12—Tensors 322Problems12.1 Does the functionTdefined byT(v) =v+cwithca constant satisfy the definition of linearity?
12.2 Let the set X be the positive integers. Let the set Y be all real numbers. Consider the following
sets and determine if they are relations between X and Y and if they are functions.
{(0,0),(1, 2 .0),(3,−π),(0, 1 .0),(− 1 ,e)}
{(0,0),(1, 2 .0),(3,−π),(0, 1 .0),(2,e)}
{(0,0),(1, 2 .0),(3,−π),(4, 1 .0),(2,e)}
{(0,0),(5, 5 .5),(5.,π) (3,− 2 .0) (7,8)}
12.3 Starting from the definition of the tensor of inertia in Eq. (12.1) and using the defining equation
for components of a tensor, compute the components ofI.
12.4 Find the components of the tensor relatingd~andF~in the example of Eq. (12.2)
12.5 The product of tensors is defined to be just the composition of functions for the second rank tensor
viewed as a vector variable. IfSandT are such tensors, then(ST)(v) =S(T(v))(by definition)
Compute the components ofSTin terms of the components ofSand ofT. Express the result both
in terms of index notation and matrices. Ans: This is matrix multiplication.
12.6 (a) The two tensors^11 T and^11 T ̃ are derived from the same bilinear functional^02 T, in the
Eqs. (12.20)–(12.22). Prove that for arbitrary~uand~v,
~u.^11 T(~v) =^11 T ̃(~u).~v
(If it’s less confusing to remove all the sub- and superscripts, do so.)
(b) If you did this by writing everything in terms of components, do it again without components
and just using the nature of these as functions. (If you did in without components, do it again using
components.)
12.7 What is the significance of a tensor satisfying the relationT ̃[T(~v)] =T[T ̃(~v)] =~vfor all~v?
Look at what effect it has on the scalar product of two vectors if you letTact on both. That is, what
is the scalar product ofT~uandT~v?
12.8 Carry out the construction indicated in section12.3to show the dielectric tensor is symmetric.
12.9 Fill in the details of the proof that the alternating tensor is unique up to a factor.
12.10 Compute the components of such an alternating tensor in two dimensions.
Ans: The matrix is anti-symmetric.
12.11 Take an alternating tensor in three dimensions, pull off one of the arguments in order to obtain
a vector valued function of two vector variables. See what the result looks like, and in particular, write
it out in component form.
Ans: You should find a cross product here.