Coordinatedependent

Coordinate-dependent is a term used to describe quantities or statements that rely on the specific choice of coordinates or reference frame. In mathematics and physics, many expressions take on different numerical values or forms when expressed in alternate coordinate systems, even though the underlying object remains the same.

In practice, coordinate dependence appears with components of geometric objects. For example, the components of a vector change when you switch from Cartesian to polar coordinates, and the entries of a metric tensor gμν(x) vary with the coordinate chart. While these components are coordinate-dependent, the geometric relation they describe—such as the length of a vector or the spacetime interval ds2—can be coordinate-invariant when formulated correctly.

A related concept is coordinate invariance: certain physical laws or geometric truths retain their essential form regardless of the coordinate system. This leads to the use of tensor calculus, where transformation rules ensure that equations describe the same reality across different coordinates. In general relativity, for instance, the metric’s components are coordinate-dependent, but the spacetime interval is an invariant quantity.

Notes: While the standard term is coordinate-dependent (often written with a hyphen), some sources may appear to use coordinatedependent as a single word. The distinction between coordinate-dependent quantities and coordinate-invariant relations is a fundamental theme in differential geometry, physics, and related fields. See also coordinate transformation, tensor, invariance, and gauge.