APM3713 Assignment 6 2025 (Accurately Solved) Year 2025

APM3713
Assignment 6
Due Year 2025

, Question 1
Consider the surface parametrized by


√ √

r(u, v) = x(u, v), y(u, v), z(u, v) = u2 + 1 cos v, u2 + 1 sin v, u ,


with coordinates x1 = u, x2 = v.


(a) Line element
√
Define a(u) = u2 + 1. Then

 
u u


ru = a
cos v, a
sin v, 1 , rv = − a sin v, a cos v, 0 .


The metric components:

1 + 2u2
g11 = , g22 = 1 + u2 , g12 = 0.
1 + u2

Hence the line element is

1 + 2u2 2
ds2 = du + (1 + u2 ) dv 2 .
1 + u2


(b) Metric and inverse metric
1 + u2
 
1 + 2u2
 
0   1 + 2u2 0
2
gij =  1 + u

, g ij =  .

2
 1 
0 1+u 0
1 + u2

(c) Christoffel symbols
1 u 1 1 + u2 u
Γ 11 = , Γ 22 = −u , Γ2 12 = Γ2 21 = .
(1 + u2 )(1 + 2u2 ) 1 + 2u2 1 + u2
All other Christoffel symbols vanish.




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