Page 66 - Intro to Tensor Calculus

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I 27. Let A ij and A ij denote absolute second order tensors. Show that λ = A ij A ij is a scalar invariant.

I 28. Assume that a ij , i, j =1, 2, 3, 4 is a skew-symmetric second order absolute tensor. (a) Show that

∂a jk ∂a ki ∂a ij
b ijk = + +
∂x i ∂x j ∂x k
is a third order tensor. (b) Show b ijk is skew-symmetric in all pairs of indices and (c) determine the number
of independent components this tensor has.

I 29. Show the linear forms A 1 x + B 1 y + C 1 and A 2 x + B 2 y + C 2 , with respect to the group of rotations
and translations x = x cos θ − y sin θ + h and y = x sin θ + y cos θ + k, have the forms A 1 x + B 1 y + C 1 and
A 2 x + B 2 y + C 2 . Also show that the quantities A 1 B 2 − A 2 B 1 and A 1 A 2 + B 1 B 2 are invariants.
02 −3/2
I 30. Show that the curvature of a curve y = f(x)is κ = ± y (1 + y ) and that this curvature remains
00
invariant under the group of rotations given in the problem 1. Hint: Calculate dy = dy dx .
dx dx dx
I 31. Show that when the equation of a curve is given in the parametric form x = x(t),y = y(t), then
y
˙ x¨ − ˙y¨
x
the curvature is κ = ± and remains invariant under the change of parameter t = t(t), where
2
2 3/2
(˙x +˙y )
˙ x = dx , etc.
dt
ij ij
I 32. Let A denote a third order mixed tensor. (a) Show that the contraction A is a first order
k i
ii
contravariant tensor. (b) Show that contraction of i and j produces A which is not a tensor. This shows
k
that in general, the process of contraction does not always apply to indices at the same level.
∂φ
N
1
2
I 33. Let φ = φ(x ,x ,... ,x ) denote an absolute scalar invariant. (a) Is the quantity ∂x i a tensor? (b)
2
∂ φ
Is the quantity j a tensor?
i
∂x ∂x
I 34. Consider the second order absolute tensor a ij ,i, j =1, 2where a 11 =1, a 12 =2, a 21 =3 and a 22 =4.
2
1
1
1
2
2
Find the components of a ij under the transformation of coordinates x = x + x and x = x − x .
I 35. Let A i , B i denote the components of two covariant absolute tensors of order one. Show that
C ij = A i B j is an absolute second order covariant tensor.
i
I 36. Let A denote the components of an absolute contravariant tensor of order one and let B i denote the
i
i
components of an absolute covariant tensor of order one, show that C = A B j transforms as an absolute
j
mixed tensor of order two.
I 37. (a) Show the sum and difference of two tensors of the same kind is also a tensor of this kind. (b) Show
that the outer product of two tensors is a tensor. Do parts (a) (b) in the special case where one tensor A i
j
is a relative tensor of weight 4 and the other tensor B is a relative tensor of weight 3. What is the weight
k
ij i j
of the outer product tensor T = A B in this special case?
k k
ij j ij
I 38. Let A denote the components of a mixed tensor of weight M. Form the contraction B = A
km m im
j
and determine how B m transforms. What is its weight?
i
I 39. Let A denote the components of an absolute mixed tensor of order two. Show that the scalar
j
i
contraction S = A is an invariant.
i