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 imperative 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 area R3, which permits us to teach a geometrical illustration of a tensor and operations on tensors. in response to this procedure, the writer offers a mathematically rigorous definition of a tensor as someone item in arbitrary linear, Riemannian and different areas for the 1st time.

It is the 1st e-book to offer a systematized thought of tensor invariants, a concept of nonlinear anisotropic tensor services and a concept of detached tensors describing the actual houses of continua.

The publication can be important 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, strong physics, common relativity, crystallophysics, quantum chemistry of solids and fabric technological know-how.

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33) we have r;; V yEijk . 35) gmignjglk· One can verify immediately that Eijkt and also ilm d sm = ujc k ciJky .. 36) . 37) ukuj , - ·'J - Here Tu are components of an arbitrary sy mmetric matrix for which T; 1 = Tji. 39) is valid immediately by writing out the sums for all the indices in the component form. 39) is representable in another form , if only one of the Levi-Civita symbols is written in the component form: 1 · (A;AJ lk d et (A i)j - 6ftjk 1 2} 3 - A~AiA~ A;AJAk A ;AJA k 1 3 2+ 2 3 1- - + A~ A{A;- A~A~Ai' ).

4 7) Finally, we obtain the formula ax (b x c)== (a· c)b- (a· b)c . 3. 4. 6). l' axb In fact, i . 3jl)a 1 a 3 + + (E2j3 + E3j2)a 2a 3) b1. == Figure 1. 6. For derivation of the properties of a vector product == 0 . 49) =- E2j l etc. • 1. 2. 4. 5. 6): S = Ialibi sin 1/1 = Ia x bl. 50) 'f Since S is a scalar invariant, its value is independent of the choice of a coordinate system where the value is determined. Consider the special coordinate system R~, where a= a 1R'1 , Then 22 lal 2 = g;ja ; a1· = glla 11 a , lbl = g22b b , and S = Jgllgnla 1 b2 l sin '1/J = Jgllg2zla 1 b2 hh- cos 2 1{1.

B. 9. 3. 13. 14. 13 shows the geometric way of constructing the vector d followed from the definition. 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).

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