Page 168 - Intro to Tensor Calculus

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I 8. Derive the Gauss equations by assuming that

r
r
r
r
r
~ uu = c 1~r u + c 2~r v + c 3 bn, ~ uv = c 4~ u + c 5~ v + c 6 bn, ~ vv = c 7~ u + c 8~ v + c 9 bn
r
r
where c 1 ,... ,c 9 are constants determined by taking dot products of the above vectors with the vectors ~r u ,~r v ,

1 2 1 2
and bn. Show that c 1 = ,c 2 = ,c 3 = e, c 4 = ,c 5 = ,c 6 = f,
11 11 12 12
2
1 2 ∂ ~r γ ∂~ r
c 7 = ,c 8 = ,c 9 = g Show the Gauss equations can be written = +b αβ bn.
α
22 22 ∂u ∂u β αβ ∂u γ
I 9. Derive the Weingarten equations
r ∗ ∗
b n u = c 1~r u + c 2~r v ~ u = c bn u + c bn v
1
2
and
r r r ∗ ∗
b n v = c 3~ u + c 4~ v ~ v = c bn u + c bn v
3 4
and show
fF − eG gF − fG fF − gE fG − gF
∗
∗
c 1 = c 3 = c = 2 c = 2
1
3
EG − F 2 EG − F 2 eg − f eg − f
eF − fE fF − gE fE − eF fF − eG
∗
∗
c 2 = c 4 = c = c =
4
2
EG − F 2 EG − F 2 eg − f 2 eg − f 2
The constants in the above equations are determined in a manner similar to that suggested in problem 8.
Show that the Weingarten equations can be written in the form
∂bn β ∂~
r
= −b α .
∂u α ∂u β
r
~ u × ~r v
I 10. Using bn = √ , the results from exercise 1.1, problem 9(a), and the results from problem 5,
EG − F 2
verify that

2 p
r
(~ u × ~r uu ) · bn = EG − F 2
11
1 p
r
r
2
2 p (~ v × ~ uv ) · bn = − EG − F
(~ u × ~ uv ) · bn = EG − F 2 21
r
r
12
1 p
r r EG − F 2
1 p 2 (~ v × ~ vv ) · bn = −
r
r
(~ v × ~ uu ) · bn = − EG − F 22
11 p
(~ u × ~ v ) · bn = EG − F 2
r
r

2 p
r
r
(~ u × ~ vv ) · bn = EG − F 2
22
and then derive the formula for the geodesic curvature given by equation (1.5.48).
~
~
dT dT α
~ ~ αδ
Hint:(bn × T) · =(T × ) · bn and a ]βγ, δ]= .
ds ds βγ