Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Scalar product in spherical coordinates

  1. Nov 1, 2012 #1

    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 [itex]\vec{x},\vec{y}[/itex] from real 3d Euclidean space.
    If we make the standard coordinate change to spherical coordinates we can calculate it just fine in terms of [itex]\left( r, \theta, \phi \right)[/itex].
    However if we compute the metric tensor it depends on [itex]r[/itex] and [itex]\theta[/itex]. So we can't use the expression [itex]\vec{x}.\vec{y} = g_{ij} x^i y^j[/itex] (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: Nov 1, 2012
  2. jcsd
  3. Nov 1, 2012 #2
    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.
  4. Nov 3, 2012 #3
    Yes. I also said that. However that doesn't make things more clear...
  5. Nov 3, 2012 #4


    User Avatar
    Science Advisor

    Do you understand how g_ij is derived by relating tangential vectors of one basis to another?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook