# Tensor calculus

The traditional name of the part of mathematics studying tensors and tensor fields (see Tensor on a vector space; Tensor bundle). Tensor calculus is divided into tensor algebra (entering as an essential part in multilinear algebra) and tensor analysis, studying differential operators on the algebra of tensor fields.

Tensor calculus is an important constituent part of the apparatus of differential geometry. In this connection it was first systematically developed by G. Ricci and T. Levi-Civita (see [1]); it has often been called the "Ricci calculus".

The term "tensor" was already used in the middle of the 19th century to describe elastic deformations of bodies (see Deformation tensor; Stress tensor). From the beginning of the 20th century onwards, the apparatus of tensor calculus has been systematically used in relativistic physics (see [2]).

#### References

[1] | G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications" Math. Ann. , 54 (1901) pp. 125–201 |

[2] | A. Einstein, M. Grossmann, "Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation" Z. Math. Physik , 62 (1914) pp. 225–261 |

#### Comments

#### References

[a1] | J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German) |

**How to Cite This Entry:**

Anti-symmetric tensor.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Anti-symmetric_tensor&oldid=43260