Coordinate transformation

A coordinate transformation is an invertible linear map, specifically a diffeomorphism, between a real Vector space and itself, defined in general as φ:RN↦RN\varphi:\mathbb{R}^{N}\mapsto \mathbb{R}^{N}, x↦x′=f(x)x\mapsto x'=f(x). The vectors xx and x′x' are different sets of coordinates for the same space.

They are primarily used to change between different systems of coordinates

Examples

Most coordinate transformations involve going from Cartesian coordinates to another set of coordinates and viceversa. For instance, when going from spherical coordinates to Cartesian coordinates, the mapping is

φ:R3↦R3,φ(ρ,θ,ϕ)=(ρcos⁡θsin⁡ϕ,ρsin⁡θsin⁡ϕ,ρcos⁡ϕ)=(x,y,z)\varphi:\mathbb{R}^{3}\mapsto \mathbb{R}^{3},\qquad \varphi(\rho,\theta,\phi)=(\rho\cos\theta\sin\phi,\rho\sin\theta\sin\phi,\rho\cos\phi)=(x,y,z)

or from polar coordinates to Cartesian

φ:R2↦R2,φ(r,θ)=(rcos⁡θ,rsin⁡θ)=(x,y)\varphi:\mathbb{R}^{2}\mapsto \mathbb{R}^{2},\qquad \varphi(r,\theta)=(r\cos \theta,r\sin \theta)=(x,y)