Why do we need polar coordinates? So, I haven't formally started learning about polar coordinates or anything yet, I'm just curious about them. I'm saying this to convey that I might not understand many technical terms.
So, I want to know how exactly polar coordinates came into existence i.e. what gave rise to the idea of polar coordinates. I would also like to know how they found their purpose/applications later on and some examples of things that they can do but Cartesian coordinates can not.
This is my first question here. Sorry if it's off topic.
Thank you!
 A: The first thing that comes to mind is for aircraft navigation. The location of a plane in the Earth's atmosphere is described by a height above land ($r$ in polar coordinates) and latitude/longitude which are two angles, $\theta$ and $\phi$. Using $x$, $y$ and $z$ would be needlessly complicated for describing where a plane is.
This is one example of where polar coordinates are beneficial over cartesian. All coordinate systems "do" the same things, but some simplify problems more than others.
A: Curvilinear coordinate systems, like polar coordinates, came into being because they simplify computations in certain situations. For example think of a body moving with constant angular velocity $\omega$ around a circle of radius one centered at the origin. Using cartesian coordinates $x, y$ its motion is described by
$$
x = \cos\omega t,\ 
y = \sin\omega t
$$
in polar coordinates $r, \theta$ it becomes much nicer
$$
r = 1,
\theta = \omega t 
$$
this can simplify other computations you might want to do that are related to this kind of motion.
A: Polar coordinates are useful in differential equations, mainly when solving problems related to variable separable form and sometimes useful in finding out wronskian
A: I am also new to this topic, but from my reading, polar coordinates are practically useful in real-world applications where distances and angles are circular or spherical. For example, if there is a spinning motor and an engineer needs to calculate details like what force or speed needs to be applied to produce certain distances or angles, then polar coordinates could make calculations easier in situations where the spinning is restricted to circular (2D) or spherical (3D). Another example I've read is when physicists study the effects of force fields (e.g., magnetic or gravitational fields) whose strengths radiates spherically from a center point and the need is to calculate distances from the center and angles from the center. There again, polar coordinates make calculations easier than Cartesian coordiantes.
