![]() A.3, where a novel approach to find the functional basis is discussed. Basic sets of invariants for different groups of symmetry transformation are presented in Sect. Several rules from tensor analysis are summarized in Sect. Section A.1 provides a brief overview of basic alge- braic operations with vectors and second rank tensors. The calculus of matrices is presented in, for example. For more comprehensive overviews on tensor calculus we recom- mend. The purpose of this Appendix is to 168 A Some Basic Rules of Tensor Calculus give a brief guide to notations and rules of the tensor calculus applied through- out this work. When solv- ing applied problems the tensor equations can be “translated” into the language of matrices for a specified coordinate system. In this work we prefer the direct tensor notation over the index one. It must be remembered that a change of the coordinate system leads to the change of the components of tensors. The introduced basis remains in the background. the sum of two vectors is computed as a sum of their coordinates ci = ai + bi. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e. akgk ≡ 3 ∑ k=1 akgk, Aikbk ≡ 3 ∑ k=1 Aikbk In the above examples k is a so-called dummy index. + cidj ) g i ⊗ g j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. gi, i = 1, 2, 3 one can write a = aigi, A = ( aibj +. ![]() The index notation deals with components or coordinates of vectors and tensors. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. The scalars, vectors and tensors are handled as invariant (independent from the choice of the coordinate system) objects. A second rank tensor A is any finite sum of ordered vector pairs A = a ⊗ b +. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. ¡Descarga A Some Basic Rules of Tensor Calculus y más Monografías, Ensayos en PDF de Cálculo solo en Docsity!A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems.
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