Page 175 - Intro to Tensor Calculus

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I 44. Show that for a smooth surface z = f(x, y) the mean curvature at a point on the surface is given by

2 2
(1 + f )f xx − 2f x f y f xy +(1 + f )f yy
y
x
H = .
2
2
2(f + f +1) 3/2
x y
I 45. Express the Frenet-Serret formulas (1.5.13) in terms of Christoffel symbols of the second kind.
I 46. Verify the relation (1.5.106).
ij
I 47. In V n assume that R ij = ρg ij and show that ρ = R where R = g R ij . This result is known as
n
Einstein’s gravitational equation at points where matter is present. It is analogous to the Poisson equation
2
∇ V = ρ from the Newtonian theory of gravitation.
I 48. In V n assume that R ijkl = K(g ik g jl − g il g jk )and show that R = Kn(1 − n). (Hint: See problem 23.)
I 49. Assume g ij =0 for i 6= j and verify the following.
(a) R hijk =0 for h 6= i 6= j 6= k
2 √ √ √ √ √
√ ∂ g ii ∂ g ii ∂ log g hh ∂ g ii ∂ log g kk
(b) R hiik = g ii − − for h, i, k unequal.
h
∂x ∂x k ∂x h ∂x k ∂x k ∂x h
 
√ √ n √ √
√ √  ∂ 1 ∂ g ii ∂ 1 ∂ g hh X ∂ g ii ∂ g hh 
(c) R hiih = g ii g hh  √ + √ +  where h 6= i.
∂x h g hh ∂x h ∂x i g ii ∂x i ∂x m ∂x m
m=1
m6=hm6=i
I 50. Consider a surface of revolution where x = r cos θ, y = r sin θ and z = f(r) is a given function of r.
0 2
2
2
2
2
(a) Show in this V 2 we have ds =(1 + (f ) )dr + r dθ where = d .
0
ds
(b) Show the geodesic equations in this V 2 are
2
0 00
d r f f dr 2 r dθ 2
+ − =0
0 2
0 2
ds 2 1+(f ) ds 1+(f ) ds
2
d θ 2 dθ dr
+ =0
ds 2 r ds ds
dθ a
(c) Solve the second equation in part (b) to obtain = . Substitute this result for ds in part (a) to show
ds r 2
p
0 2
a 1+(f )
dθ = ± √ dr which theoretically can be integrated.
2
r r − a 2