Answer to Question #195916 in Differential Geometry | Topology for Saba Umer

Question

Find the first fundamental form of surface of revolution?

Answer 1

 The First Fundamental Form

(note : bold symbol denote to vector, example X denote X vector)

Suppose M is a surface determined by X(u, v) ⊂ E³ and suppose α(t) is a curve on M, t ∈ [a, b]. Then we can write α(t) = X(u(t), v(t)) (then (u(t), v(t)) is a curve in R² whose image under X is α). Then

α'(t) = (∂X/∂u)(du/dt) + (∂X/∂v)(dv/dt) = u' X₁ + v' X₂

If s(t) represents the arc length along α (with s(a) = 0) then

s(t) = ∫ || α^' (r) || dr

where lower limit is a

and upper limit is t

and

(ds/dt) = || α^' (r) ||

so

(ds/dt)² = || α^' (r) || ² = α^' . α^' = ( u' X₁ + v' X₂ ).( u' X₁ + v' X₂ )

= u'² ( X₁ . X₁) + 2 u' v' ( X₁ . X₂) + v'² ( X ₂. X₂)

Following Gauss’ notation (briefly) we denote 

E = X₁ . X₁

F =X₁ . X₂

G = X ₂. X₂

and have

(ds/dt)² =E (du/dt)² +2F (du/dt)(dv/dt) + G (dv/dt)²

or in differential notation

(ds)² =E (du)² +2F (du)(dv) + G (dv)²

it is represent " first fundamental form of surface of revolution "