Want to clarify polar, spherical coordinates.

[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 [itex] 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 $(r,\phi,\theta)$ 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.

 

[jvicens]

Hello,

You are correct in your understanding of the difference between polar, cylindrical, and spherical coordinates in vector calculus and vector fields used in electromagnetics. In vector calculus, we use these coordinates to represent a point in space and to perform calculations such as surface integrals. On the other hand, in vector fields, these coordinates are used to represent the direction and magnitude of a vector at a specific point in space.

One key difference between the two is that in vector calculus, the coordinates are fixed and do not change, while in vector fields, the coordinates can vary depending on the point in space where the vector is being represented.

It is important to understand the distinction between these two uses of coordinates in order to properly apply them in different situations. I hope this helps clarify any confusion you may have had. Please let me know if you have any further questions.