What is the meaning of tensor calculus?

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Discussion Overview

The discussion revolves around the meaning of tensor calculus, exploring its definitions and interpretations. Participants examine the term from various perspectives, including computational methods, algebraic properties, topological aspects, and analytical connections to other mathematical concepts.

Discussion Character

  • Exploratory
  • Conceptual clarification

Main Points Raised

  • Some participants suggest that "tensor calculus" can be understood as computational methods involving tensors.
  • Others propose that it encompasses a range of perspectives, including algebraic views related to tensor algebras and their categorical properties.
  • A different viewpoint considers the topological aspects of tensor fields, which are defined similarly to vector fields.
  • Analytical connections between tensors, tangent spaces, curvatures, and differential forms are also mentioned as part of tensor calculus.
  • One participant notes the ambiguity of the term "calculus" in different languages, suggesting it may refer to a system of calculation methods rather than strictly to analysis.

Areas of Agreement / Disagreement

Participants express differing interpretations of tensor calculus, with no consensus on a singular definition or understanding. Multiple competing views remain on the scope and meaning of the term.

Contextual Notes

The discussion highlights potential limitations in definitions and interpretations of "calculus" across languages, as well as the varying perspectives on what constitutes tensor calculus.

Vance Grey
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Vance Grey said:
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Not sure, whether this is correct what I'm going to say, because "calculus" as an English word is often used quite synonymous to "analysis" whereas in my language it means "system of calculation methods". So I interpret it in the sense of the latter and understand "tensor calculus" as computational methods where tensors are used. This could simply be how to deal with tensors, or a rather algebraic point of view if we consider tensor algebras and their categorical properties, or a topological point of view if we consider tensor fields, which are similar defined as vector fields, or analytically - and this is more likely - the connection of tensors on one hand, and tangent spaces, curvatures and differential forms on the other hand. Personally I would all of them summarize under "tensor calculus" and only spent a different amount of space for the different perspectives.
 
Vance Grey said:
https://www.physicsforums.com/attachments/205736
Link is broken...
 
fresh_42 said:
Not sure, whether this is correct what I'm going to say, because "calculus" as an English word is often used quite synonymous to "analysis" whereas in my language it means "system of calculation methods". So I interpret it in the sense of the latter and understand "tensor calculus" as computational methods where tensors are used. This could simply be how to deal with tensors, or a rather algebraic point of view if we consider tensor algebras and their categorical properties, or a topological point of view if we consider tensor fields, which are similar defined as vector fields, or analytically - and this is more likely - the connection of tensors on one hand, and tangent spaces, curvatures and differential forms on the other hand. Personally I would all of them summarize under "tensor calculus" and only spent a different amount of space for the different perspectives.
Similar in English, actually, Calculus, sometimes 'Algebra' is used, as the rules used to operate with/on the objects.
 

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