0forms

In differential geometry, a 0-form is a smooth function on a manifold. Equivalently, 0-forms are the sections of the exterior algebra Λ0(T*M), which is naturally identified with the space of smooth real-valued functions on the manifold, denoted C∞(M).

Locally, a 0-form f has no differential components by itself. The exterior derivative d maps 0-forms to 1-forms, so df is a 1-form. In a local coordinate system x1, ..., xn, df = sum_i (∂f/∂xi) dxi. Thus df encodes the differential of f, and in Euclidean space it corresponds to the gradient information of f.

0-forms form an algebra under pointwise addition and multiplication: if f and g are 0-forms, then (f + g)(p) and (fg)(p) are defined at each point p ∈ M. Wedge product with a 0-form is simply multiplication: ω ∧ f = f ω for any differential form ω.

In the de Rham complex, d: Ω0 → Ω1 satisfies d^2 = 0. A 0-form f is closed if df = 0, which on a connected manifold means f is locally constant and, globally, constant on each connected component. A 0-form is exact if f = dg for some g, which is always true for g = f in the case of differential systems, but in the de Rham sense exactness refers to the existence of a primitive. The zeroth de Rham cohomology group H0(M) captures the connected components of M.

Examples include smooth functions on the real line or higher-dimensional Euclidean spaces. Conceptually, 0-forms are the scalar fields on a manifold and form the foundation for building higher-degree differential forms via the exterior derivative.