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Published in:
2025 | OriginalPaper | Chapter
Fundamentals of Tensor Calculus
Author : Udo F. Meissner
Published in: Tensor Calculus with Object-Oriented Matrices for Numerical Methods in Mechanics and Engineering
Publisher: Springer Nature Switzerland
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Tensor calculus, with its consistent formulation of invariants and transformations, has acquired a high significance in engineering over the years. Thus, early works on the mechanics of load-bearing structures, such as [Green/Zerna 1954], which renewed the theory’s formulation, and mathematical compendia on tensor calculus in index notation, such as [Duschek/Hochrainer 1968], can serve as a basis at this point. In this chapter, we recapitulate the most important basics of tensor algebra in this context, especially to make the approach of object-oriented matrix calculus comprehensible and to provide the ability to apply the presented methodology consistently. This is because the index notation allows the arithmetic operations of tensor calculus to be seamlessly transformed into the practical numeric aspects of programming tools using the new object-oriented classes and methods of matrix calculus. This capability is demonstrated with the tensor class BASIS and its application in the numeric function METRIC.