What is the meaning of tensor calculus?

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SUMMARY

Tensor calculus is defined as a set of computational methods that utilize tensors, encompassing various perspectives such as algebraic, topological, and analytical approaches. The discussion highlights the relationship between tensors, tangent spaces, curvatures, and differential forms, emphasizing that tensor calculus integrates these concepts. Participants noted that the term "calculus" can be synonymous with "analysis," but in this context, it refers to a systematic method of calculation involving tensors. Overall, tensor calculus serves as a foundational tool in advanced mathematics and physics.

PREREQUISITES
  • Understanding of tensor algebra
  • Familiarity with differential forms
  • Knowledge of vector fields
  • Basic concepts of topology
NEXT STEPS
  • Study the properties of tensor algebras
  • Learn about the application of tensors in differential geometry
  • Explore the relationship between tensors and curvature
  • Investigate the role of tangent spaces in tensor calculus
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Mathematicians, physicists, and students in advanced mathematics who are looking to deepen their understanding of tensor calculus and its applications in various fields.

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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