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Jan2-12, 09:00 AM
Sci Advisor
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Quote Quote by Gadhav View Post
These terms are confusing since some books insist that Contravariant component is parallel to axes and Covariant component is perpendicular.
Quote Quote by Fredrik View Post
I don't think I've seen this claim, and I don't understand it.
He's probably talking about the ability to identify forms and tangent vectors in the presence of a metric (lowering and raising indices). Something like "As can be seen, the jth contravariant component consists of the projection of P onto the jth axis parallel to the other axis, whereas the jth covariant component consists of the projection of P into the jth axis perpendicular to that axis."

That's a different approach from what you've taken where you define forms and vectors first without a metric. Then only with a metric are we allowed to identify forms and vectors. In this approach, the metric always exists, which is an assumption of classical GR.