Length, area, and volume are all calculated differently in different orthogonal coordinate systems. The gradient of a function and the divergence and curl of vector fields, likewise, are calculated according to the orthogonal coordinates which the function or vector field is given in.

For example, in polar coordinates the point (r, a) is the point (rcos(a), rsin(a)) in Cartesian coordinates. Hence the distance from the point (r, a) to the point (s, b) given in polar coordinates is equal to (rcos(a) - scos(b)) ² + (rsin(a) - ssin(b))²)^½. But the distance from (w,x) to (y,z) in Cartesian coordinates is ((w-y)² + (x-z) ²)^½.

Let u₁, u₂, and u₃ be functions defining a three-dimensional orthogonal coordinate system. Then at every point (x, y, z) in space, the surfaces defined by the equations u₁ = x, u₂ = y, and u₃ = z...