Want to clarify polar, spherical coordinates.

1. Aug 24, 2010

yungman

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 $x = r cos(\phi)sin(\theta),\; y = r sin(\phi)sin(\theta),\; z = rcos(\theta) [/tex] $$\vec{r} = r cos(\phi)sin(\theta) \hat{x} + r sin(\phi)sin(\theta) \hat{y} + rcos(\theta)\hat{z}$$ $$\hbox { Surface intergal }\; \int_{\Gamma} f(a,\phi, \theta) dA \hbox { where }\; dA = a^2sin(\theta)d\theta d\phi$$ 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)$ represents a vector ( vector field ) at a single point P. P can be $P(x,y,z) \hbox { or }\; P(r,\phi,\theta)$ 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 $\; r,\phi,\theta \;$ 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.