Page 67 - Intro to Tensor Calculus

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I 40. Let A = A (x ,x ,...,x ) denote the components of an absolute contravariant tensor. Form the
i
quantity B = ∂A i i
∂x j and determine if B transforms like a tensor.
j j
∂A i ∂A j
I 41. Let A i denote the components of a covariant vector. (a) Show that a ij = − are the
∂x j ∂x i
∂a ij ∂a jk ∂a ki
components of a second order tensor. (b) Show that + + =0.
∂x k ∂x i ∂x j
i
I 42. Show that x = Ke ijk A j B k ,with K 6= 0 and arbitrary, is a general solution of the system of equations
i
i
A i x =0,B i x =0,i =1, 2, 3. Give a geometric interpretation of this result in terms of vectors.
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I 43. Given the vector A = y b e 1 + z b e 2 + x b e 3 where b e 1 , b e 2 , b e 3 denote a set of unit basis vectors which
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define a set of orthogonal x, y, z axes. Let E 1 =3 b e 1 +4 b e 2, E 2 =4 b e 1 +7 b e 2 and E 3 = b e 3 denote a set of
basis vectors which define a set of u, v, w axes. (a) Find the coordinate transformation between these two
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~ 1 ~ 3 ~ 3
sets of axes. (b) Find a set of reciprocal vectors E , E , E . (c) Calculate the covariant components of A.
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(d) Calculate the contravariant components of A.
I 44. Let A = A ij b e i b e j denote a dyadic. Show that
A : A c = A 11 A 11 + A 12 A 21 + A 13 A 31 + A 21 A 12 + A 22 A 22 + A 23 A 32 + A 31 A 13 + A 32 A 23 + A 23 A 33
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~ ~
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I 45. Let A = A i b e i , B = B i b e i , C = C i b e i , D = D i b e i denote vectors and let φ = AB, ψ = CD denote
dyadics which are the outer products involving the above vectors. Show that the double dot product satisfies
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φ : ψ = AB : CD =(A · C)(B · D)
I 46. Show that if a ij is a symmetric tensor in one coordinate system, then it is symmetric in all coordinate
systems.
I 47. Write the transformation laws for the given tensors. (a) A k (b) A ij (c) A ijk
ij k m
∂x j ∂x j
I 48. Show that if A i = A j i ,then A i = A j ∂x i . Note that this is equivalent to interchanging the bar
∂x
and unbarred systems.
I 49.
(a) Show that under the linear homogeneous transformation
1 2
x 1 =a x 1 + a x 2
1 1
1 2
x 2 =a x 1 + a x 2
2
2
the quadratic form
2
2
Q(x 1 ,x 2 )= g 11 (x 1 ) +2g 12x 1 x 2 + g 22 (x 2 ) 2 becomes Q(x 1 , x 2 )= g (x 1 ) +2g x 1 x 2 + g (x 2 ) 2
22
11
12
j
j
i
j i
j i
i
where g ij = g 11 a a + g 12(a a + a a )+ g 22 a a .
1 2
1 2
1 1
2 2
2
(b) Show F = g 11 g 22 −(g 12 ) is a relative invariant of weight 2 of the quadratic form Q(x 1 ,x 2 ) with respect
2
to the group of linear homogeneous transformations. i.e. Show that F =∆ F where F = g g −(g ) 2
12
11 22
2 1
1 2
and ∆ = (a a − a a ).
1 2
1 2