Exercises

1. Show that a general second-order tensor $a_{ij}$ can be decomposed into three tensors
   $$a_{ij} = u_{ij} + v_{ij} + s_{ij},$$
   where $u_{ij}$ is symmetric (i.e., $u_{ji}=u_{ij}$ ) and traceless (i.e., $u_{ii}=0$ ), $v_{ij}$ is isotropic, and $s_{ij}$ only has three independent components.

2. Use tensor methods to establish the following vector identities:
   1. ${\bf a}\cdot{\bf b}\times {\bf c}={\bf a}\times {\bf b}\cdot{\bf c}=-{\bf b}\cdot{\bf a}\times{\bf c}$ .
   2. $({\bf a}\times {\bf b})\times {\bf c} = ({\bf a}\cdot{\bf c})\,{\bf b} - ({\bf c}\cdot{\bf b})\,{\bf a}$ .
   3. $({\bf a}\times {\bf b})\cdot({\bf c}\times {\bf d})= ({\bf a}\cdot{\bf c})\,({\bf b}\cdot{\bf d}) - ({\bf a}\cdot{\bf d})\,({\bf b}\cdot{\bf c})$ .
   4. $({\bf a}\times {\bf b})\times ({\bf c}\times {\bf d})= ({\bf a}\times {\bf b}\cdot{\bf d})\,{\bf c} - ({\bf a}\times {\bf b}\cdot{\bf c})\,{\bf d}$ .
   5. $\nabla\cdot(\phi\,{\bf a}) = \phi\,\nabla\cdot {\bf a} + {\bf a}\cdot\nabla \phi$ .
   6. $\nabla\times (\phi\,{\bf a})= \phi\,\nabla\times {\bf a} + \nabla\phi\times {\bf a}$ .
   7. $\nabla\times({\bf a}\times {\bf b}) = ({\bf b}\cdot\nabla)\,{\bf a} - ({\bf a}... ...nabla)\,{\bf b} + (\nabla\cdot {\bf b})\,{\bf a} -(\nabla\cdot{\bf a})\,{\bf b}$ .
   8. $\nabla ({\bf a}\cdot{\bf a}) = 2\,{\bf a}\times (\nabla\times{\bf a}) + 2\,({\bf a}\cdot\nabla)\,{\bf a}$ .
   9. $\nabla\times(\nabla\times {\bf a}) = \nabla(\nabla\cdot{\bf a}) - \nabla^{\,2} {\bf a}$ .
   Here, $[({\bf b}\cdot{\nabla}){\bf a}]_i= b_j\,\partial a_i/\partial x_j$ , and $(\nabla^{\,2} {\bf a})_i = \nabla^{\,2} a_i$ .

3. A quadric surface has an equation of the form
   $$a\,x_1^{\,2} + b\,x_2^{\,2}+c\,x_3^{\,2} + 2\,f\,x_1\,x_2+2\,g\,x_1\,x_3+2\,h\,x_2\,x_3=1.$$
   Show that the coefficients in the previous expression transform under rotation of the coordinate axes like the components of a symmetric second-order tensor. Hence, demonstrate that the equation for the surface can be written in the form
   $$x_i\,T_{ij}\,x_j = 1,$$
   where the $T_{ij}$ are the components of the aforementioned tensor.

4. The determinant of a second-order tensor $A_{ij}$ is defined
   $${\rm det}(A) = \epsilon_{ijk}\,A_{i1}\,A_{j2}\,A_{k3}.$$
   1. Show that
      $${\rm det}(A) = \epsilon_{ijk}\,A_{1i}\,A_{2j}\,A_{3k}$$
      is an alternative, and entirely equivalent, definition.
   2. Demonstrate that ${\rm det}(A)$ is invariant under rotation of the coordinate axes.
   3. Suppose that $C_{ij} = A_{ik}\,B_{kj}$ . Show that
      $${\rm det}(C) = {\rm det}(A)\,{\rm det}(B).$$

5. If
   $$A_{ij}\,x_j = \lambda\,x_i$$
   then $\lambda$ and $x_j$ are said to be eigenvalues and eigenvectors of the second-order tensor $A_{ij}$ , respectively. The eigenvalues of $A_{ij}$ are calculated by solving the related homogeneous matrix equation
   $$(A_{ij}-\lambda\,\delta_{ij})\,x_j = 0.$$
   Now, it is a standard result in linear algebra that an equation of the previous form only has a non-trivial solution when (Riley 1974)
   $${\rm det}(A_{ij}-\lambda\,\delta_{ij}) = 0.$$
   Demonstrate that the eigenvalues of $A_{ij}$ satisfy the cubic polynomial
   $$\lambda^{\,3} - {\rm tr}(A) \,\lambda^{\,2} + {\mit\Pi}(A)\,\lambda - {\rm det}(A) = 0,$$
   where ${\rm tr}(A) = A_{ii}$ and ${\mit\Pi}(A)= (A_{ii}\,A_{jj}-A_{ij}\,A_{ji})/2$ . Hence, deduce that $A_{ij}$ possesses three eigenvalues--$\lambda_1$ , $\lambda_2$ , and $\lambda_3$ (say). Moreover, show that

   $${\rm tr}(A) =\lambda_1+\lambda_2+\lambda_3,$$

   $${\rm det}(A) =\lambda_1\,\lambda_2\,\lambda_3.$$

   
   

6. Suppose that $A_{ij}$ is a (real) symmetric second-order tensor: that is, $A_{ji}=A_{ij}$ .
   1. Demonstrate that the eigenvalues of $A_{ij}$ are all real, and that the eigenvectors can be chosen to be real.
   2. Show that eigenvectors of $A_{ij}$ corresponding to different eigenvalues are orthogonal to one another. Hence, deduce that the three eigenvectors of $A_{ij}$ are, or can be chosen to be, mutually orthogonal.
   3. Demonstrate that $A_{ij}$ takes the diagonal form $A_{ij}=\lambda_i\,\delta_{ij}$ (no sum) in a Cartesian coordinate system in which the coordinate axes are each parallel to one of the eigenvectors.

7. In an isotropic elastic medium under stress, the displacement $u_i$ satisfies

   $$\frac{\partial\sigma_{ij}}{\partial x_j} =\rho\,\frac{\partial^{\,2} u_i}{\partial t^{\,2}},$$

   $$\sigma_{ij} =c_{ijkl}\,\frac{1}{2}\!\left(\frac{\partial u_k}{\partial x_l}+ \frac{\partial u_l}{\partial x_k}\right),$$

   
   
   where $\sigma_{ij}$ is the stress tensor (note that $\sigma_{ji}=\sigma_{ij}$ ), $\rho$ the mass density (which is a uniform constant), and
   $$c_{ijkl} = K\,\delta_{ij}\,\delta_{kl} + \mu\,[\delta_{ik}\,\delta_{jl}+\delta_{il}\,\delta_{jk} - (2/3)\,\delta_{ij}\,\delta_{kl}].$$
   the isotropic stiffness tensor. Here, $K$ and $\mu$ are the bulk modulus and shear modulus of the medium, respectively. Show that the divergence and the curl of ${\bf u}$ both satisfy wave equations. Furthermore, demonstrate that the characteristic wave velocities of the divergence and curl waves are $[(K+4\mu/3)/\rho]^{1/2}$ and $(\mu/\rho)^{1/2}$ , respectively.