Tensor

A tensor is a multilinear object that encodes relations among vectors and covectors. On a manifold, tensor fields assign such objects smoothly to each point.

Vector space: A set of vectors equipped with addition and scalar multiplication over a field. Tensors are defined from vector spaces and their duals.
Tensor rank: The number of vector and covector slots in a tensor. Rank counts the arguments and determines the shape of a component array.
Tensor field: A smoothly varying assignment of a tensor to each point of a manifold. It extends pointwise tensors into geometric quantities defined throughout a space.
Matrix: A rectangular array of scalars representing a linear map or data in chosen bases. A matrix may represent a tensor, but an arbitrary matrix lacks tensor transformation behavior.
Dual space: The vector space of linear maps from a vector space to its underlying field. Covectors are elements of this space and provide tensor inputs.
Tensor components: The scalar entries representing a tensor relative to chosen bases. Components make abstract tensors calculable while depending on the selected basis.
Metric tensor: A symmetric, nondegenerate bilinear form on each tangent space of a manifold. It measures lengths and angles and converts vectors into covectors.
Tensor density: A coordinate-dependent quantity that transforms like a tensor with an additional determinant factor. Its extra transformation factor separates it from an ordinary tensor.
Multilinear map: A function of several vector arguments that is linear in each argument when the others are fixed. A tensor can be represented as a multilinear map on vectors and covectors.
Einstein notation: A convention that sums over repeated upper and lower indices in tensor expressions. It compresses the index contractions used to calculate with tensors.

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