- #1

- 713

- 5

some time ago I read that if we know the metric tensor [itex]g_{ij}[/itex] associated with a change of coordinates [itex]\phi[/itex], it is possible to calculate the (Euclidean?) inner product in a way that is invariant to the parametrization. Essentially the inner product was defined in terms of the metric tensor as: [tex]g_{ij}a^i b^j[/tex] using Einstein notation (see here).

I understand Einstein summation convention, but the problem that I have here is that this formula looks totally ambiguous to me. In fact, the article speaks about

*curvilinear*coordinates, thus the metric tensor [itex]g_{ij}[/itex] is inevitably position-dependent. When I see [itex]g_{ij}[/itex] I interpret it as [itex]g_{ij}(u_1,\ldots,u_n)=g_{ij}(\mathbf{u})[/itex]. The formula above does not say *what* coordinates we have to plug into

**u**.

I would really like to see how someone uses the formula above to calculate the inner product between two vectors in ℝ

^{2}expressed in polar coordinates: [itex](r_1,\theta_1)=(1,0)[/itex] and [itex](r_2, \theta_2)=(1,\frac{\pi}{2})[/itex]. The result should be 0.