scalarfield

A scalar field is a function that assigns a single real value to every point in a space. Formally, if Ω is a region in n-dimensional space or a differentiable manifold, a scalar field is a function f: Ω → R. The value f(x) represents a quantity defined at the point x, such as temperature, elevation, or potential energy.

In Euclidean space, common examples include a temperature distribution in a room, atmospheric pressure in the atmosphere, or a gravitational or electric potential field in space. Scalar fields can be visualized by level sets or isosurfaces: for a constant c, the set {x ∈ Ω | f(x) = c} consists of points sharing the same scalar value.

Analytically, scalar fields have a gradient ∇f, a vector field that points in the direction of steepest ascent. The gradient is normal to level sets. In coordinates, ∇f is the vector of partial derivatives: (∂f/∂x1, ..., ∂f/∂xn). The Laplacian ∇²f = div(∇f) is another operator that measures how f spreads from a point; functions with ∇²f = 0 are harmonic and arise in potential theory and physics.

Regularity describes smoothness: a scalar field may be continuous (C^0), differentiable (C^1), or smooth (C^∞). On manifolds, scalar fields are understood as smooth functions, i.e., smooth sections of the trivial line bundle.

Applications of scalar fields span physics, engineering, meteorology, geography, computer graphics, and computational science. In mathematics, scalar fields are foundational in differential geometry and analysis, serving as simple yet powerful objects to study geometry and topology.