Physics- Vibrations and Waves by Crowell B.

By Crowell B.

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C) What are components of Riik... if Rijk... are components of a tensor of nth order? 45 The components of an arbitrary vector a and an arbitrary second tensor T are related by a triply subscripted quantity Rijk in the manner ai ¼ Rijk Tjk for any rectangular Cartesian basis {ei}. Prove that Rijk are the components of a third-order tensor. 46 For any vector a and any tensor T, show that (a) a Á TA a ¼ 0 and (b) a Á Ta ¼ a Á TS a, where TA and TS are antisymmetric and symmetric part of T, respectively.

3)] and Eq. 7) [or Eq. 8)] are the transformation laws relating components of the same tensor with respect to different Cartesian unit bases. Again, it is important to note that in Eqs. 7), [T] and [T]0 are different matrices of the same tensor T. We note that the equation ½TŠ 0 ¼ ½QŠT ½TŠ½QŠ differs from T 0 ¼ QT TQ in that the former relates the components of the same tensor T whereas the latter relates the two different tensors T and T0. 17-1). Solution Since e10 ¼ e2 ; e20 ¼ Àe1 and e30 ¼ e3 ; by Eq.

54), where I1 is the first scalar invariant of the rotation tensor. 57 Let F be an arbitrary tensor. (a) Show that FTF and FFT are both symmetric tensors. (b) If F ¼ QU ¼ VQ, where Q is orthogonal, show that U2 ¼ FT F and V2 ¼ FFT . (c) If l and n are eigenvalue and the corresponding eigenvector for U, find the eigenvalue and eigenvector for V. Tii Tjj Tij Tji À . 59 A tensor T has a matrix [T] given below. (a) Write the characteristic equation and find the principal values and their corresponding principal directions.

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