TENSORS
26.3 In section 26.3 the transformation matrix for a rotation of the coordinate axes
was derived, and this approach is used in the rest of the chapter. An alternative
view is that of taking the coordinate axes as fixed and rotating the components
of the system; this is equivalent to reversing the signs of all rotation angles.
Using this alternative view, determine the matrices representing (a) a positive
rotation ofπ/4 about thex-axis and (b) a rotation of−π/4 about they-axis.
Determine the initial vectorrwhich, when subjected to (a) followed by (b),
finishes at (3, 2 ,1).
26.4 Show how to decompose the Cartesian tensorTijinto three tensors,
Tij=Uij+Vij+Sij,
whereUijis symmetric and has zero trace,Vijis isotropic andSijhas only three
independent components.
26.5 Use the quotient law discussed in section 26.7 to show that the array
y^2 +z^2 −x^2 − 2 xy − 2 xz
− 2 yx x^2 +z^2 −y^2 − 2 yz
− 2 zx − 2 zy x^2 +y^2 −z^2
forms a second-order tensor.
26.6 Use tensor methods to establish the following vector identities:
(a) (u×v)×w=(u·w)v−(v·w)u;
(b) curl (φu)=φcurlu+ (gradφ)×u;
(c) div (u×v)=v·curlu−u·curlv;
(d) curl (u×v)=(v·grad)u−(u·grad)v+udivv−vdivu;
(e) grad^12 (u·u)=u×curlu+(u·grad)u.26.7 Use result (e) of the previous question and the general divergence theorem for
tensors to show that, for a vector fieldA,
∫
S[
A(A·dS)−^12 A^2 dS]
=
∫
V[AdivA−A×curlA]dV ,whereSis the surface enclosing volumeV.
26.8 A column matrixahas componentsax,ay,azandAis the matrix with elements
Aij=−ijkak.
(a) What is the relationship between column matricesbandcifAb=c?
(b) Find the eigenvalues ofAand show thatais one of its eigenvectors. Explain
why this must be so.26.9 Equation (26.29),
|A|lmn=AliAmjAnkijk,is a more general form of the expression (8.47) for the determinant of a 3× 3
matrixA. The latter could have been written as|A|=ijkAi 1 Aj 2 Ak 3 ,
whilst the former removes the explicit mention of 1, 2 ,3 at the expense of an
additional Levi–Civitasymbol. As stated in the footnote on p. 942, (26.29) can
bereadilyextendedtocoverageneralN×Nmatrix.
Use the form given in (26.29) to prove properties (i), (iii), (v), (vi) and (vii)
of determinants stated in subsection 8.9.1. Property (iv) is obvious by inspection.
For definiteness takeN= 3, but convince yourself that your methods of proof
would be valid for any positive integerN.