Page 106 - Intro to Tensor Calculus

101

EXERCISE 1.3

I 1.
a
∂x ∂x b
(a) From the transformation law for the second order tensor g ij = g ab i j
∂x ∂x
solve for the g ab in terms of g .
ij
(b) Show that if g ij is symmetric in one coordinate system it is symmetric in all coordinate systems.
p √ x
2 x
(c) Let g = det(g )and g = det(g ij )and show that g = gJ ( ) and consequently g = gJ( ). This
ij x x
√
shows that g is a scalar invariant of weight 2 and g is a scalar invariant of weight 1.
I 2. For
i
∂y m ∂y m ij ∂x ∂x j
g ij = show that g =
i
∂x ∂x j ∂y m ∂y m
I 3. Show that in a curvilinear coordinate system which is orthogonal we have:
(a) g = det(g ij )= g 11 g 22 g 33
(b) g mn = g mn =0 for m 6= n
1
NN
(c) g = for N =1, 2, 3 (no summation on N)
g NN
i 2
∂y 2
I 4. Show that g = det(g ij )= = J , where J is the Jacobian.
∂x j
∂~ ∂~ ∂~ r
r
r
I 5. Define the quantities h 1 = h u = | |, h 2 = h v = | |, h 3 = h w = | | and construct the unit
∂u ∂v ∂w
vectors
1 ∂~r 1 ∂~r 1 ∂~r
b e u = , b e v = , b e w = .
h 1 ∂u h 2 ∂v h 3 ∂w
(a) Assume the coordinate system is orthogonal and show that
∂x 2 ∂y 2 ∂z 2
2
g 11 = h = + + ,
1
∂u ∂u ∂u
∂x 2 ∂y 2 ∂z 2
2
g 22 = h = + + ,
2
∂v ∂v ∂v
2 2 2
∂x ∂y ∂z
2
g 33 = h = + + .
3
∂w ∂w ∂w
(b) Show that d~ can be expressed in the form d~ = h 1 b e u du + h 2 b e v dv + h 3 b e w dw.
r
r
r
(c) Show that the volume of the elemental parallelepiped having d~ as diagonal can be represented
√ ∂(x, y, z)
dτ = gdudvdw = Jdudvdw = dudvdw.
∂(u, v, w)
Hint:
A 1 A 2 A 3
~ ~ ~
|A · (B × C)| = B 1 B 2 B 3

C 1 C 2 C 3