Best way to teach myself differential forms?

[MichaelBack12]

Any suggestions? Online courses or videos?

 

[jedishrfu]

Here's an earlier thread that asks the same question:

https://www.physicsforums.com/threads/best-books-for-learning-differential-forms.592560/

and I found this recent book on Amazon with good reviews:

https://www.amazon.com/dp/0957389477/?tag=pfamazon01-20

 

[jedishrfu]

And there Harley Flanders book:

https://www.google.com/books/editio...rential+forms+book&pg=PA4&printsec=frontcover

and this video sequence on Youtube by Metzler:

 

[fresh_42]

Any suggestions? Online courses or videos?

It depends a bit on where you are heading to. We have physics, analysis, algebraic topology, category theory, abstract algebra, and differential geometry on our list.

 

[martinbn]

It is hard to say what is best for you if we don't know much about you. What do you know already and what are your goals.

 

[MichaelBack12]

Masters in electrical engineering, focus on EM fields. But been a very long time since I graduated.

 

[vanhees71]

Then perhaps a nice way to get introduced to differential forms is

F. W. Hehl, Y. N. Obkunov, Foundations of classical electrodynamics, Springer (2003)

 

[MichaelBack12]

Great recommendation. Could you also suggest a differential geometry text for someone with zero background?

 

[fresh_42]

Great recommendation. Could you also suggest a differential geometry text for someone with zero background?

This is a nice one:
https://www.amazon.de/dp/0387903577/

 

[ergospherical]

Ted Shifrin's notes are good, being hands-on and example-heavy (with "mainstream" texts it's easy to get lost in very abstract concepts [not necessarily a bad thing, but...]). You can probably sprint/skip through the a fair amount of the first couple of chapters depending on how familiar you are with the general theory of curves and surfaces.

http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

 

[Mondayman]

I am not familiar with the topic myself. But there is a new book that looks good by Tristan Needham. Seems well written from the preview.

https://www.amazon.ca/dp/0691203709/

 

[MichaelBack12]

I had not noticed that one. Looking at the preview pages right now. Thanks very much.

 

[robphy]

There are some nice videos by eigenchris on YouTube
https://www.physicsforums.com/threads/videos-eigenchris-relativity-tensors.1011495/

and

"Volume 4" Fields by Prof Robert Ghrist on YouTube
https://www2.math.upenn.edu/~ghrist/BLUE.html



and...

probably not for beginners...
but may be enlightening ( might be good to skim periodically as you learn )
"Div Grad and Curl are Dead" by William Burke ( https://www.ucolick.org/~burke/home.html )
https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf

 

[jedishrfu]

There are some nice videos by eigenchris on YouTube
https://www.physicsforums.com/threads/videos-eigenchris-relativity-tensors.1011495/

and

"Volume 4" Fields by Prof Robert Ghrist on YouTube
https://www2.math.upenn.edu/~ghrist/BLUE.html



and...

probably not for beginners...
but may be enlightening ( might be good to skim periodically as you learn )
"Div Grad and Curl are Dead" by William Burke ( https://www.ucolick.org/~burke/home.html )
https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf

Perusing WIlliam Burke's website, I found he had passed away unexpectedly due to a cervical fracture in 1996. It's ironic and sad that he was drafting a book that never got published but at least we have it thanks to the internet and his college for maintaining his presence.

There's another book titled Div Grad Curl and All That by HM Schey which covers topics in EM theory but sadly doesn't cover Differential Forms.

https://archive.org/details/H.M.ScheyDivGradCurlAndAllThat

 

[atyy]

Great recommendation. Could you also suggest a differential geometry text for someone with zero background?

Fecko's book is quite nice.
https://www.amazon.com/dp/0521187966/?tag=pfamazon01-20

 

[robphy]

Perusing WIlliam Burke's website, I found he had passed away unexpectedly due to a cervical fracture in 1996. It's ironic and sad that he was drafting a book that never got published but at least we have it thanks to the internet and his college for maintaining his presence.

There's another book titled Div Grad Curl and All That by HM Schey which covers topics in EM theory but sadly doesn't cover Differential Forms.

https://archive.org/details/H.M.ScheyDivGradCurlAndAllThat

Schey's "Div Grad Curl and All That" is a famous reference for students looking to learn vector calculus
and for scientists looking to use the "and All That" in the title of their own presentations.

Burke was advocating to move on from vector calculus

In 1978 Walter Thirring wrote in his multivolume series on theoretical physics: --
"The best and latest mathematical methods to appear on the market have been used whenever possible.
In doing this many an old and trusted favorite of the older generation as been forsaken, as I deemed it best not to hand dull and worn-out tools down to the next generation."

It is now 1993, and we are still teaching and using the old clumsy tools.
As much as I would like to, I am not trying to bury vector calculus.
Vector calculus will be longer lived than the typewriter keyboard or Fortran.
My goal is to provide support for those students who want to learn the modern methods,
but whose textbooks and teachers can provide no help.

Some of Burke's ideas on differential forms have been published
https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20 "Applied Differential Geometry"
https://www.amazon.com/dp/0486845583/?tag=pfamazon01-20 "Spacetime, Geometry, Cosmology" https://doi.org/10.1063/1.525603 Manifestly parity invariant electromagnetic theory and twisted tensors - J. Math. Phys. 24, 65 (1983)
and (although not about differential forms) https://scott.physics.ucsc.edu/pdf/primer.pdf - Special Relativity Primer

I first got interested in differential forms from the presentation in Gravitation (by Misner Thorne Wheeler [MTW]).
Burke's books and articles helped me better understand that presentation from MTW
by giving constructions for drawing and calculating with differential forms.
You can find a link to an old poster of mine "Visualizing Tensors" (with references)
http://www.opensourcephysics.org/CPC/posters/salgado-talk.pdf
at http://www.opensourcephysics.org/CPC/abstracts_contributed.html
(In advocating the use of differential forms, I am actively trying to show how to visualize and calculate with them in physics (e.g. relativity, electromagnetism, mechanics, and thermodynamics).)

