Source : ct|01.06.13

< Mathematics and physics

Coordinate systems

"All geometry problems can easily be reduced to such terms that there is solely need to know the length of a few straight lines to build them."
René Descartes (1596-1650), mathematician, physicist and French philosopher - Discourse on Method (1637)

"Coordinates are numbers used to determine the position of a point in space relative to a reference system".
Dictionary Larousse (1992)

There are many coordinate systems. The following systems are frequently used in mathematics and physics :

• Cartesian coordinate system (adapted to objects of cubic shape);
• Cylindrical coordinate system (adapted to objects cylindrical shape);
• Spherical coordinate system (adapted to objects spherical shape).

1) Cartesian coordinate system

With the Cartesian coordinate system, a point M which is located in a fixed coordinate system (O, i, j, k) is determined by its abscissa x, by its ordinate y and by its altitude z with respect to an origin O in the system.

OM = x i + y j + z k

A vector V=MN whose origin is in M is determined by its components Vx, Vy, Vz measured on the axes i, j et k.
V(x, y, z)= Vx(x,y,z) i Vy(x,y,z) j + Vz(x,y,z) k

With two infinitely close points M(x,y,z) and M'(x+dx,y+dy, z+dz), MM' = dx.i+dy.j+dz.k and dV=dx.dy.dz

2) Cylindrical coordinate system

With the cylindrical coordinate system, a point M which is located in a fixed coordinate system (O,i,j,k) is determined by its distance r from the origin O, by its altitude z relative to the plane (O,i,j) and by an angle θ in the plane (O,i,j) (polar angle).

In Cartesian coordinates OM (x,y,z) = x.i + y.j + z.k with x=r.cosθ and y=r.sinθ (see diagram below).
Hence, in cylindrical coordinates OM (r,θ,z) = r.cosθ i + r.sinθ j + z k

The orientation of the vector k is fixed. The orientation of the vectors u et v (unit vectors) is not fixed in the fixed system (O,i,j), but varies depending on the position of the point M.

A vector V=MN having an origin M is determined by its components Vr(r,θ,z), Vθ(r,θ,z) and Vz(r,θ,z) measured on the axes u, v (moving axes) and on the axe k ( fixed axe).
V(x, y, z)= Vr(r,θ,z) u + Vθ(r,θ,z) v + Vz(r,θ,z) k

With two infinitely close points M(r,θ,z) and M'(r+dr,θ+dθ, z+dz), MM'=dr.u+rdθ.v+dz.k and dV=dr.rdθ.dz

2) Spherical coordinate system

With the spherical coordinate system, a point M which is located in a fixed coordinate system (O,i,j,k) is determined by its distance r (relative to the origin O) and by two angles θ and φ (longitude and latitude).

In Cartesian coordinates, OM (x,y,z) = x i + y j + z k avec x=rsinθcosφ, y=rsinθsinφ et z=rcosθ (see diagram below).
Hence, in spherical coordinates, OM (r,θ,φ) = r sinθ cosφ i + r sinθ sinφ j + r cosθ k

The orientation of vectors u,v and w is not fixed in the fixed coordinate system (O,i,j,k), but varies depending on the position of point M.

A vector V=MN having an origin M is determined by its components Vr(r,θ,φ), Vθ(r,θ,φ) and Vφ(r,θ,φ) measured on the axes u, v,w (moving axes).
V(r,θ,φ)= Vr(r,θ,φ) u + Vθ(r,θ,φ)v + Vφ(r,θ,φ) w

With two infinitely close points M(r,θ,φ) and M'(r+dr,θ+dθ, φ+dφ), MM'=dr.u+rdθ.v+rsinθdφ.w and dV=r²sinθdr.dθ.dφ

With this coordinate system we can quickly calculate the area S and the volume V of a sphere based on the value of its radius R.