Problems for Appendix B

 

Brief Introduction to Tensors

 

 

 

 

A.      

B.1    Let  be a Cartesian basis. Vector u has components  in this basis, while tensors S and T have components

 

B.1.1                    Calculate the components of the following vectors and tensors

     

B.1.2                    Find the eigenvalues and the components of the eigenvectors of T.

B.1.3                    Denote the three (unit) eigenvectors of T by , ,  (It doesn’t matter which eigenvector is which, but be sure to state your choice clearly).

B.1.4                     Let  be a new Cartesian basis.  Write down the components of T in .  (Don’t make this hard: in the new basis, T must be diagonal, and the diagonal elements must be the eigenvalues.  Do you see why this is the case? You just need to get them in the right order!)

B.1.5                    Calculate the components of S in the basis . 

 

B.2    Let S and T be tensors with components

 

B.2.1                    Calculate  and  

B.2.2                    Calculate trace(S) and trace(T)

 

B.3    Let   be a Cartesian basis, and let , ,  be three unit vectors.   The components of a tensor T in  are

 

B.3.1                    Verify that  is also Cartesian basis.

B.3.2                    Calculate the components of T in  

 

 

B.4    Let  be a Cartesian basis, and let , ,  be three vectors.

B.4.1                    Calculate the components in  of the tensor T that satisfies .

B.4.2                    Calculate the eigenvalues and eigenvectors of T.

B.4.3                    Calculate the components of T in a basis of unit vectors parallel to , , .

 

 

B.5    Let S be a tensor, and let

    

denote the components of S in Cartesian bases  and , respectively.   Show that

the trace of S is invariant to a change of basis, i.e. show that .

 

B.6    Show that the inner product of two tensors is invariant to a change of basis.

 

B.7    Show that the outer product of two tensors is invariant to a change of basis.

 

B.8    Show that the eigenvalues of a tensor are invariant to a change of basis.  Are the eigenvectors similarly invariant?

 

B.9    Let S be a real symmetric tensor with three distinct eigenvalues  and corresponding eigenvectors .   Show that  for .

 

B.10   Let S be a real symmetric tensor with three distinct eigenvalues  and corresponding normalized eigenvectors  satisfying .  Use the results of B.9 to show that

 

for any arbitrary vector b, and hence deduce that

 

 

B.11  Use the results of B.10. to find a way to calculate the square root of a real, symmetric tensor.

 

B.12 Let

 

Find expressions for the eigenvalues and eigenvectors of T in terms of its components  

 

 

 

(c) A.F. Bower, 2008
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