Page 111 - Intro to Tensor Calculus

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I 38. For A ijk a Cartesian tensor, determine if a contraction on the indices i and j is allowed. That
is, determine if the quantity A k = A iik , (summation on i) is a tensor. Hint: See part(c) of the previous
problem.
j k
k
j
I 39. Prove the e-δ identity e ijk e imn = δ δ − δ δ .
n m
m n
I 40. Consider the vector V k ,k =1, 2, 3 and define the matrix (a ij ) having the elements a ij = e ijk V k ,
where e ijk is the e−permutation symbol.
(a) Solve for V i in terms of a mn by multiplying both sides of the given equation by e ijl and note the e − δ
identity allows us to simplify the result.
(b) Sum the given expression on k and then assign values to the free indices (i,j=1,2,3) and compare your
results with part (a).
(c) Is a ij symmetric, skew-symmetric, or neither?

I 41. It can be shown that the continuity equation of fluid dynamics can be expressed in the tensor form

1 ∂ √ r ∂%
√ r ( g%V )+ =0,
g ∂x ∂t
r
where % is the density of the fluid, t is time, V ,with r =1, 2, 3 are the velocity components and g = |g ij |
is the determinant of the metric tensor. Employing the summation convention and replacing the tensor
components of velocity by their physical components, express the continuity equation in
(a) Cartesian coordinates (x, y, z) with physical components V x ,V y ,V z .
(b) Cylindrical coordinates (r, θ, z) with physical components V r ,V θ ,V z .
(c) Spherical coordinates (ρ, θ, φ) with physical components V ρ ,V θ ,V φ .

2
3
1
1
3
2
I 42. Let x ,x ,x denote a set of skewed coordinates with respect to the Cartesian coordinates y ,y ,y .
~
~
~
3
2
1
Assume that E 1 , E 2 , E 3 are unit vectors in the directions of the x ,x and x axes respectively. If the unit
vectors satisfy the relations
~
~
~
~
E 1 · E 1 =1 E 1 · E 2 =cos θ 12
~
~
~
~
E 2 · E 2 =1 E 1 · E 3 =cos θ 13
~
~
~
~
E 3 · E 3 =1 E 2 · E 3 =cos θ 23 ,
ij
then calculate the metrices g ij and conjugate metrices g .
I 43. Let A ij ,i, j =1, 2, 3, 4 denote the skew-symmetric second rank tensor
0 a b c
 
 −a 0 d e 
−b −d 0 f
A ij =   ,
−c −e −f 0
where a, b, c, d, e, f are complex constants. Calculate the components of the dual tensor
1 ijkl
ij
V = e A kl .
2