Why do we have three different coordinate systems (Cartesian, cylindrical, and spherical)?

Question

I am studying the usage and conversion between these coordinates systems in EMF Theory, but don't really understand why we need all 3 of these?

Answer 1

There are many more coordinate systems than that, though these are the most common. Things are much simpler if the symmetry of the coordinate system matches the symmetry of the problem. If we are studying the field of a point charge at the origin, the problem is spherically symmetric. In spherical coordinates the field is $\frac 1{r^2}\hat r$. In Cartesian coordinates it is $\frac {x\hat x + y\hat y + z\hat z}{(x^2+y^2+z^2)^{3/2}}$

Answer 2

Things often have symmetry. If an object is the same in all directions, for example a cone, or a cylinder, or even more so a sphere, then equations become simpler. Instead of three variables $x,y,z$, you can get by with $r,z$ because $\theta$ doesn't appear due to the symmetry.

Answer 3

The three common 3D geometric shapes are cube, cylinder and sphere. Other 3D shapes can often be viewed as variants of the three.

In order to qualify the three common shapes, for example, computing their volumes, it is most convenient to use their respective coordinate systems, as illustrated below.

The volume of a unit cube

$$ \int_0^1\int_0^1\int_0^1dxdydz=1$$

The volume of a unit sphere,

$$ \int_0^{2\pi}\int_0^{\pi}\int_0^1 r^2\sin\theta drd\theta d\phi=\frac 43 \pi$$

The volume of cylinder with unit height and unit circular base

$$ \int_0^1\int_0^{2\pi} \int_0^1 rdr d\theta dz=\pi$$

It would have been awkward if the volumes are not integrated with their own coordinate systems. In many other problems, it may not be even possible to use any system other than the ‘native’ one.