potentiaalifunktion

Potentiaalifunktion is a scalar field that serves as a potential for a given vector field. In mathematical terms, a vector field F defined on a region D in Euclidean space is called conservative if there exists a scalar function φ: D → R such that F = ∇φ, the gradient of φ. The scalar φ is then called a potential function for F. A key consequence is that the line integral of F between two points depends only on the endpoints: ∫C F · dr = φ(b) − φ(a), for any path C from a to b.

Existence and uniqueness: If F is smooth and curl F = 0 on a simply connected domain, then a global potential function φ exists (unique up to an additive constant). On domains with nontrivial topology, curl-free does not guarantee a single global potential; multiple local potentials may exist.

Physical interpretation: In physics, potential functions often represent potential energy. For a force field F, the relation F = −∇V expresses that the force is the negative gradient of a potential energy V. In electrostatics, the electric field E satisfies E = −∇φ, where φ is the electric potential. In gravity, the gravitational field corresponds to the gradient of the gravitational potential.

Applications: In optimization and dynamical systems, a potential function describes an energy landscape; gradient-based methods seek minima of the potential. In geometry and physics, potential functions help analyze conservative fields, stability (via Lyapunov functions), and integrals of motion.

Examples: For F(x, y) = (2x, 3y), a potential function is φ(x, y) = x^2 + (3/2)y^2, since ∇φ = F. A non-conservative field such as F(x, y) = (−y, x) has no global potential in R^2.