Scalar product in spherical coordinates

[batboio]

Hello!

I seem to have a problem with spherical coordinates (they don't like me sadly) and I will try to explain it here. I need to calculate a scalar product of two vectors $\vec{x},\vec{y}$ from real 3d Euclidean space.
If we make the standard coordinate change to spherical coordinates we can calculate it just fine in terms of $\left( r, \theta, \phi \right)$.
However if we compute the metric tensor it depends on $r$ and $\theta$. So we can't use the expression $\vec{x}.\vec{y} = g_{ij} x^i y^j$ (Einstein summation used). And here is my problem. It seems that our space has transformed from flat to curved.

Now I think I understand on an intuitive level that this is like defining an atlas for a manifold but I would appreciate a little rigour. Also why one of the ways gives an answer not depending on the point we are calculating our scalar product at while the other depends on it? I suspect that it is just hidden in the bad notation and definition of the first one...

I guess I haven't written my question very clearly but any answers will be appreciated and I will try to clear things a bit if needed :)

 

[Muphrid]

In Cartesian coordinates, the basis vectors do not change with position. In spherical coordinates, they do, so to take a scalar product of two vectors, you must know what location this is taking place at.

The expression for the scalar product in terms of the metric--$x \cdot y = g_{ij} x^i y^j$--takes this into account, as the metric components are functions of position.

 

[batboio]

Yes. I also said that. However that doesn't make things more clear...

 

[chiro]

Do you understand how g_ij is derived by relating tangential vectors of one basis to another?