Page 108 - Intro to Tensor Calculus

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I 13. Calculate the metric and conjugate metric tensor associated with the toroidal surface coordinates
(ξ, η) illustrated in the figure 1.3-19, where

x =(a + b cosξ)cos η a>b> 0
y =(a + b cosξ)sin η 0 <ξ < 2π

z = b sin ξ 0 <η < 2π

Figure 1.3-19. Toroidal surface coordinates

I 14. Calculate the metric and conjugate metric tensor associated with the spherical surface coordinates
(θ, φ), illustrated in the figure 1.3-20, where

x = a sin θ cos φ a> 0 is constant
y = a sin θ sin φ 0 <φ < 2π
π
z = a cos θ 0 <θ <
2

I 15. Consider g ij ,i, j =1, 2
(a) Show that g 11 = g 22 , g 12 = g 21 = −g 12 , g 22 = g 11 where ∆ = g 11 g 22 − g 12 g 21 .
∆ ∆ ∆
k
(b) Use the results in part (a) and verify that g ij g ik = δ , i,j,k =1, 2.
j
I 16. Let A x ,A y ,A z denote the constant components of a vector in Cartesian coordinates. Using the
transformation laws (1.2.42) and (1.2.47) to find the contravariant and covariant components of this vector
upon changing to (a) cylindrical coordinates (r, θ, z). (b) spherical coordinates (ρ, θ, φ) and (c) Parabolic
cylindrical coordinates.

I 17. Find the relationship which exists between the given associated tensors.
(a) A pqk and A pq (c) A i.j. and A .s.p
r. rs .l.m r.t.
(b) A p and A pq (d) and A ij
.mrs ..rs A mnk ..k