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The values of the "putative function" do not depend on the chosen coordinate system. This is the essence of notions like scalar, vector, tensor, that they do n
by adrian_b 2y ago
The values of the "putative function" do not depend on the chosen coordinate system.
This is the essence of notions like scalar, vector, tensor, that they do not depend on the chosen coordinate system.
Only their numeric representations associated with a chosen coordinate system do depend on that system.
If you compute some arbitrary functions of the numeric components of a tensor in a certain coordinate system, in most cases the array of numbers that composes the result will not be a tensor, precisely because the result will really be different in any other coordinate system, while a tensor must be invariant.
All physical laws are formulated only using various kinds of tensors, including vectors and scalars, precisely because they must be invariant at the choice of the coordinate system.
- cfgauss2718 2y agoHere here! Functions do not depend on your choice of coordinates, only the components of tensors do! I think this is why it’s important to keep covariance and contravariance in mind. While tensor(fields) do not depend on coordinates intrinsically, the way we represent them when doing calculations most certainly does, and this is usefully characterized by co/contravariance.
- Koshkin 2y ago"Here here!" is indeed topological. "Hear, hear!", on the other hand, is conversational.
- cfgauss2718 2y agoI love when grammatical mistakes become unintentional puns
- prof-dr-ir 2y agoIn the end tensor fields are sections of a bundle. I insist that calling them "just functions" is simplistic. In fact, I'd say that the complexity of your elaborations kind of proves my point. Note that I deliberately use "simplistic" and not "wrong", since a section is a function of sorts.