Page 167 - Intro to Tensor Calculus

162

EXERCISE 1.5

~
~ ~ ~
~
I 1. Let κ = δ ~ T · N and τ = δ ~ N · B. Assume in turn that each of the intrinsic derivatives of T, N, B are
δs δs
~ ~ ~
some linear combination of T, N, B and hence derive the Frenet-Serret formulas of differential geometry.
I 2. Determine the given surfaces. Describe and sketch the curvilinear coordinates upon each surface.
2
2uv 2 2u v
r
r
r
(a) ~(u, v)= u b e 1 + v b e 2 (b) ~(u, v)= u cos v b e 1 + u sin v b e 2 (c) ~(u, v)= 2 2 b e 1 + 2 2 b e 2 .
u + v u + v
I 3. Determine the given surfaces and describe the curvilinear coordinates upon the surface. Use some
graphics package to plot the surface and illustrate the coordinate curves on the surface. Find element of
area dS in terms of u and v.
r
(a) ~(u, v)= a sin u cos v b e 1 + b sin u sin v b e 2 + c cos u b e 3 a, b, c constants 0 ≤ u, v ≤ 2π
u u u
r
(b) ~(u, v)= (4 + v sin )cos u b e 1 +(4+ v sin )sin u b e 2 + v cos b e 3 − 1 ≤ v ≤ 1, 0 ≤ u ≤ 2π
2 2 2
r
(c) ~(u, v)= au cosv b e 1 + bu sin v b e 2 + cu b e 3
r
(d) ~(u, v)= u cosv b e 1 + u sin v b e 2 + αv b e 3 α constant
r
(e) ~(u, v)= a cosv b e 1 + b sin v b e 2 + u b e 3 a, b constant
2
r
(f) ~(u, v)= u cosv b e 1 + u sin v b e 2 + u b e 3

E F
I 4. Consider a two dimensional space with metric tensor (a αβ )= . Assume that the surface is
F G
i
i
described by equations of the form y = y (u, v) and that any point on the surface is given by the position
i
r
vector ~ = ~(u, v)= y b e i . Show that the metrices E, F, G are functions of the parameters u, v and are given
r
by
∂~ r ∂~ r
r
r
r
r
r
r
r
E = ~ u · ~r u , F = ~ u · ~ v , G = ~ v · ~ v where ~ u = and ~ v = .
∂u ∂v
I 5. For the metric given in problem 4 show that the Christoffel symbols of the first kind are given by
r r r r r
[1 1, 1] = ~ u · ~r uu [1 2, 1] = [2 1, 1] = ~ u · ~ uv [2 2, 1] = ~ u · ~ vv
r r r r r r
[1 1, 2] = ~ v · ~ uu [1 2, 2] = [2 1, 2] = ~ v · ~ uv [2 2, 2] = ~ v · ~ vv
2
r
∂ ~r ∂~
which can be represented [αβ,γ]= · , α,β,γ =1, 2.
α
∂u ∂u β ∂u γ
I 6. Show that the results in problem 5 can also be written in the form
1 1 1
[1 1, 1] = E u [1 2, 1] = [2 1, 1] = E v [2 2, 1] = F v − G u
2 2 2
1 1 1
[1 1, 2] = F u − E v [1 2, 2] = [2 1, 2] = G u [2 2, 2] = G v
2 2 2
where the subscripts indicate partial differentiation.
I 7. For the metric given in problem 4, show that the Christoffel symbols of the second kind can be

γ γδ
expressed in the form = a [αβ,δ], α,β,γ =1, 2 and produce the results
αβ

1 GE u − 2FF u + FE v 1 1 GE v − FG u 2 2EF u − EE v − FE u
= = = =
2
2
2
11 2(EG − F ) 12 21 2(EG − F ) 11 2(EG − F )

1 2GF v − GG u − FG v 2 2 EG u − FE v 2 EG v − 2FF v + FG u
= = = =
2
2
2
22 2(EG − F ) 12 21 2(EG − F ) 22 2(EG − F )
where the subscripts indicate partial differentiation.