What is a tensor and why is it useful?
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?
mathwonk wrote a long post on tensors here:
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