I was watching a lecture in which the professor states that by taking the derivative of a vector function, one can find the basis of the coordinate system being studied. My question is, how can a vector function define a coordinate system? Does it not need the coordinate system in the first place so that we can study it?
I know vectors are not the numerical representations we assign to them, they are geometric entities of their own, and a vector valued function goes around every point in a vector space and assigns it a vector, but how can this function work if its arguments are described by the coordinate system? What is the explanation that makes this seemingly circular logic go away?
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Barkimedes
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I didn't watch the video, just looked at a couple of stills in the middle, but it looks like what he's doing is taking coordinate functions for a manifold (in this case the manifold is just the plane and the coordinates are polar coordinates) and using them to give a basis for the tangent space to the manifold at a given point.
Since the tangent space to the plane at a point is a two-dimensional vector space looking very much like the plane itself, it's easy to get the two mixed up. But they're fundamentally different sorts of things.
Daniel McLaury
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