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I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term is more common). Basis vectors are simple enough, it's just a carry over from vector spaces in linear algebra. Cobasis vectors, on the other hand, don't seem well motivated to me. I understand that they're dual to the basis vectors, but why exactly do we need them? When should they be used over the normal basis vectors?

I'm familiar with the treatment of vectors as elements of a tangent space of a manifold and dual vectors as linear operators on vectors, if that's the sort of explanation that motivates the cobasis most effectively, although I don't have a good intuition with that sort of thing quite yet.

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I think maybe you are just wondering why we need covectors at all. Once you accept that we need covectors, it makes sense to have cobases for your covectors just like you have bases for your vectors.

Ok so why do we need covectors. I will give one application---taking a derivative. Often in differential geometry you want to take a derivative. Suppose for example you have a function defined on your manifold. What kind of object should the derivative be? Well you should give it a tangent vector to the manifold and it gives you a number saying how quickly the function changes when you move at the velocity given by that tangent vector. This relationship between tangent vector and rate of change of the function value should be linear. Therefore it must be that the derivative of our function is a covector field.

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  • $\begingroup$ You're exactly right, understanding the need for covectors is all I need! Forgive me if this is a silly question, but just to make sure I understand: in your example, would the derivative be a directional derivative in the direction of the tangent vector? That is, would the covector be the gradient of the function, which we can then multiply by the tangent vector to get the number we're looking for? $\endgroup$ – EtaZetaTheta May 14 '14 at 2:36
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    $\begingroup$ @EtaZetaTheta yes. A covector is a function that takes in vectors and spits out numbers. The gradient is most naturally viewed as (the linear extension of) a function which takes in unit vectors and spits out the rate of change of the function in that direction, which makes it a covector. $\endgroup$ – Robert Mastragostino May 14 '14 at 3:03
  • $\begingroup$ Perfect, thank you! I've always read that the gradient of a function is sort of the prototypical covariant tensor, and your example helped connect that with the role of a covector as a thing that eats vectors and spits out numbers, so that was especially helpful. $\endgroup$ – EtaZetaTheta May 14 '14 at 3:06

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