Calculations with tensors in modern notation

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Discussion Overview

The discussion revolves around the topic of performing calculations with tensors in modern notation, exploring different approaches and resources for understanding tensor analysis. Participants express interest in both theoretical and applied aspects of tensor calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about books that focus on calculations with tensors in modern notation.
  • Another participant recommends a textbook by @Orodruin, highlighting its chapter on tensors and suggesting the "Look Inside" feature on Amazon for further exploration.
  • A question is posed regarding the definition of modern notation, indicating a lack of clarity on the term.
  • A participant expresses uncertainty about the definition of modern notation, suggesting it may relate to a modern formulation of differential geometry, while emphasizing a preference for practical calculations over proofs.
  • Another participant suggests the first few chapters of "The Geometry of Physics" by Frankel as a potential resource.
  • A later reply contrasts the modern approach to tensor analysis through Cartan theory with the Ricci calculus commonly used by physicists, noting the advantages and disadvantages of both methods.
  • The participant mentions that the representation-free formulations of mathematicians may simplify certain calculations, while the Ricci calculus focuses on components and standard calculus applications.

Areas of Agreement / Disagreement

Participants express varying opinions on the definition of modern notation and the approaches to tensor analysis, indicating that multiple competing views remain without a consensus on the best method or resource.

Contextual Notes

There is uncertainty regarding the specific characteristics of modern notation and its implications for tensor calculations. The discussion reflects differing preferences for theoretical versus applied approaches, as well as the potential limitations of each method mentioned.

Who May Find This Useful

This discussion may be of interest to students and professionals in physics and mathematics who are looking to deepen their understanding of tensor calculations and explore different methodologies in tensor analysis.

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

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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.
 
caz said:
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.
 
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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.
 
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