By Yuriy I. Dimitrienko

**Tensor research and Nonlinear Tensor Functions** embraces the elemental fields of tensor calculus: tensor algebra, tensor research, tensor description of curves and surfaces, tensor necessary calculus, the root of tensor calculus in Riemannian areas and affinely hooked up areas, - that are utilized in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc.

The ebook indicates a brand new method of definition of a tensor in house R^{3}, which permits us to teach a geometrical illustration of a tensor and operations on tensors. in line with this procedure, the writer supplies a mathematically rigorous definition of a tensor as a person item in arbitrary linear, Riemannian and different areas for the 1st time.

It is the 1st booklet to give a systematized concept of tensor invariants, a idea of nonlinear anisotropic tensor capabilities and a conception of detached tensors describing the actual houses of continua.

The booklet could be worthy for college kids and postgraduates of mathematical, mechanical engineering and actual departments of universities and in addition for investigators and educational scientists operating in continuum mechanics, sturdy physics, normal relativity, crystallophysics, quantum chemistry of solids and fabric technology.

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1 ijk Wjgnk R Rn -_ - V9 ijk nspR i 0 R k _ i 0 2 V9t. 204) 46 CHAPTER 1. 36). A Find eigenvalues of a skew-symmetric tensor. 206) we find the eigenvalues ( 1. 3. 17, a tensor 0 is called orthogonal, if or= o-1. 208) For an orthogonal tensor, the following relationship is always satisfied: Or · 0 =0 · Or = E. >:' u k. 212), there are six relationships for nine components of the tensor 0. Therefore, an arbitrary orthogonal tensor has not more than three independent components. 213) 1. 6. Symmetric, Skew-Symmetric and Orthogonal Tensors hence, det (0) = ±1.

I, then the complex-conjugate root A2 = 5. 1 = A~ - iA~ would correspond to A1, and the following complex-conjugate eigenvectors would correspond to the eigenvalues: 0 e1 = 1 0 If 0 I ;;)(e 1 + ie 1), v2 0 e2 = 1 0 I 0 II rr;(e 1 - ie 1), v2 0 I , II le 1 I= 1. 186) On scalar multiplication of the eigenvectors one by the other, we have ~ 1 · ~ 2 = 1. This conflicts with the mutual orthogonality of eigenvectors: ~ 1 · ~ 2 = 0. 188) CHAPTER 1. 14. For a symmetric positive-definite tensor T, all its eigenvalues are positive: ~In Aa > 0.

Prove that the cyclic permutation does not change the result of the mixed product: (ax b)· c = (c x a)· b = (b x c) · a. 2. 7. Show that ax b = -b x a , ax a= 0. 2. 8. Show that if ei are three orthogonal vectors of the unit length then i #j#k# i. 9. 38). 3. 1. GRAPHIC REPRESENTATION OF VECTORS AND OPERATIONS ON THEM For vectors (called first-order tensors) there is an easy-to-interpret graphic representation. 1 (see Introduction) shows a vector a connecting points 0 and M. The presence of an arrow at the point M indicates that the point 0 is initial , and the point M is end.