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A quantity that does not change under a translation or rotation of the coordinate axes but changes its sign when the direction of each axis is reversed. As an example of a pseudo-scalar one could take the mixed triple scalar product of three vectors (cf. Mixed product), or the inner product $(\mathbf{a},\mathbf{b})$, where $\mathbf{a}$ is an axial vector and $\mathbf{b}$ is a general vector (based at the origin).


Pseudo-scalars are e.g. used in the context of the Clifford algebra based approach to the foundations of geometry and physics; cf. e.g. various articles in [a1] and [a2]. In the terminology of [a3], a pseudo-scalar as defined above is a $W$-scalar (a $W$-tensor of valency $0$).


[a1] J.S.R. Chisholm, A.K. Common, "Clifford algebras and their applications in mathematical physics" , Reidel (1986)
[a2] D. Hestenes, "New foundations for classical mechanics" , Reidel (1986)
[a3] J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) pp. 11ff (Translated from German)
How to Cite This Entry:
Pseudo-scalar. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article