Transcription of A Basic Operations of Tensor Algebra - Springer
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A Basic Operations of Tensor Algebra - Springer
A Basic Operations of Tensor AlgebraThe 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. In general, there are two possibilities for the representation of the tensors andthe tensorial equations: the direct (symbolic, coordinate-free) notation and the index (component) notationThe direct notation operates with scalars, vectors and tensors as physical objectsdefined in the three-dimensional space (in this book we are limit ourselves to thiscase). A vector (first rank Tensor )aaais considered as a directed line segment ratherthan a triple of numbers (coordinates).
172 A Basic Operations of Tensor Algebra For a given basis e i any vector a can be represented as follows a = a1e1 +a2e2 +a3e3 ≡ aie i The numbers ai are called the coordinates of the vector aa for the basis e i.In order to compute the coordinates ai the dual (reciprocal) basis ek is introduced in such a way that ek ·· e i = δ k = 1, k = i, 0, k = i δk i is the Kronecker symbol. The ...
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