Vector calculus

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An obsolete name for the branch of mathematics dealing with the properties of operations carried out on vectors (cf. Vector). Vector calculus comprises vector algebra and vector analysis. In vector algebra linear operations (addition of vectors and multiplication of vectors by numbers) as well as various vector products (scalar, pseudo-scalar, vector, mixed, double and triple vector products) are studied. The subject of vector analysis are vectors which are functions of one or more scalar arguments.

Vector calculus originated in the 19th century in connection with the needs of mechanics and physics, when operations on vectors began to be performed directly, without their previous conversion to coordinate form [1], [2], [3]. More advanced studies of the properties of mathematical and physical objects which are invariant with respect to the choice of coordinate systems led to a generalization of vector calculus — tensor calculus.


[1] C. Wessel, Arch. for Math. og Naturvid. , 18 (1896)
[2] W.R. Hamilton, "Elements of quaternions" , Chelsea, reprint (1969)
[3] J.W. Gibbs, E.B. Wilson, "Vector analysis" , Yale Univ. Press (1913)
[4] N.E. Kochin, "Vector calculus and fundamentals of tensor calculus" , Moscow (1965) (In Russian)
[5] Ya.S. Dubnov, "Fundamentals of vector calculus" , 1–2 , Moscow-Leningrad (1950–1952) (In Russian)



[a1] A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958)
[a2] B. Spain, "Tensor calculus" , Oliver & Boyd (1960)
How to Cite This Entry:
Vector calculus. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098