Appendix: Exercises... 395
Due toc^3 =−(a^3 + 3 a^2 b+ 3 ab^2 +b^3 ), the diagonal elements are equal toa^3 +a^2 b+
ab^2 =a(a^2 +ab+b^2 ),b^3 +a^2 b+ab^2 =b(a^2 +ab+b^2 ), and−a^3 −b^3 − 2 a^2 b−
2 ab^2. On the other hand, the 11-element ofa(a:a)is equal to(a^2 +b^2 +c^2 )a=
2 a^3 + 2 b^3 + 2 a^2 b= 2 a(a^2 +ab+b^2 ). Similarly, one finds for the 22-element
2 b(a^2 +ab+b^2 ). The 33-element is(a^2 +b^2 +c^2 )c=− 2 (a+b)(a^2 +ab+b^2 )=
− 2 (a^3 +b^3 + 2 a^2 b+ 2 ab^2 ). Comparison of the diagonal matrix elements shows
the validity of the relation (5.51).
Exercises Chapter 7
7.1 Divergence, Rotation and the Symmetric Traceless Part of the Gradient
Tensor for the Vector Fields iv to vi of Sect.7.2.1(p.90)
(iv)Uniaxial Squeeze-stretch Field
vμ= 3 eμeνrν−rμ.The gradient is∇νvμ= 3 eμeν−δμν. One finds
∇·v= 0 , ∇×v= 0 , ∇νvμ = 3 eνeμ.(v)Planar Squeeze-stretch Field
vμ=eμuνrν+uμeνrν,whereeanduare two orthogonal unit vectors,e·u=0. Since here∇νvμ=
eμuν+eνuμ,
∇·v= 0 , ∇×v= 0 , ∇νvμ = 2 eνuμ,is found. When the coordinate axes are rotated by 45◦, this vector field reads
vμ=eμeνrν−uμuνrν.Now one finds
∇·v= 0 , ∇×v= 0 , ∇νvμ =eνeμ −uνuμ.(vi)Solid-like Rotation or Vorticity Field. A circular flow with a constant angular
velocityw:
vμ=εμκλwκrλ.