Page 107 - Intro to Tensor Calculus

102

Figure 1.3-18 Oblique cylindrical coordinates.

r
I 6. For the change d~ given in problem 5, show the elemental parallelepiped with diagonal d~ has:
r
q
(a) the element of area dS 1 = g 22 g 33 − g 2 dvdw in the u =constant surface.
23
q
(b) The element of area dS 2 = g 33 g 11 − g 2 13 dudw in the v =constant surface.
q
2
(c) the element of area dS 3 = g 11 g 22 − g 12 dudv in the w =constant surface.

(d) What do the above elements of area reduce to in the special case the curvilinear coordinates are orthog-
q
~
~
~
~
~
~
|A × B| = (A × B) · (A × B)
onal? Hint: q .
~ ~
~
~ ~
~ ~ ~
= (A · A)(B · B) − (A · B)(A · B)
I 7. In Cartesian coordinates you are given the affine transformation. x i = ` ij x j where
1 1 1
x 1 = (5x 1 − 14x 2 +2x 3 ), x 2 = − (2x 1 + x 2 +2x 3 ), x 3 = (10x 1 +2x 2 − 11x 3)
15 3 15
(a) Show the transformation is orthogonal.
~
(b) A vector A(x 1 ,x 2 ,x 3 ) in the unbarred system has the components
2 2 2
A 1 =(x 1 ) , A 2 =(x 2 ) A 3 =(x 3 ) .
Find the components of this vector in the barred system of coordinates.

I 8. Calculate the metric and conjugate metric tensors in cylindrical coordinates (r, θ, z).
I 9. Calculate the metric and conjugate metric tensors in spherical coordinates (ρ, θ, φ).
I 10. Calculate the metric and conjugate metric tensors in parabolic cylindrical coordinates (ξ, η, z).
I 11. Calculate the metric and conjugate metric components in elliptic cylindrical coordinates (ξ, η, z).
I 12. Calculate the metric and conjugate metric components for the oblique cylindrical coordinates (r, φ, η),
illustrated in figure 1.3-18, where x = r cos φ, y = r sin φ + η cos α, z = η sin α and α is a parameter
π π
0 <α ≤ . Note: When α = cylindrical coordinates result.
2 2