I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that displacements are contravariant vectors and gradients of scalar fields are covariant vectors, but in A Student's Guide to Vectors and Tensors by Fleisch I found this statement:
[I]t's not the vector itself that is contravariant or covariant, it's the set of components that you form through its parallel or perpendicular projections. (p.121)
If physical concepts can be represented as either type of mathematical object, I don't understand how a displacement could be represented as a covector. Wouldn't its components transform the wrong way if the coordinates were changed?
If covariantness/contravariantness is part of the definition of a physical concept, I don't understand how force is classified. The gradient of a potential would have dimensions of length in the denominator, making it a covariant vector. Mass times acceleration has dimensions of length in the numerator, making it contravariant.
Edit: I read through the accepted answer to Forces as One-Forms and Magnetism. One thing I don't understand is whether in relativistic spacetime any vector quantity can also be represented as a 1 form (because there is a metric) or whether its classification as a 1 form or vector depends on how its components change under a coordinate transformation. Doesn't a displacement have to be a vector and not a 1 form?
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