Can anybody please explain why distinguish contra-variant and covariant vectors? I don't want the description of how they transform, but I would like to know why consider the transformations at all? If someone could explain this without resorting to how they transform, but why consider them anyways, that would be incredibly helpful.

Also, I am not looking for the geometrical intuition, but I would like to know how it comes up, too?

  • $\begingroup$ the presence of these type of tensors is because we have vector duality to distinguish between vectors of a vectorspace and linear functionals on a vectorspace $\endgroup$ – janmarqz Jul 15 at 17:23

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