Tensor calculus

Part of a series of articles about
Calculus
Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem
Differential 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 Inverse General Leibniz Faà di Bruno's formula
Integral Lists of integrals Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green's Kelvin–Stokes Stokes
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Specialized Fractional Malliavin Stochastic Variations
Glossary of calculus Glossary of calculus

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 elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory.

See also

• Vector analysis
• Matrix calculus
• Ricci calculus
• Tensors in curvilinear coordinates
• Multilinear subspace learning

Further reading

• Dullemond, Kees; Peeters, Kasper (1991–2010). "Introduction to Tensor Calculus" (PDF). Retrieved 17 May 2018.CS1 maint: Date format (link)

Books

• Dimitrienko, Yuriy (2002). Tensor Analysis and Nonlinear Tensor Functions. Kluwer Academic Publishers (Springer). ISBN 1-4020-1015-X.
• Sokolnikoff, Ivan S (1951). Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua. Wiley. ISBN 0471810525.
• A.I. Borisenko & I.E. Tarapov (1979). Vector and Tensor Analysis with Applications. 0486638332; 2nd edition. ISBN 0486638332.CS1 maint: Uses authors parameter (link)
• Itskov, Mikhail (2015). Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics. Springer; 2nd edition. ISBN 9783319163420.
• 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.