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I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x + y + z + t, in order to recover the Cartesian basis vectors of flat spacetime? Or am I missing the point?

Thanks!

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# Partial derivatives as basis vectors?

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