A coordinate system is a system of surfaces in n-dimensional space which allows the association of every point in that space point with an ordered set of numbers (ξ_{1}, ξ_{2}, ... , ξ_{n}) (in customary three-dimensional space—three numbers (ξ_{1}, ξ_{2}, ξ_{3}). The proper choice of a coordinate system in a mathematical description of a process in some volume can significantly simplify its mathematical formulation and problem solution.

*Cartesian (rectangular) coordinates* are formed by three families of self-perpendicular planes: x = *const*; y = *const*; z = *const*. If one puts three other families of surfaces, which are not parallel with each other, on this system, then the location of every point (x,y,z) can be defined by a crossing of three surfaces from these families, which form a new system of curvilinear coordinates. If the surfaces of curvilinear coordinates are described by the equations ξ(x,y,z) = *const*, ξ_{2}(x,y,z) = *const*, ξ_{3}(x,y,z) = *const*, the location of some point is fixed by coordinates (ξ_{1}, ξ_{2}, ξ_{3}), as well as by coordinates (x,y,z). The Jacobian transformation (x,y,z) → (ξ_{1}, ξ_{2}, ξ_{3}), which is a coefficient of an elementary volume change through the transformation, isn't equal to zero.

The distance between two close points is predicted as:

where H_{ij} are the Lamé coefficients, describing the metrics of the coordinate system.

For orthogonal coordinate systems with self-perpendicular coordinate surfaces, H_{ij} = 0 for i ≠ j and

where H_{i} = H_{ii}. Then, the values of surface and volume elements are:

In orthogonal coordinate systems, a function gradient is written as:

where
, are unit vectors normal to surfaces ξ_{i} = *const*

A vector divergence is written as:

a Laplacian function

Components of a vector rotor are:

In the Cartesian coordinate system, x,y,z H_{1} = H_{2} = H_{3} = 1.

In a cylindrical coordinate system, ρ, φ, z (x = ρ cos φ, y = ρ sin φ, z = z), H_{1} = 1, H_{2} = r. H_{3} = 1 (an equivalent coordinate system on a plane ρ, φ is called the polar coordinate system).

In a spherical coordinate system, r, θ, φ (x = ρ sin θ cos φ, y = ρ sin θ sin φ, z = ρ cos θ) H_{i} = 1, H_{2} = r, H_{3} = r sin θ.

For some problems characterized by particular volume geometry, some other coordinate systems can be employed, such as ellipsoidal (elliptic), paraboloidal, bicylindrical (bipolar), conical, etc.