feltkoordinaterne

Feltkoordinaterne, or field coordinates, denote the component functions that describe a field once a coordinate system or local frame is chosen on a manifold. In mathematics and physics, a field assigns to each point a quantity — a real number, a vector, or a higher‑rank tensor — and the feltkoordinaterne are the coordinate expressions of that assignment relative to a basis.

For a scalar field f on a space M, the feltkoordinaterne are simply the values f(p) at each point p in a chart; in practice one writes f in coordinates as f(x). For a vector field V, one writes V = sum_i V^i e_i, where the V^i are the field coordinates with respect to the coordinate basis {e_i}. Higher‑rank tensor fields have components that play the role of field coordinates, transforming under coordinate changes by the appropriate tensor transformation laws.

Under a change of coordinates, the components mix according to the Jacobian of the transformation, while the underlying geometric object remains unchanged. This separation between the geometric object and its coordinate representation is central in differential geometry and in physical theories.

In physics, field coordinates are used to express physical quantities in a chosen frame, such as the components of the electromagnetic field tensor F_{\mu\nu} or the metric tensor g_{\mu\nu} in spacetime coordinates. The concept emphasizes how observations depend on the chosen coordinates, while laws of physics are formulated covariantly.

See also: coordinate chart, tangent bundle, tensor fields, field theory.