Page 144 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Generalities about surfaces 131

(ii) We say that Σ is a nonparametric surface if
© 3 ª
Σ = v (x, y)= (x, y, u (x, y)) ∈ R :(x, y) ∈ Ω

2
with u : Ω → R continuous and where Ω ⊂ R is a domain.
m
(iii) A parametric surface is said to be regular of class C ,(m ≥ 1 an
integer) if, in addition, v ∈ C m ¡ Ω; R 3 ¢ and v x × v y 6=0 for every (x, y) ∈ Ω
3
(where a × b stands for the vectorial product of a, b ∈ R and v x = ∂v/∂x,
v y = ∂v/∂y). We will in this case write
v x × v y
e 3 = .
|v x × v y |
Remark 5.2 (i) In many cases we will restrict our attention to the case where
¡ ¢
Ω is the unit disk and Σ = v Ω will then be called a surface of the type of the
disk.
(ii) In the sequel we will let

¡ ¢ 0 ¡ 3 ¢ 1,2 ¡ 3 ¢
M Ω = C Ω; R ∩ W Ω; R .
(iii) For a regular surface the area will be defined as
ZZ
J (v)= Area (Σ)= |v x × v y | dxdy .
Ω
¡ ¢
It can be shown, following Mac Shane and Morrey, that if v ∈ M Ω ,thenthe
above formula still makes sense (see Nitsche [78] pages 195-198).
(iv) In the case of nonparametric surface v (x, y)= (x, y, u (x, y)) we have
ZZ
q
2
2
Area (Σ)= J (v)= I (u)= 1+ u + u dxdy .
x y
Ω
2
Note also that, for a nonparametric surface, we always have |v x × v y | =1 +
2
2
u + u 6=0.
x y
We now introduce the different notions of curvatures.
m
Definition 5.3 Let Σ be a regular surface of class C , m ≥ 2,welet
2 2 v x × v y
E = |v x | ,F = hv x ; v y i ,G = |v y | ,e 3 =
|v x × v y |
L = he 3 ; v xx i ,M = he 3 ; v xy i ,N = he 3 ; v yy i
3
where h.; .i denotes the scalar product in R .