Tensor field

A tensor field assigns a tensor to each point of a smooth manifold, with components varying smoothly in local coordinates. It includes familiar fields such as vector fields and metrics.

Tensor: A multilinear object that transforms according to specified rules under changes of coordinates or basis. Each point receives one of these multilinear objects.
Tensor bundle section: A map from a manifold to a fiber bundle that assigns each point an element of its fiber. This is the bundle-theoretic description of a tensor field.
Vector field: A smooth assignment of a tangent vector to each point of a manifold. It is the simplest non-scalar tensor field and describes directions varying over space.
Tensor calculus: The methods for manipulating tensors and tensor fields using algebraic operations and differentiation. It studies the calculations performed on tensor fields.
Smooth manifold: A space locally modeled on Euclidean space, with smoothly compatible coordinate charts. Its points are the locations where the tensor assignment is defined.
Coordinate chart: A homeomorphism from part of a manifold to an open subset of Euclidean space, used to assign coordinates. Charts let tensor fields be written as component functions.
Differential form: A smooth field of alternating covariant tensors on a manifold. Differential forms are tensor fields suited to integration and exterior differentiation.
Differential geometry: The study of smooth spaces using tools from calculus, geometry, and linear algebra. Tensor fields are among its central tools for expressing geometric structure.
Tangent space: A vector space of tangent directions at a point of a smooth manifold. Vectors and covectors in a tensor at a point are drawn from tangent spaces and their duals.
Tensor transformation law: The rule specifying how tensor components change under a change of coordinates or basis. It ensures local component descriptions represent the same field on overlapping charts.

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