What is a tensor and why is it useful?

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What is a tensor and why is it useful?

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HallsofIvy
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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.

mathwonk