Coordinate transformations Spherical to Cartesian

  1. Hi,
    I would like to transform a vector from Spherical to cartesian coordinate system. But the question is probably not that straight forward. :(

    I have a vector say [tex]E = E_r~\hat{r}+E_{\theta}~\hat{\theta}+E_{\phi}~\hat{\phi}[/tex].

    But I know only the cartesian coordinate from where it starts, say [tex](x,y,z)[/tex] and I do not know where it ends. So I am unable to find angles [tex]\theta[/tex] and [tex]\phi[/tex] for computing the transformation matrix [tex]R[/tex] that transforms the vector [tex]E[/tex] to cartesian system. This [tex]R[/tex] is the usual matrix with sines and cosines of [tex]\theta[/tex] and [tex]\phi[/tex] and can be seen here.
    http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates

    So how do I go about it. Is there even a way to do this. Once again this is not a homework question and is for a small project that I am doing. There aren't any homeworks at this time of the year. :)

    Appreciate any form of help.

    Kedar
     
  2. jcsd
  3. CompuChip

    CompuChip 4,299
    Science Advisor
    Homework Helper

    What do you mean, you don't know where it ends?
    Isn't the (r, theta, phi) system relative to (x, y, z)?
     
  4. May be I am missing something very simple here. But I do not know the [tex](r,~\theta,~\phi)[/tex] as well. I did try to do it that way though to start with.

    All I know is the magnitudes in the [tex]\hat{r},~\hat{\theta}[/tex] and [tex]\hat{\phi}[/tex] directions and the starting point. And nothing else.

    Can the [tex](r,~\theta,~\phi)[/tex] be found out from the magnitudes in the [tex]\hat{r},~\hat{\theta}[/tex] and [tex]\hat{\phi}[/tex] ([tex]E_r,~E_{\theta},~E_{\phi}[/tex] as above)?

    It can be really simple. But I cannot just think about it right.
     
  5. CompuChip

    CompuChip 4,299
    Science Advisor
    Homework Helper

    Yes, you might be missing something or I might.
    But aren't the coordinate values simply the coefficients of the unit vectors?
    Like, in a Cartesian system you can write either (3, 0, -2) for the coordinates of a point, or you can describe it by a vector [itex]3 \hat x + 0 \hat y - 2 \hat z[/itex].
    When writing down a tuplet of numbers like (3, 0, -2), we are implicitly assuming that we have these three basis vectors [itex]\hat x, \hat y, \hat z[/itex] and we are using them to fix our point.
     
  6. Thanks Compuchip. I got your point. So stupid of me. The (x,y,z) can be used to find theta, phi. My bad.
     
  7. CompuChip

    CompuChip 4,299
    Science Advisor
    Homework Helper

    You're welcome. We all get confused sometimes.
     
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