details, but the good news is that the text gives enough information to make this possible in all cases. ... ... "Can you comment ... is the point made above about short and slick methods true and fair ... ?

I am holding off my purchase pending your advice ... ...

Peter

 

[Fantini]

The whole set is expensive, so perhaps if you stick to your decision of acquiring it you should buy only volume one at first.

As for the proofs, his proofs and methods are not all short and slick. In fact, he is rarely short and slick. He prepares the chain rule for three sections before getting into the actual result, and by then it will feel very naturally. The book has plenty of computational examples. He makes frequent interludes to help the reader acquire the necessary background knowledge if he does not have it already. For instance, the chapter on differentiable mappings opens up with a whole section on linear algebra.

This is the chapter breakdown of volume one:

Chapter 1 deals with continuity, which is basically topology in $\mathbb{R}^n$. It is short by its nature, most proofs of the basic results are very simple. He still presents nice examples and computations.

Chapter 2 deals with differentiable mappings. He takes his time to develop linear algebra tools and every proposition so that the whole idea of differentiation becomes natural. He discusses higher-order derivatives and a thorough examination of the interchange of limit operations (limits and derivatives, limits and integration, integration and derivatives) with a very nice set of examples.

Chapter 3 deals with the inverse and implicit function theorems. It follows a nice format: one section for motivation, one section for the proof (which is fairly drawn out) and one section for applications. Thus one learns not only why the theorems are important, but the proofs are detailed and you learn to apply it in various conditions.

Chapter 4 deals with manifolds in $\mathbb{R}^n$. He defines manifolds in four ways and exemplifies the situations where each description is most useful. He follows the format of the third chapter when he discusses the immersion and submersion theorems. His discussion of Morse's lemma is okay.

Chapter 5 deals with tangent spaces. This chapter is basically an introduction to differential geometry. It opens up by discussing tangent spaces and giving NINE examples of construction of nontrivial tangent spaces (such as the cycloid and Descartes's folium). He proves and applies the method of Lagrange multipliers and revisits critical points. Then you deal with more differential geometry specific topics such as Gaussian curvature, curvature and torsion of curves, linear Lie groups and Lie algebras.

The problem sets for each chapter are varied and very complete. Most difficult problems are very detailed and broken down, as much as eleven items. He fits nice discussions and investigations into the exercise chapters, such as:

Hilbert's space-filling curve, Weierstrass's approximation theorem on $\mathbb{R}$, an analog of Rolle's theorem, Casimir and Euler operators, Cartan decomposition, Laplace integrals, Airy function, partial derivatives in arbitrary coordinates, confocal coordinates, moving frame, gradient in arbitrary coordinates, divergence in arbitrary coordinates, Laplacian in arbitrary coordinates, the cardioid, quadrics, the rotation group of $\mathbb{R}^n$, the special linear group, the Hopf fibration, stereographic projection, Steiner's Roman surface, Cayley's surface, Whitney's umbrella, the tangent bundle of a submanifold, the lemniscate, the astroid, Diocles's cissoid, conchoid and trisection of angles, Villarceau's circles, spherical trigonometry, Mercator projection of sphere onto cylinder, Steiner's hypocycloid, and much more.

That's my argument.

I browsed my library and I found another book which, for differential forms, might be better suited to you: Steven Weintraub Differential Forms, Theory and Practice. :)

 

[Math Amateur]

The whole set is expensive, so perhaps if you stick to your decision of acquiring it you should buy only volume one at first.

As for the proofs, his proofs and methods are not all short and slick. In fact, he is rarely short and slick. He prepares the chain rule for three sections before getting into the actual result, and by then it will feel very naturally. The book has plenty of computational examples. He makes frequent interludes to help the reader acquire the necessary background knowledge if he does not have it already. For instance, the chapter on differentiable mappings opens up with a whole section on linear algebra.

This is the chapter breakdown of volume one:

Chapter 1 deals with continuity, which is basically topology in $\mathbb{R}^n$. It is short by its nature, most proofs of the basic results are very simple. He still presents nice examples and computations.

Chapter 2 deals with differentiable mappings. He takes his time to develop linear algebra tools and every proposition so that the whole idea of differentiation becomes natural. He discusses higher-order derivatives and a thorough examination of the interchange of limit operations (limits and derivatives, limits and integration, integration and derivatives) with a very nice set of examples.

Chapter 3 deals with the inverse and implicit function theorems. It follows a nice format: one section for motivation, one section for the proof (which is fairly drawn out) and one section for applications. Thus one learns not only why the theorems are important, but the proofs are detailed and you learn to apply it in various conditions.

Chapter 4 deals with manifolds in $\mathbb{R}^n$. He defines manifolds in four ways and exemplifies the situations where each description is most useful. He follows the format of the third chapter when he discusses the immersion and submersion theorems. His discussion of Morse's lemma is okay.

Chapter 5 deals with tangent spaces. This chapter is basically an introduction to differential geometry. It opens up by discussing tangent spaces and giving NINE examples of construction of nontrivial tangent spaces (such as the cycloid and Descartes's folium). He proves and applies the method of Lagrange multipliers and revisits critical points. Then you deal with more differential geometry specific topics such as Gaussian curvature, curvature and torsion of curves, linear Lie groups and Lie algebras.

The problem sets for each chapter are varied and very complete. Most difficult problems are very detailed and broken down, as much as eleven items. He fits nice discussions and investigations into the exercise chapters, such as:

Hilbert's space-filling curve, Weierstrass's approximation theorem on $\mathbb{R}$, an analog of Rolle's theorem, Casimir and Euler operators, Cartan decomposition, Laplace integrals, Airy function, partial derivatives in arbitrary coordinates, confocal coordinates, moving frame, gradient in arbitrary coordinates, divergence in arbitrary coordinates, Laplacian in arbitrary coordinates, the cardioid, quadrics, the rotation group of $\mathbb{R}^n$, the special linear group, the Hopf fibration, stereographic projection, Steiner's Roman surface, Cayley's surface, Whitney's umbrella, the tangent bundle of a submanifold, the lemniscate, the astroid, Diocles's cissoid, conchoid and trisection of angles, Villarceau's circles, spherical trigonometry, Mercator projection of sphere onto cylinder, Steiner's hypocycloid, and much more.

That's my argument.

I browsed my library and I found another book which, for differential forms, might be better suited to you: Steven Weintraub Differential Forms, Theory and Practice. :)

Thanks for for the detailed assessment ... Most helpful ... based on what you have said I will definitely add the books to my library ...

Thanks again for the help in this matter ...

Peter

 

[Math Amateur]

The whole set is expensive, so perhaps if you stick to your decision of acquiring it you should buy only volume one at first.

As for the proofs, his proofs and methods are not all short and slick. In fact, he is rarely short and slick. He prepares the chain rule for three sections before getting into the actual result, and by then it will feel very naturally. The book has plenty of computational examples. He makes frequent interludes to help the reader acquire the necessary background knowledge if he does not have it already. For instance, the chapter on differentiable mappings opens up with a whole section on linear algebra.

This is the chapter breakdown of volume one:

Chapter 1 deals with continuity, which is basically topology in $\mathbb{R}^n$. It is short by its nature, most proofs of the basic results are very simple. He still presents nice examples and computations.

Chapter 2 deals with differentiable mappings. He takes his time to develop linear algebra tools and every proposition so that the whole idea of differentiation becomes natural. He discusses higher-order derivatives and a thorough examination of the interchange of limit operations (limits and derivatives, limits and integration, integration and derivatives) with a very nice set of examples.

Chapter 3 deals with the inverse and implicit function theorems. It follows a nice format: one section for motivation, one section for the proof (which is fairly drawn out) and one section for applications. Thus one learns not only why the theorems are important, but the proofs are detailed and you learn to apply it in various conditions.

Chapter 4 deals with manifolds in $\mathbb{R}^n$. He defines manifolds in four ways and exemplifies the situations where each description is most useful. He follows the format of the third chapter when he discusses the immersion and submersion theorems. His discussion of Morse's lemma is okay.

