Spherical & Cylindrical Coordinates

[intwo]

Are spherical and cylindrical coordinate systems only a physical tool or is there some mathematical motivation behind them? I assume that they can be derived mathematically, but multivariable calculus texts introduce them and state their important properties without much background information.

I have taken two courses on linear algebra and we built up the tools for coordinate transformations in rectangular coordinates, but we did not discuss other coordinate systems. Is there such a thing as curvilinear algebra?

Thanks!

 

[Mark44]

Are spherical and cylindrical coordinate systems only a physical tool or is there some mathematical motivation behind them? I assume that they can be derived mathematically, but multivariable calculus texts introduce them and state their important properties without much background information.

Coordinate systems other than the usual rectangular (or Cartesian) coordinate system are really just different ways to describe the location of points. Polar (and by extension to $\mathbb R^3$, cylindrical) coordinate systems are useful from a mathematics perspective, allowing one to write the equation of a curve or surface in a possibly simpler way. These alternate coordinate systems make it possible to evaluate integrals where it might be very difficult or impossible to do so in rectangular coordinates.

Polar coordinates come into play naturally in such applications as radar, in which a plane's coordinates are given in terms of its distance from the radar set (r) and the angle relative to some base angle ($\theta$).