NonCartesian

Non-Cartesian coordinates are coordinate systems that are not based on a fixed orthonormal grid of perpendicular axes as in Cartesian coordinates. In such systems, a point in space is described by a set of coordinates (u1, u2, u3) that parameterize position, often chosen to match the geometry or symmetry of a problem. Curvilinear coordinates, including both orthogonal and non-orthogonal systems, are typical examples.

Common examples include polar coordinates (plane), cylindrical coordinates (r, θ, z), and spherical coordinates (r, θ, φ). These systems can simplify descriptions of boundaries and symmetries, such as circles and spheres, but they introduce metric elements that vary with position. In general, one uses a position vector r(u1, u2, u3) and defines basis vectors ei = ∂r/∂ui. The metric tensor gij = ei · ej encodes how distances and angles are measured in the chosen coordinates, and the line element is ds² = gij dui duj. The Jacobian determinant J = det(∂(x,y,z)/∂(u1,u2,u3)) relates volume elements between coordinate systems.

Applications span mathematics, physics, and engineering. Non-Cartesian coordinates are especially useful when problems exhibit circular or spherical symmetry, when boundaries are curved, or when numerical methods employ curvilinear grids. They can complicate expressions for differential operators, which depend on the metric, and may introduce coordinate singularities (for example, r = 0 in polar coordinates or the axis in cylindrical coordinates).

Historically, Cartesian coordinates form a standard reference point, while non-Cartesian systems arose to exploit problem-specific geometry. In physics and relativity, curved and non-orthonormal coordinates are common, reflecting the underlying space rather than a fixed grid.