Scalar field

A function that assigns a scalar value to each point in a domain. Its values may vary across space or another set of inputs.

Scalar: A single quantity represented by a number, rather than by a direction or multiple components. Each point in a scalar field is assigned one scalar value.
Gradient: A vector of partial derivatives that points toward the steepest local increase of a scalar function. The gradient turns local changes in a scalar field into a direction and rate.
Temperature field: A scalar function assigning temperature to each point in a region or at each time and position. Temperature is a familiar measured quantity represented as a field.
Differential geometry: The study of smooth spaces using calculus and geometric structures. Scalar fields on manifolds are basic objects in differential geometry.
Function: A rule that assigns each permitted input exactly one output. A scalar field is a function whose outputs are scalar values.
Directional derivative: The rate at which a function changes at a point along a chosen direction. It measures how the field changes along a specified direction.
Gravitational potential: A scalar potential whose spatial gradient determines the gravitational field, up to sign convention. It represents gravity through a scalar field rather than directly through vectors.
Tensor field: A function assigning a tensor to each point of a space. Scalar fields are the rank-zero case within the broader framework of tensor fields.
Domain of a function: The set of inputs for which a function is defined. The domain supplies the points at which the field has values.
Level set: The set of points where a function has one fixed value. Level sets show the geometry of equal values within a scalar field.

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