Math Newb Wants to know what a Tensor is

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Discussion Overview

The discussion revolves around the concept of tensors, exploring their definitions, properties, and applications. Participants provide various explanations and examples, addressing both mathematical and physical contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants describe tensors as generalizations of vectors, emphasizing that they change homogeneously under coordinate transformations.
  • It is noted that a vector is a tensor of rank 1, while scalars can be viewed as tensors of rank 0.
  • One participant explains that tensors must obey specific relationships when changing coordinate systems, depending on their covariant and contravariant orders.
  • Another viewpoint suggests that tensors can be thought of as mathematical machines that take vectors and produce numbers or other vectors, with examples of different ranks provided.
  • One participant mentions the metric tensor as a specific example, highlighting its role in the dot product of vectors and its complexity in curved spaces.
  • There is a discussion about the linearity of tensors and the existence of different types, such as affine tensors and Cartesian tensors, which transform under specific conditions.
  • Some participants express appreciation for the clarity of explanations provided, while others share additional resources for further reading.

Areas of Agreement / Disagreement

Participants generally agree on the foundational aspects of tensors but present multiple competing views on their definitions, properties, and classifications. The discussion remains unresolved regarding the nuances of tensor types and their applications.

Contextual Notes

Limitations include the omission of detailed discussions on one-forms versus vector fields and the specific transformation properties of various tensor types. Some participants acknowledge leaving out certain complexities intentionally.

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