@mathwonk what do you mean by thinking geometrically? Could you explain somehow? I dont understand how a geometrical picture can be associated to any space higher than 3 dimensions. And how can you get a geometric picture for some algebraic structures? I dont see any geometry in that appendix...
@mathwonk I read in many places about the importance of geometric intuition in advanced mathematics. If you read some articles and interviews with Misha Gromov, or historical articles on Rochlins and Alexandrovs geometric schools, they mention there are different ways of approaching non...
I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
Oh okay, so differential forms are a subset of multilinear forms. And more specifically differential forms are totally antisymmetric multilinear forms. So can we write any Tensor as multilinear form? Why are differential forms much more important to differential geometry than multilinear forms...
Robphy, that is a great analogy with complex numbers. So then from what I understand, a Tensor with a non-zero symmetric part cannot be written in differential forms (because forms are antisymmetric). So from my understanding, you can rewrite the antisymmetric type (0,n) tensor in a differential...
i don’t understand. Could you maybe provide an example.Maybe what the author was trying to say was a linear combination of a symmetric and antisymmetric form of itself?
Warhammer,
I am sure you will find some of the top theoretical physicists in the list of ISI highly cited researchers. Of course this may not include younger physicists but it will give you a list of more established physicists whose research has made impact. I haven't gone through it but you...
Yes differential forms are defined as antisymmetric tensors. I write here the definition 5.4.1 (page 52) in Michio Nakahara's book. " A differential form of order r or an r-form is a totally anti-symmetric tensor of type (0, r ).". Now if we quote also Gravitation by Wheeler, on page 83 they say...
I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12).
And in Topology, Geometry and Physics by Michio...
Thanks George, I appreciate your feedback. What do you mean by everything can be treated with bundles? Are bundles a kind of unifying mathematical tool? Do any of the books above take the approach to bundles?
Could you provide recommendations for a good modern introductory textbook on differential geometry, geared towards physicists. I know physicists and mathematicians do mathematics differently and I would like to see how it is done by a physicists standard. I have heard Chris Ishams “Modern Diff...
What I wrote was exactly from his book, I just paraphrased a little and didn't put it all in quotation marks. I just thought there are so many books teaching physics from a purely differential form viewpoint. But you are right, I am obviously confused for no reason.
I definitely understand that differential forms are tensors. Kip Thorne in his book "modern classical physics" uses Tensors with indices throughout. I know these are coordinate-free. He says also that he does not use differential forms in the book and leaves this "richer mathematics" to more...
Martin I want to show that my confusion isn’t unreasonable.
Many authors mention they will introduce modern differential geometry using the approach of differential forms. I mean if you go and read Kip Thorne book “Modern Classical Physics” he explicitly says he will be using Tensors throughout...
Maybe? I’m not very experienced so if you can enlighten me that would be great. I thought Tensors and Differential Forms were coordinate free until you choose a coordinate system for them. Would you say that all the books I mentioned above use coordinate free approaches. Is this the trend in...
Hi Martin, you are certainly right. Maybe my question gave the impression that I wasn’t aware of this. The main point of the question is that there are differing approaches to theoretical physics such as by differential forms and the tensors with indices. I know it sounds strange to speak of...
There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...