For a given set of coordinate axes the components of a 2 nd rank tensor are given by .
(a) Show that if is an eigenvalue of the matrix with elements then it is also an eigenvalue of the matrix of the components of in any other coordinate frame.
Show that if is a symmetric tensor then the multiplicity of the eigenvalues of the matrix of components of is independent of coordinate frame.
A symmetric tensor in three dimensions has eigenvalues , with .
Show that the components of can be written in the form
where are the components of a unit vector.
(b) The tensor is defined by
where is the surface of the unit sphere, is the position vector of a point on , and is a constant.
Deduce, with brief reasoning, that the components of can be written in the form (1) with . [You may quote any results derived in part (a).]
Using suitable spherical polar coordinates evaluate and .
Explain how to deduce the values of and from and . [You do not need to write out the detailed formulae for these quantities.]