Other alternatives:
Intended as an advanced introductory course, it might be too advanced for a beginner:
Bamberg and Sternberg - A course in Mathematics for Students of Physics (Vols 1 and 2)
https://www.amazon.com/dp/0521406498/?tag=pfamazon01-20
https://www.amazon.com/dp/0521406501/?tag=pfamazon01-20
[I'm trying to absorb the ideas in the Thermodynamics and Circuit Analysis chapters.]

https://scholarsarchive.byu.edu/facpub/669/
Teaching electromagnetic field theory using differential forms
(by Karl F. Warnick, Richard H. Selfridge, David V. Arnold )
see also their course notes: http://eceformsweb.groups.et.byu.net/

A Visual Introduction to Differential Forms and Calculus on Manifolds (by Fortney)
https://www.amazon.com/dp/3319969919/?tag=pfamazon01-20

 

[Frabjous]

Bachman
https://www.amazon.com/dp/0817683038/?tag=pfamazon01-20

 

[robphy]

Bachman
https://www.amazon.com/dp/0817683038/?tag=pfamazon01-20

Related:
https://www.physicsforums.com/threa...to-differential-forms-by-david-bachman.67268/

 

[Dr Transport]

An interesting take on this, i bought it last week because I needed another item for free shipping. The end of the book deals with differential forms in physics

https://www.amazon.com/dp/B08Y5DVT62/?tag=pfamazon01-20

 

[Frabjous]

An interesting take on this, i bought it last week because I needed another item for free shipping. The end of the book deals with differential forms in physics

https://www.amazon.com/dp/B08Y5DVT62/?tag=pfamazon01-20

What are your thoughts on the book in general? While it looks tempting, I have never seen a rigorous review.

 

[Dr Transport]

What are your thoughts on the book? While it looks tempting, I have never seen a rigorous review.

So new, and I've only had the time to read thru chapters 1 & 2.

 

[MichaelBack12]

Thanks very much. All good suggestions.

An interesting take on this, i bought it last week because I needed another item for free shipping. The end of the book deals with differential forms in physics

https://www.amazon.com/dp/B08Y5DVT62/?tag=pfamazon01-20

that does look good. Just bought it.

 

[Coelum]

Any suggestions? Online courses or videos?

Hi Michael,
I am in your position, too: I graduated in Engineering many years ago. Last year I decided to understand differential forms. After some research, I settled on Shifrin's video lectures () and bought his book (https://www.amazon.com/dp/047152638X/?tag=pfamazon01-20), which is perfectly aligned with the lectures and contains a lot of good exercises, many with solutions. A wonderful experience! HIH...

 

[MichaelBack12]

Really appreciate it. Any review or prep you would recommend before I start.

 

[Coelum]

Well, it depends on how much you remember of multivariate calculus (differentiation and integration in 3D). I started straight away with differential forms, which is 24 lectures into the course, because I had already reviewed multivariate calculus. If you need a refresh, you can start the course from the beginning. In that case, the prereq is univariate calculus only. Enjoy!

 

[Coelum]

Well, it depends on how much you remember of multivariate calculus (differentiation and integration in 3D). I started straight away with differential forms, which is 24 lectures into the course, because I had already reviewed multivariate calculus. If you need a refresh, you can start the course from the beginning. In that case, the prereq is univariate calculus only. Enjoy!

I forgot the review: one of the best classes I ever had! Clear, easy to follow, well modulated exercises. Perfect for engineers and physicists. Next step may be Shifrin's notes on Differential Geometry (http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf), that I started one week ago. No video lecture, unfortunately.

 

[jbergman]

Great recommendation. Could you also suggest a differential geometry text for someone with zero background?

I'd recommend Tu's Introduction to Manifolds to first learn about Smooth Manifolds and forms followed by Tu or Lee's book on Riemannian Geometry.

 

[MichaelBack12]

Thanks. Lee looks great.

 

[jbergman]

Thanks. Lee looks great.

Just one warning, Lee's Smooth Manifolds books is loong, though, great, which is why I would recommend Tu's book for forms and Manifolds.

Lee's Riemannian Geometry book doesn't have that problem.

 

[MichaelBack12]

Do you think I need an analysis course before reading either book?

 

[jbergman]

Do you think I need an analysis course before reading either book?

Not really. Topology is more needed than hard analysis. I believe the Tu book has an appendix, though, that should be sufficient.

 

[mathwonk]

here is another free set of lectures from a mathematician who tried to present forms to his undergraduates in an unsophisticated way:

https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf

 

[martinbn]

here is another free set of lectures from a mathematician who tried to present forms to his undergraduates in an unsophisticated way:

https://www.math.purdue.edu/~arapura/preprints/diffforms.pdf

When I saw the name of the author I expected to is an intro to differential forms in algebraic geometry, but it is actually a nice introduction to differential forms for someone with basic calculus knowledge.

 

[mathwonk]

well, (cough, cough) algebraic geometers are famous for knowing everything.
(maybe make that needing to know.)

 

[martinbn]

well, (cough, cough) algebraic geometers are famous for knowing everything.
(maybe make that needing to know.)

So, what Mumford said in the preface of his Curves and Their Jacobians book was not a joke.