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In the following notes, http://www.maths.manchester.ac.uk/~wparnell/MT20401/MT20401_lecture3.pdf

$\frac{\partial \vec{r}} {\partial q_i}$ forms a basis set for the vector space. How does this happen? (proof sketch(?))

And how does one justify this equation from classical Mechanics using the method above ? (from goldstein'e classical mechanics book)

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