gradientformen

Gradientformen, or gradient forms, denote the differential 1-form associated with the gradient of a scalar field on a Riemannian manifold. Formally, for a smooth function f on a Riemannian manifold (M, g), the gradient ∇f is the unique vector field satisfying g(∇f, X) = df(X) for all vector fields X. The gradient form is the metric dual of ∇f, namely the 1-form df. In coordinates, on Euclidean space with the standard metric, df = ∑i (∂f/∂x_i) dx_i and ∇f = (∂f/∂x_1, ..., ∂f/∂x_n).

Relation to other notions: df is a 1-form, the covariant dual of the gradient vector field. The Laplace–Beltrami operator is Δf = div(∇f), and in weak formulations this relation is often expressed using the gradient form via integration by parts. The Hessian of f, a (0,2)-tensor, can be obtained from the covariant derivative of df: Hess_f(X, Y) = ∇_X(df)(Y).

Applications and context: gradient forms appear across differential geometry, geometric analysis, and partial differential equations. They provide a covariant way to encode local rate of change that interacts naturally with the metric. In Euclidean space, gradient forms reduce to the standard differential df, which acts on a direction v to give the directional derivative df(v) = v(f).

Terminology: some authors use gradient form specifically to mean df, while others distinguish the gradient vector field ∇f from the gradient form df. The concept is central to relating differential, metric, and variational viewpoints in smooth manifolds.