Page 33 - Intro to Tensor Calculus

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I 10. For A =(1, −1, 0) and B =(4, −3, 2) find using the index notation,
(a) C i = e ijk A j B k , i =1, 2, 3

(b) A i B i
(c) What do the results in (a) and (b) represent?

dy 1 dy 2
I 11. Represent the differential equations = a 11 y 1 + a 12 y 2 and = a 21 y 1 + a 22 y 2
dt dt
using the index notation.
I 12.
Let Φ = Φ(r, θ)where r, θ are polar coordinates related to Cartesian coordinates (x, y) by the transfor-
mation equations x = r cos θ and y = r sin θ.
2
∂Φ ∂ Φ
(a) Find the partial derivatives , and
∂y ∂y 2
(b) Combine the result in part (a) with the result from EXAMPLE 1.1-18 to calculate the Laplacian
2
2
∂ Φ ∂ Φ
2
∇ Φ= +
∂x 2 ∂y 2
in polar coordinates.
I 13. (Index notation) Let a 11 =3, a 12 =4, a 21 =5, a 22 =6.
Calculate the quantity C = a ij a ij ,i, j =1, 2.

I 14. Show the moments of inertia I ij defined by
ZZZ ZZZ
2
2
I 11 = (y + z )ρ(x, y, z) dτ I 23 = I 32 = − yzρ(x, y, z) dτ
R R
ZZZ ZZZ
2
2
I 22 = (x + z )ρ(x, y, z) dτ I 12 = I 21 = − xyρ(x, y, z) dτ
R R
ZZZ ZZZ
2
2
I 33 = (x + y )ρ(x, y, z) dτ I 13 = I 31 = − xzρ(x, y, z) dτ,
R R
ZZZ

m m
i j
can be represented in the index notation as I ij = x x δ ij − x x ρdτ, where ρ is the density,
R
2
3
1
x = x, x = y, x = z and dτ = dxdydz is an element of volume.
I 15. Determine if the following relation is true or false. Justify your answer.
b e i · ( b e j × b e k )= ( b e i × b e j ) · b e k = e ijk , i,j,k =1, 2, 3.
Hint: Let b e m =(δ 1m ,δ 2m ,δ 3m ).
I 16. Without substituting values for i, l =1, 2, 3 calculate all nine terms of the given quantities

il
i
i
(a) B =(δ A k + δ A j )e jkl (b) m k k m
i
i
j k A il =(δ B + δ B )e mlk
i
m n
i
I 17. Let A mn x y = 0 for arbitrary x and y , i =1, 2, 3, and show that A ij = 0 for all values of i, j.