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# Exercises

1. Show that a general second-order tensor can be decomposed into three tensors

where is symmetric (i.e., ) and traceless (i.e., ), is isotropic, and only has three independent components.

2. Use tensor methods to establish the following vector identities:
1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
Here, , and .

3. A quadric surface has an equation of the form

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

where the are the components of the aforementioned tensor.

4. The determinant of a second-order tensor is defined

1. Show that

is an alternative, and entirely equivalent, definition.
2. Demonstrate that is invariant under rotation of the coordinate axes.
3. Suppose that . Show that

5. If

then and are said to be eigenvalues and eigenvectors of the second-order tensor , respectively. The eigenvalues of are calculated by solving the related homogeneous matrix equation

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)

Demonstrate that the eigenvalues of satisfy the cubic polynomial

where and . Hence, deduce that possesses three eigenvalues-- , , and (say). Moreover, show that

6. Suppose that is a (real) symmetric second-order tensor: that is, .
1. Demonstrate that the eigenvalues of are all real, and that the eigenvectors can be chosen to be real.
2. Show that eigenvectors of corresponding to different eigenvalues are orthogonal to one another. Hence, deduce that the three eigenvectors of are, or can be chosen to be, mutually orthogonal.
3. Demonstrate that takes the diagonal form (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 satisfies

where is the stress tensor (note that ), the mass density (which is a uniform constant), and

the isotropic stiffness tensor. Here, and are the bulk modulus and shear modulus of the medium, respectively. Show that the divergence and the curl of both satisfy wave equations. Furthermore, demonstrate that the characteristic wave velocities of the divergence and curl waves are and , respectively.

Next: Non-Cartesian Coordinates Up: Cartesian Tensors Previous: Isotropic Tensors
Richard Fitzpatrick 2016-01-22