Chapter 5 deals with tangent spaces. This chapter is basically an introduction to differential geometry. It opens up by discussing tangent spaces and giving NINE examples of construction of nontrivial tangent spaces (such as the cycloid and Descartes's folium). He proves and applies the method of Lagrange multipliers and revisits critical points. Then you deal with more differential geometry specific topics such as Gaussian curvature, curvature and torsion of curves, linear Lie groups and Lie algebras.

The problem sets for each chapter are varied and very complete. Most difficult problems are very detailed and broken down, as much as eleven items. He fits nice discussions and investigations into the exercise chapters, such as:

Hilbert's space-filling curve, Weierstrass's approximation theorem on $\mathbb{R}$, an analog of Rolle's theorem, Casimir and Euler operators, Cartan decomposition, Laplace integrals, Airy function, partial derivatives in arbitrary coordinates, confocal coordinates, moving frame, gradient in arbitrary coordinates, divergence in arbitrary coordinates, Laplacian in arbitrary coordinates, the cardioid, quadrics, the rotation group of $\mathbb{R}^n$, the special linear group, the Hopf fibration, stereographic projection, Steiner's Roman surface, Cayley's surface, Whitney's umbrella, the tangent bundle of a submanifold, the lemniscate, the astroid, Diocles's cissoid, conchoid and trisection of angles, Villarceau's circles, spherical trigonometry, Mercator projection of sphere onto cylinder, Steiner's hypocycloid, and much more.

That's my argument.

I browsed my library and I found another book which, for differential forms, might be better suited to you: Steven Weintraub Differential Forms, Theory and Practice. :)

Hi Fantini ...

Interestingly ... you write ...

" ... ... I browsed my library and I found another book which, for differential forms, might be better suited to you: Steven Weintraub Differential Forms, Theory and Practice. ... ... "I do have a copy of Weintraub's book on differential forms ... and was browsing it and wondering whether to use it on my first approach to differential forms ... then I found that Weintraub made the unconventional decision not to use the wedge in his notation ... I was a bit put off by this decision as I wanted a conventional notational approach ... at least for my first approach ... but your remarks have got me wondering whether I need worry about his notational approach ... maybe he just drops the wedge from the notation as it is superfluous as he says in Remark 1.1.8 on pages 9-10 ... as follows:

https://www.physicsforums.com/attachments/5244
https://www.physicsforums.com/attachments/5245What do you think? Is this something that matters ... even for a first approach ... or do you need conventional notation for a first approach so you can recognise the theory in other books ... ?

Peter*** NOTE *** I note that Weintraub has a large number of very explicit and helpful looking computational examples ... which is a BIG plus in my opinion ...

 

[Fantini]

Sorry for not answering earlier, I didn't see you answered the thread until two days ago. The answer to your question is no. I don't think the wedge makes much of a difference, notation-wise. You know it is there and what you must pay attention is to operate with it according to the wedge rules. Otherwise it's fine.

 

[Math Amateur]

Sorry for not answering earlier, I didn't see you answered the thread until two days ago. The answer to your question is no. I don't think the wedge makes much of a difference, notation-wise. You know it is there and what you must pay attention is to operate with it according to the wedge rules. Otherwise it's fine.

So, I think you are saying that it is fine to use Weintraub as a text even though he does not use an explicit wedge in his notation ... is that right ...

Peter

 

[Fantini]

Yes, that is what I'm saying. It is fine to use Weintraub even as he doesn't use the wedge notation. :)

 

[steenis]

I am probably too late with my contribution to this discussion, but you might consider, or have a look, at these books:

Bachman - A Geometric Approach to Differential Forms (2nd,2012)

Bressoud - Second Year Calculus; From Celestial Mechanics to Special Relativity (1991) (I like this book)

Edwards - Advanced Calculus; A Differential Forms Approach (reprint,2014) (easy)

Flanders - Differential Forms with Applications to the Physical Sciences (1963) (Maybe too difficult)