By van der Laan M., Bryen J.

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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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