Scalar product in spherical coordinates

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SUMMARY

The discussion focuses on calculating the scalar product of two vectors in spherical coordinates, specifically addressing the challenges posed by the metric tensor's dependence on the coordinates \( r \) and \( \theta \). The participants highlight that in Cartesian coordinates, basis vectors remain constant, while in spherical coordinates, they vary with position, complicating the scalar product calculation. The expression for the scalar product, \( \vec{x} \cdot \vec{y} = g_{ij} x^i y^j \), incorporates the metric tensor, which is essential for accurate computation in curved spaces. The conversation emphasizes the need for a rigorous understanding of how the metric tensor is derived and its implications for scalar product calculations.

PREREQUISITES
  • Understanding of spherical coordinates and their representation in 3D space.
  • Familiarity with the concept of a metric tensor in differential geometry.
  • Knowledge of the Einstein summation convention for tensor notation.
  • Basic grasp of vector calculus in both Cartesian and spherical coordinates.
NEXT STEPS
  • Study the derivation of the metric tensor in spherical coordinates.
  • Learn about the implications of curved spaces in differential geometry.
  • Explore the relationship between tangent vectors and basis transformations.
  • Investigate the application of scalar products in various coordinate systems.
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, physics, and engineering, particularly those dealing with vector calculus, differential geometry, and applications involving spherical coordinates.

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 :)
 
Last edited:
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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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