The following is a component-based "classical" treatment of tensors. See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two. In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalars, vectors, and linear operators. ... Note: This is a fairly abstract mathematical approach to tensors. ... Note: The following is a modern component-based treatment of tensors (sometimes called the classical treatment of tensors). ...

The Einstein notation is used throughout this page. For help with notation, refer to the table of mathematical symbols. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ... In mathematics, a set of symbols is frequently used in mathematical expressions. ...

A tensor is a generalization of the concepts of vectors and matrices. Tensors allow one to express physical laws in a form that applies to any coordinate system. For this reason, they are used extensively in continuum mechanics and the theory of relativity.-1... Wikisource has original text related to this article: Relativity: The Special and General Theory Albert Einsteins theory of relativity is a set of Two scientific theories in physics: special relativity and general relativity. ...

A tensor is an invariant multi-dimensional transformation, that takes forms in one coordinate system into another. It takes the form: In mathematics, an invariant is something that does not change under a set of transformations. ...

The new coordinate system is represented by being 'barred'(), and the old coordinate system is unbarred(xⁱ).

The upper indices [i₁,i₂,i₃,...i_n] are the contravariant components, and the lower indices [j₁,j₂,j₃,...j_n] are the covariant components. Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ...

Contravariant and covariant tensors

A contravariant tensor of order 1(Tⁱ) is defined as:

A covariant tensor of order 1(T_i) is defined as:

General tensors

A multi-order (general) tensor is simply the tensor product of single order tensors: In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

such that:

This is sometimes termed the tensor transformation law.

More about tensors

• tensor product
• tensor derivative
• absolute differentiation
• curvature
• Riemannian geometry

In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... The introduction of this article does not provide enough context for readers unfamiliar with the subject. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ...

Further reading

• Schaum's Outline of Tensor Calculus
• Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949