I am currently learning about the spherical coordinate system in class, but I do not know its advantages or even if it is advantageous in using this coordinate system over another.
I am very comfortable in using the rectangular coordinate system and the cylindrical coordinate system (polar coordinate system but just in 3D), as the rectangular coordinate system is just the cartesian coordinate system with another dimension and the cylindrical coordinate system is just the polar coordinate system with an additional dimension.
However, the concept of spherical coordinates come out of nowhere (that I know of) and I am unable to see its advantages. For example, if wanting to calculate an integral in the first octant, you can just restrict to $x>0$, $y>0$, and $z>0$ for the rectangular coordinate system. And for cylindrical coordinates, you can restrict $z>0$, $0<\theta<\frac{\pi}{2}$, and $r$ its corresponding boundary conditions.
My question is are there ever cases when using spherical coordinates are more intuitive than using cylindrical or rectangular coordinates?
Best Answer
You want to choose a coordinate system that matches symmetry of the problem at hand. That makes everything easier. Spherical coordinates work well for situations with spherical symmetry, like the field of a point charge. Cylindrical coordinates work well for situations with cylindrical symmetry, like the field of a long wire. Usually it is obvious which set (and there are many more for more complex geometries) you should use.