By Yuriy I. Dimitrienko
Tensor research and Nonlinear Tensor Functions embraces the fundamental fields of tensor calculus: tensor algebra, tensor research, tensor description of curves and surfaces, tensor indispensable calculus, the root of tensor calculus in Riemannian areas and affinely attached areas, - that are utilized in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc.
The booklet indicates a brand new method of definition of a tensor in house R3, which permits us to teach a geometrical illustration of a tensor and operations on tensors. in line with this process, the writer supplies a mathematically rigorous definition of a tensor as somebody item in arbitrary linear, Riemannian and different areas for the 1st time.
It is the 1st publication to give a systematized idea of tensor invariants, a idea of nonlinear anisotropic tensor features and a idea of detached tensors describing the actual houses of continua.
The e-book can be helpful 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, stable physics, basic relativity, crystallophysics, quantum chemistry of solids and fabric technological know-how.
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Extra info for Tensor Analysis and Nonlinear Tensor Functions
3. 2. 1. 37). 2. Prove that 1. 2. 3. 4. 5. Show that R; X = 0. 2. 6. 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.
Symmetric , Skew- Symmetric and Orthogonal Tensors Here the eigenvalues Aa and Ba are different; if Aa B coincide A= B. = Ba, then the tensors A and The scalar product of coaxial tensors A and B gives the symmetric tensor (A · B)T =A· B. e. permutable: A· B == B ·A . 2. 26. 201) As follows from the definition, the diagonal components Qij are equal to zero: naa = 0, and the tensor 0 has only three independent components. 15. A skew-symmetric tensor and a vector accompanying the tensor are connected as follows: n = w x E.
14 shows the geometric construction of the vector f. C. 10. 15) . 4. 11 . Components of a tensor T are the double scalar products of the tensor T from the left and from the right by basis vectors er : ( 1. 70) 22 CHAPTER 1. 15. 68). i1e2) = ii1. 16. Geometric representation of components of a second-order tensor (1. 72) (Note that since the basis ei is a Cartesian one, all iii and iii coincide, respectively). In a similar way, we obtain Thus, the following theorem has been proved here. 6. Components TIJ of a tensor T coinnde with corresponding components of its individual vectors iii and bJ.