Page 109 - Intro to Tensor Calculus

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Figure 1.3-20. Spherical surface coordinates

I 18. Given the fourth order tensor C ikmp = λδ ik δ mp + µ(δ im δ kp + δ ip δ km )+ ν(δ im δ kp − δ ip δ km )where λ, µ
and ν are scalars and δ ij is the Kronecker delta. Show that under an orthogonal transformation of rotation of

axes with x i = ` ij x j where ` rs ` is = ` mr ` mi = δ ri the components of the above tensor are unaltered. Any
tensor whose components are unaltered under an orthogonal transformation is called an ‘isotropic’ tensor.
Another way of stating this problem is to say “Show C ikmp is an isotropic tensor.”

I 19. Assume A ijl is a third order covariant tensor and B pqmn is a fourth order contravariant tensor. Prove
that A ikl B klmn is a mixed tensor of order three, with one covariant and two contravariant indices.

I 20. Assume that T mnrs is an absolute tensor. Show that if T ijkl + T ijlk = 0 in the coordinate system x r
r
then T ijkl + T ijlk = 0 in any other coordinate system x .
I 21. Show that

g ir g is g it

ijk rst = g jr g js g jt

g kr g ks g kt
Hint: See problem 38, Exercise 1.1
I 22. Determine if the tensor equation mnp mij + mnj mpi = mni mpj is true or false. Justify your answer.
ij
I 23. Prove the epsilon identity g ipt jrs = g pr g ts − g ps g tr . Hint: See problem 38, Exercise 1.1
1 mn
rs
I 24. Let A denote a skew-symmetric contravariant tensor and let c r = rmn A where
2
√
rmn = ge rmn. Show that c r are the components of a covariant tensor. Write out all the components.
1 rmn rmn 1 rmn
r
I 25. Let A rs denote a skew-symmetric covariant tensor and let c = A mn where = √ e .
2 g
r
Show that c are the components of a contravariant tensor. Write out all the components.