0vorm

0vorm, or zero-form, is a basic concept in differential geometry describing a scalar field on a differentiable manifold. It is the simplest type of differential form.

On a smooth manifold M, a 0-form is a smooth function f: M → R, often denoted f ∈ C∞(M). The exterior derivative d maps 0-forms to 1-forms, so df is a 1-form. In local coordinates (x^1, ..., x^n), df = ∑i (∂f/∂x^i) dx^i. The operation satisfies d^2 = 0, and it extends to higher forms to form the de Rham complex.

Algebraically, 0-forms behave like ordinary scalar fields. Wedge products with 0-forms act as multiplication: α ∧ f = f α for a k-form α. Thus 0-forms serve as coefficients that scale higher-degree forms. In particular, the space of 0-forms on M, Ω^0(M), is the algebra of smooth real-valued functions on M.

Geometric and physical interpretations are common: a 0-form assigns a real value to each point of the manifold, recording a scalar quantity such as temperature or potential. The exterior derivative df captures the rate of change of this scalar field and relates to the gradient in coordinates.

In Dutch mathematical usage, the term is often called nulvorm or 0-vorm. Related concepts include 1-forms (covectors), higher-degree forms, and the broader framework of exterior calculus and de Rham cohomology.