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?

 

[blue_raver22]

Hi there,

Thank you for reaching out with your question about scalar product in spherical coordinates. I can understand how it can be confusing to work with curved coordinates, but I will do my best to provide a clear explanation for you.

Firstly, it is important to note that the scalar product is a mathematical operation that is defined in terms of a metric tensor. The metric tensor is a mathematical object that allows us to define distances and angles in a curved space. In a flat space, the metric tensor is simply the identity matrix, but in a curved space, it becomes more complicated.

In spherical coordinates, the metric tensor is indeed dependent on both r and θ, as you have correctly noted. This is because the curvature of the space is not constant, but varies depending on the distance from the origin and the angle of measurement. This is why the expression for the scalar product in spherical coordinates cannot be written as \vec{x}.\vec{y} = g_{ij} x^i y^j, as it would be in a flat space.

Instead, we must use the expression \vec{x}.\vec{y} = g_{ij} x^i y^j \sin\theta, where the extra factor of sinθ accounts for the curvature of the space. This is why the scalar product in spherical coordinates depends on the point at which it is calculated, as the curvature of the space varies at different points.

I hope this explanation helps to clarify any confusion you may have had. Please let me know if you have any further questions or if I can provide any additional information. Keep up the good work in your studies!