Tensor Algebra

Antonio Romano () and Mario Mango Furnari ()
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Antonio Romano: Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”
Mario Mango Furnari: Consiglio Nazionale delle Ricerche, Istituto di Cibernetica

Chapter 1 in The Physical and Mathematical Foundations of the Theory of Relativity, 2026, pp 3-49 from Springer

Abstract: Abstract In this chapter the reader is supposed to be acquainted with the algebraic structure of an n-dimensional abstract vector space $$E_n$$ E n . We here show how to generate many other algebraic structures starting from a vector space. We begin by defining the dual vector space, the contravariant, covariant, and mixed tensors. We then collect all these new entities in the global structure of tensor algebra $$T(E_n)$$ T ( E n ) on $$E_n$$ E n . Finally, the exterior algebra, that is the subalgebra of $$T(E_n)$$ T ( E n ) of skew-symmetric covariant tensors, and its geometric meaning are analyzed.

Date: 2026
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DOI: 10.1007/978-3-032-23179-6_1

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