Tensor calculus

Part of a series of articles about

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula

Integral

Definitions
Lists of integrals
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Divergence Gradient Green's Stokes'

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix

Specialized

In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).

Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his theory of general relativity. Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.

Tensor calculus has many real-life applications in physics and engineering, including stress analysis, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), and general relativity (see mathematics of general relativity).

Books

Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
J.R. Tyldesley (1973). An introduction to Tensor Analysis: For Engineers and Applied Scientists. Longman. ISBN 0-582-44355-5.
D.C. Kay (1988). Tensor Calculus. Schaum’s Outlines, McGraw Hill (USA). ISBN 0-07-033484-6.
P.Grinfeld (2014). Introduction to Tensor Analysis and the Calculus of Moving Surfaces. Springer. ISBN 1-4614-7866-9.

Tensors

Glossary of tensor theory

Scope

Mathematics	coordinate system multilinear algebra Euclidean geometry tensor algebra differential geometry exterior calculus tensor calculus

Physics Engineering	continuum mechanics electromagnetism transport phenomena general relativity computer vision

Notation

Tensor definitions

Operations

Related abstractions

Notable tensors

Mathematics	Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor

Physics	moment of inertia angular momentum tensor spin tensor Cauchy stress tensor stress–energy tensor EM tensor gluon field strength tensor Einstein tensor metric tensor (GR)

Mathematicians

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Tensor calculus

See also

Books