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Want to clarify polar, spherical coordinates.

  1. Aug 24, 2010 #1
    I am always a little confuse in polar, cylindrical and spherical coordinates in vector calculus vs cylinderical and spherical coordinates in vector fields used in Electromagnetics. I want to clarify what my finding and feel free to correct me and add to it.

    A) Vector calculus:

    We use [itex] x = r cos(\phi)sin(\theta),\; y = r sin(\phi)sin(\theta),\; z = rcos(\theta) [/tex]

    [tex]\vec{r} = r cos(\phi)sin(\theta) \hat{x} + r sin(\phi)sin(\theta) \hat{y} + rcos(\theta)\hat{z}[/tex]

    [tex]\hbox { Surface intergal }\; \int_{\Gamma} f(a,\phi, \theta) dA \hbox { where }\; dA = a^2sin(\theta)d\theta d\phi [/tex]

    All these are just simply rectangular coordinate presented in polar and spherical coordinate value.

    B) Vector field in Spherical and cylindrical coordinates:

    This is "true" Spherical or cylindrical coordinates in [itex] (r,\phi,\theta)[/itex] represents a vector ( vector field ) at a single point P. P can be [itex] P(x,y,z) \hbox { or }\; P(r,\phi,\theta)[/itex] respect to the origin.

    So in conclusion, I think it is very very different between the two, where in the first case A), it is very much like the [itex]\; r,\phi,\theta \;[/itex] representation of (x,y,z). The second case B) really about vector fields where you set up a coordinate system at a point P and use the coordinate system to represent the direction and magnitude of the vector field at that point. Therefore the two are not the same.
  2. jcsd
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