12—Tensors 392Problems12.1 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 Instead of the coordinate-free proof of the representation theorem for linear functionals, Eq. ( 5 ), give a
proof in terms of the usual orthonormal basisxˆ,ˆy,zˆby lettingfact on them and constructingA~in terms of its
components. Perhaps guess an answer and show that it works.
12.4 Starting from the definition of the tensor of inertia in Eq. ( 1 ) and using the defining equation for components
of a tensor, compute the components of I.
12.5 Find the components of the tensor relatingd~andF~ in the example of Eq. ( 2 )
12.6 The product of tensors is defined to be just the composition of functions for the second rank tensor viewed as
a vector variable. IfSandTare 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.
12.7 The two tensors^11 T and^11 T ̃are derived from the same bilinear functional^02 T. Prove that for arbitrary~u
and~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.)