12. THE FIRST FUNDAMENTAL FORM

Explanation 103 : We have to consider differentials first.
If f = f(x) then df = f '(x) dx ; for example, if f = cos(x) then df = -sin(x) dx ; df is the first order approximation of Δf := f(x+dx) - f(x) (calculate df and Δf in the example, with x = π/2, dx = 0.001).
If f = f(x,y) then df = f_xdx + f_ydy ('total differential') ; for example, if f = x/y then df = (1/y)dx -(x/y²)dy ; df is the first order approximation of Δf := f(x+dx,y+dy) - f(x,y).

Explanation 104 : Now we have to consider curves on the surface.

The image curve has parametrisation x(u₁(t),u₂(t)), short x(t).
The arc length s of x(t) satisfies ds/dt = ||x '(t)|| = ||x_u1 du₁/dt + x_u2 du₂/dt|| ; hence it follows that (ds)² = x_u1 du₁ . x_u1 du₁ + 2 x_u1 du₁ . x_u2 du₂ + x_u2 du₂ . x_u2 du₂.

Definition 105 : We define the first fundamental form of the surface by (ds)² = a_{1 1} (du₁)² + 2 a_{1 2} du₁ du₂ + a_{2 2} (du₂)², where a_{i j} = x_ui . x_uj.

Examples 106 :

Plane: x(u,v) = (u,v,0). Then (ds)² = (du)² + (dv)² ; this corresponds to the theorem of Pythagoras.
Sphere: x(u,v) = (sin u cos v, sin u sin v, cos u) ; here (ds)² = (du)² + sin²u (dv)² ; see the picture at the end of this section 12.
Graph of a function: x(u,v) = (u,v,f(u,v); then (ds)² = (1+f_u²) (du)² + 2 f_uf_v du dv + (1+f_v²) (dv)².

For each surface we have: a_{1 2} = 0 in the points where the parameter lines are perpendicular to each other.

Explanation 107 : We will now see that the first fundamental form determines the metrics of the surface:

i) The arc length of a curve on the surface, say x(u₁(t),u₂(t)), t ∈ (t₀,t₁), is _s(t0)∫^s(t₁) ds = _t0∫^t₁ ds/dt dt = _t0∫^t₁ √(Σ a_{i j} du_i/dt du_j/dt) dt.
ii) The angle between two curves on the surface, say x(a₁(t),a₂(t)) and x(b₁(t'),b₂(t')), is φ with
cos(φ) = (x_u1 da₁/dt + x_u2 da₂/dt) . (x_u1 db₁/dt' + x_u2 db₂/dt' ) / ( || x_u1 da₁/dt + x_u2 da₂/dt || || x_u1 db₁/dt' + x_u2 db₂/dt' || ) =
= ( a_{1 1} + a_{1 2} db₂/db₁ + a_{2 1} da₂/da₁ + a_{2 2} da₂/da₁ db₂/db₁ ) / ( √(a_{1 1} + 2 a_{1 2} da₂/da₁ + a_{2 2} (da₂/da₁)²) √(a_{1 1} + 2 a_{1 2} db₂/db₁ + a_{2 2} (db₂/db₁)²) ).

iii) The area of a domain on the surface, say x(u₁,u₂) ((u₁,u₂) ∈ G), is
∫ ∫ _x(G) dσ = ∫ ∫ _G || x_u1 ⊗ x_u2 || du₁du₂, where the integrand is also equal to √(a_{1 1} a_{2 2} - a_{1 2}²).

So arc length, angle and area are variables we calculate with the help of the coefficients of the first fundamental form and of variables we calculate in the u₁,u₂-plane ; therefore we call them 'intrinsic variables'.
From the following definition and explanation it will become clear why we call them 'invariant under bending' as well.

Definition 108 : Consider the following picture (x¹ and x² are differentiable):

If the surfaces x¹(u₁,u₂) and x²(u₁,u₂) have the same first fundamental form, we call them bents of each other, and we call the mapping x² o (x¹) ^-1 an isometry.

Explanation 109 : Check by calculation that the right circular cylinder x(u,v) = (cos u, sin u, v) is a bent of the plane (u,v,0); if we shade on a piece of paper a domain and draw two curves on it, and roll up the piece of paper so that it becomes a right circular cylinder, then the area of the shaded domain, the lengths of the curves, and the angle between the curves in the intersection point don't change.

Problem 110 : Calculate the angle between the curves (t² cos(t³), t² sin(t³), 2t²) and (t³ cos(t²), t³ sin(t²), 2t³) on the surface of revolution (u cos v, u sin v, 2u) in the point with parameter value t=1.
Do this in the direct way first, and then with the help of the first fundamental form.

Problem 111 : Consider the surface V with parametrisation (u+v, u²+v², uv). Prove that the points on V where the parameter lines are perpendicular to each other are lying on a parabola.

Problem 112 : Calculate the area of the part of the sphere with center (0,0,0) and radius 1 that lies between the parallels of latitude θ = θ₁ and θ = θ₂ by making use of the first fundamental form.

Problem 113 : Show that the right circular cone u(cos v, sin v, 1) (a kind of conical cap) is a bent of a plane (first use for the parametrisation of the plane polar coordinates u and v, and then make some little adaptations).

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