Geometry Calculations with tensors in modern notation

[Frabjous]

Is there a book that emphasizes performing calculations with tensors in modern notation?

 

[berkeman]

Have you looked through the textbook written by @Orodruin yet? It has a nice chapter on tensors, as well as lots of other good subjects and treatments. Check out his Insights article about writing it:

https://www.physicsforums.com/insights/the-birth-of-a-textbook/

You can use the "Look Inside" feature at Amazon to check out the Table of Contents:

https://www.amazon.com/dp/113805688X/?tag=pfamazon01-20

 

[dextercioby]

How do you define modern notation?

 

[Frabjous]

How Socratic. I actually do not know. I have an index intensive understanding and am looking for something more. I am guessing it is a modern formulation of differential geometry, but I do not know for sure. I am an applied sort of person, so I am more interested in learning how to do calculations than in proofs.

 

[George Jones]

How Socratic. I actually do not know. I have an index intensive understanding and am looking for something more. I am guessing it is a modern formulation of differential geometry, but I do not know for sure. I am an applied sort of person, so I am more interested in learning how to do calculations than in proofs.

I am not sure what will work for you, but maybe the first few chapters of "The Geometry of Physics" by Frankel.

 

[vanhees71]

Well, that's my question too. The modern approach to tensor analysis is through Cartan theory, i.e., using (differential alternating) forms and coordinate free formulations, while physicists usually use the Ricci calculus using components and upper and lower indices. I'd say, both have their advantages and disadvantages. I'd say using the mathematicians' representation free formulations has advantages as far as formal developments are concerned (e.g., there is basically only one integral theorem, the Stokes's theorem for differential forms of arbitrary rank) and also some calculational tasks are simplified (e.g., when calculating the curvature tensor in GR; see Misner, Thorne, Wheeler), while the Ricci calculus just deals with the components and thus real functions so that you can use standard calculus.