Scalar product in spherical coordinates

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Discussion Overview

The discussion revolves around the calculation of the scalar product of two vectors in spherical coordinates, particularly focusing on the implications of using a metric tensor that varies with position. Participants explore the differences between Cartesian and spherical coordinates in this context, and the challenges that arise from the curvature of the space represented in spherical coordinates.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in calculating the scalar product of vectors in spherical coordinates, noting that the metric tensor depends on the coordinates, which complicates the calculation.
  • Another participant points out that in Cartesian coordinates, basis vectors remain constant, whereas in spherical coordinates, they vary with position, which necessitates knowing the location for the scalar product.
  • A participant acknowledges the previous point but indicates that it does not clarify their confusion regarding the dependence of the scalar product on the point of calculation.
  • Another participant questions whether the derivation of the metric tensor (g_ij) is understood in terms of relating tangential vectors of different bases.

Areas of Agreement / Disagreement

Participants generally agree on the fact that the metric tensor's dependence on position is crucial for calculating the scalar product in spherical coordinates. However, there remains some disagreement or lack of clarity regarding the implications of this dependence and the derivation of the metric tensor.

Contextual Notes

The discussion highlights limitations in understanding the relationship between the scalar product and the metric tensor in spherical coordinates, as well as the need for clarity on the derivation of the metric tensor itself.

batboio
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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 :)
 
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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.
 
Yes. I also said that. However that doesn't make things more clear...
 
Do you understand how g_ij is derived by relating tangential vectors of one basis to another?
 

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