12—Tensors 39312.8 What is the significance of a tensor satisfying the relationT ̃[T(~v)] =T[T ̃(~v)] =~vfor all~v?
12.9 Carry out the construction indicated in section12.3to show the dielectric tensor is symmetric.
12.10 Fill in the details of the proof that the alternating tensor is unique up to a factor.
12.11 Compute the components of such an alternating tensor in two dimensions.
12.12 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.
12.13 Take a basis~e 1 = 2ˆx,~e 2 =ˆx+ 2ˆy. Compute the reciprocal basis. Find the components of the vectors
A~=xˆ−ˆyandB~=yˆin in each basis and computeA~.B~ several different ways.
O A CBDv12.14 Show that if the direct basis vectors have unit length along the directions of
−→
OA
and
−−→
OBthen the components of~vin the direct basis are the lengthsOAandOB. What
are the components in the reciprocal basis?
12.15 Derive the equations ( 37 ).
12.16 What happens to the components of the alternating tensor when a change of basis is
made? Show that the only thing that can happen is that all the components are multiplied
by the same number (defined to be the determinant of the transformation). Compute this
explicitly in two dimensions. Ans: The determinant
12.17 If a tensor (viewed as a function of two vector variables) has the property that it equals zero whenever
the two arguments are the same,T(~v,~v) = 0, then it is antisymmetric. This is also true if it is a function of
more than two variables and the above equation holds on some pair of arguments. Consider~v=α~u+β~w
12.18 If the components of the alternating tensor are (in three dimensions)eijkwheree 123 = 1, Computeeijk.
Compute
eijkemk, eijkejk, eijkeijk