Page 34 - Intro to Tensor Calculus

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I 18.
m n
(a) For a mn ,m,n =1, 2, 3 skew-symmetric, show that a mn x x =0.
m n
i
(b) Let a mn x x =0, m, n =1, 2, 3 for all values of x ,i =1, 2, 3and show that a mn must be skew-
symmetric.

I 19. Let A and B denote 3 × 3 matrices with elements a ij and b ij respectively. Show that if C = AB is a
matrix product, then det(C)= det(A) · det(B).
Hint: Use the result from example 1.1-9.

I 20.
2
3
3
1
2
1
2
3
1
(a) Let u ,u ,u be functions of the variables s ,s ,s . Further, assume that s ,s ,s are in turn each

m 1 2 3
∂u ∂(u ,u ,u )
3
1
2
functions of the variables x ,x ,x . Let = denote the Jacobian of the u s with
0
∂x ∂(x ,x ,x )
n 1 2 3
0
respect to the x s. Show that

i i j i j
∂u ∂u ∂s ∂u ∂s
·
= = .
∂x m ∂s ∂x m ∂s j ∂x m
j
i
∂x ∂¯x j ∂x i i x ¯ x x
(b) Note that = = δ m and show that J( )·J( ) = 1, where J( ) is the Jacobian determinant
j
∂¯x ∂x m ∂x m ¯ x x ¯ x
of the transformation (1.1.7).
I 21. A third order system a `mn with `, m, n =1, 2, 3 is said to be symmetric in two of its subscripts if the
components are unaltered when these subscripts are interchanged. When a `mn is completely symmetric then
a `mn = a m`n = a `nm = a mn` = a nm` = a n`m . Whenever this third order system is completely symmetric,
then: (i) How many components are there? (ii) How many of these components are distinct?
Hint: Consider the three cases (i) ` = m = n (ii) ` = m 6= n (iii) ` 6= m 6= n.
I 22. A third order system b `mn with `, m, n =1, 2, 3 is said to be skew-symmetric in two of its subscripts
if the components change sign when the subscripts are interchanged. A completely skew-symmetric third
order system satisfies b `mn = −b m`n = b mn` = −b nm` = b n`m = −b `nm. (i) How many components does
a completely skew-symmetric system have? (ii) How many of these components are zero? (iii) How many
components can be different from zero? (iv) Show that there is one distinct component b 123 and that
b `mn = e `mn b 123 .
Hint: Consider the three cases (i) ` = m = n (ii) ` = m 6= n (iii) ` 6= m 6= n.
I 23. Let i, j, k =1, 2, 3 and assume that e ijk σ jk = 0 for all values of i. What does this equation tell you
about the values σ ij , i, j =1, 2, 3?

m n
m n
I 24. Assume that A mn and B mn are symmetric for m, n =1, 2, 3. Let A mn x x = B mn x x for arbitrary
i
values of x ,i =1, 2, 3, and show that A ij = B ij for all values of i and j.
m n
i
I 25. Assume B mn is symmetric and B mn x x = 0 for arbitrary values of x ,i =1, 2, 3, show that B ij =0.