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let's assume we have an admissible change of coordinates [itex]\phi:U\rightarrow \mathbb{R}^n[/itex]. I would like to know how the inner product on ℝ

^{n}changes under this transformation. In other words, what is [itex]\left\langle \phi (u), \phi (v) \right\rangle[/itex] for some [itex]u,v \in U[/itex] ?

I thought that if we consider the two corresponding vectors [itex]x(u)[/itex] and [itex]y(v)[/itex] where [itex]x(u) = \phi_1(u)e_1 + \ldots \phi_n(u)e_n[/itex] we obtain: [tex]\left\langle x,y \right\rangle = \left\langle \phi (u), \phi (v) \right\rangle = \left\langle \phi_1(u)\phi_1(v) + \ldots \phi_n(u)\phi_n(v) \right\rangle[/tex] but for some reason (that I would like to know) this does not seem to be the usual approach.