Vector field

A vector field assigns a vector to each point in a space. It can represent quantities such as force, velocity, or direction that vary across space.

Gradient: A differential operator that points in the direction of greatest increase of a scalar field. It turns a scalar field into a vector field that points uphill.
Tangent space: A vector space of possible tangent directions at a point on a manifold. On curved spaces, field vectors belong to the tangent space at each point.
Gravitational field: A vector field describing the gravitational force per unit mass at each point. It models the force experienced by a test mass throughout space.
Scalar field: A function assigning a scalar value to each point in a space. Unlike a vector field, it assigns magnitude alone, without direction.
Divergence: A scalar measure of the net outward flow of a vector field from a point. It detects local sources and sinks in the field.
Vector bundle: A space built from vector spaces attached continuously to points of a base space. Tangent spaces assemble into a bundle that can host a vector field.
Electric field: A vector field giving the electric force per unit positive charge at each point. It maps how electric charges exert force through space.
Tensor field: A field assigning a tensor to each point of a space. It generalizes vector fields by assigning multilinear objects rather than vectors.
Curl: A differential operator measuring the local rotation of a vector field. It identifies where the field circulates around a point.
Manifold: A space that locally resembles Euclidean space and supports geometric notions such as tangent directions. Vector fields extend from ordinary regions to curved spaces.

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