Page 110 - Intro to Tensor Calculus

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I 26. Let A pq B qs = C s where B qs is a relative tensor of weight ω 1 and C s is a relative tensor of weight
r pr r pr
ω 2 . Prove that A pq is a relative tensor of weight (ω 2 − ω 1 ).

i
i
I 27. When A is an absolute tensor prove that √ gA is a relative tensor of weight +1.
j
j
1
i
i
I 28. When A is an absolute tensor prove that √ A is a relative tensor of weight −1.
j g j
I 29.
(a) Show e ijk is a relative tensor of weight +1.
1
(b) Show ijk = √ e ijk is an absolute tensor. Hint: See example 1.1-25.
g
3
2
1
I 30. The equation of a surface can be represented by an equation of the form Φ(x ,x ,x )= constant.
Show that a unit normal vector to the surface can be represented by the vector
g ij ∂Φ j
i
n = ∂x 1 .
(g mn ∂Φ ∂Φ n ) 2
∂x m ∂x
ij
I 31. Assume that g = λg ij with λ a nonzero constant. Find and calculate g ij in terms of g .
ij
I 32. Determine if the following tensor equation is true. Justify your answer.

r
r
r
r
rjk A + irk A + ijr A = ijk A .
i
r
k
j
Hint: See problem 21, Exercise 1.1.
i
j
i
I 33. Show that for C i and C associated tensors, and C = ijk A j B k , then C i = ijk A B k
I 34. Prove that ijk and ijk are associated tensors. Hint: Consider the determinant of g ij .
k
j
i
I 35. Show ijk A i B j C k = ijk A B C .
i
I 36. Let T ,i, j =1, 2, 3 denote a second order mixed tensor. Show that the given quantities are scalar
j
invariants.
(i) I 1 = T i
i
1 i 2 i m
(ii) I 2 = (T ) − T T
i
m i
2
i
(iii) I 3 = det|T |
j
I 37.
ij
ij
(a) Assume A and B ,i, j =1, 2, 3 are absolute contravariant tensors, and determine if the inner product
ij
C ik = A B jk is an absolute tensor?
j
∂x ∂x j
(b) Assume that the condition = δ nm is satisfied, and determine whether the inner product in
n
∂x ∂x m
part (a) is a tensor?
(c) Consider only transformations which are a rotation and translation of axes y = ` ij y j +b i , where ` ij are
i
∂y ∂y j
j
direction cosines for the rotation of axes. Show that = δ nm
∂y n ∂y m