What is a tensor and why is it useful?

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Discussion Overview

The discussion centers on the concept of tensors, their definitions, and their significance in various contexts, particularly in relation to coordinate systems and the formulation of natural laws. The scope includes theoretical explanations and conceptual clarifications regarding tensors and their properties.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant defines a tensor as a generalization of a vector, explaining the hierarchy of tensors from scalars to higher-order tensors and their component structures.
  • The same participant emphasizes the importance of tensors in maintaining the validity of equations across different coordinate systems, asserting that if a tensor has zero components in one system, it will have zero components in all systems.
  • Another participant notes the extensive prior discussion on the topic, suggesting that a search for existing content on the forum may be beneficial.
  • A further participant references a specific long post on tensors, indicating that there is additional detailed information available on the forum.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the definition and implications of tensors, as the discussion includes varying levels of detail and references to previous discussions without resolving the complexities involved.

Contextual Notes

The discussion may be limited by the assumptions underlying the definitions of tensors and the context in which they are applied, as well as the potential for differing interpretations of their significance in various fields.

relativitydude
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What is a tensor and why is it useful?
 
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A "Tensor" is a generalization of "vector". In three dimensional space, a "scalar" (zero order tensor) has one component (number), a vector (first order tensor) has 3, a second order tensor has 9, etc. But they can't be just any numbers. The basic concept is that if you change coordinate systems, the components of the tensor change "homogeneously"- basically that means each component in the new coordinates is a sum of products of numbers with the components of the tensor in the old coordinates.
The point of that is: If a tensor has all its components 0 in one coordinate system, then (since we are multiplying all those numbers by the 0 components) it has all components 0 in any coordinate system.

Why is that important? Because coordinate systems are not "natural"- we make them up ourselves- so any equations representing "natural laws" should be independent of the coordinates system. If I can write a "natural law" as A= B where A and B are both tensors, that is the same as A-B= 0. And if that is true in one coordinates system, then it is true in all. That is: if a tensor equation is true in any coordinate system, then it is true in all coordinate systems.
 
young person, there have been approximately 100,000 words written on this question in the last 9 months here. is it possible to search the site on this topic?
 
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