Page 120 - Vector Analysis

116 CHAPTER 4. Vector Calculus

Since a natural way to write Lv, where v = ae1 + be2 P R2, is
[ ] [ ][ ]
[ ]...[ψ,2 ] a ∇ψ a
Lv = [ψ,1 ] b = b ,

sometimes we also use ∇ψ to denote L, and then write ?g as det(∇ψ) (even though [∇ψ]
is a 3 ˆ 2 matrix) and call ?g the Jacobian of the map ψ.

Example 4.46. Let Σ be the sphere centered at the origin with radius R. Consider the local
parametrization ψ(θ, ϕ) = (R cos θ sin ϕ, R sin θ sin ϕ, R cos ϕ) with (θ, ϕ) P V ” (0, 2π) ˆ
(0, π). Then

ψ,1 (θ, ϕ) = ψθ(θ, ϕ) = (´R sin θ sin ϕ, R cos θ sin ϕ, 0) ,
ψ,2 (θ, ϕ) = ψϕ(θ, ϕ) = (R cos θ cos ϕ, R sin θ cos ϕ, ´R sin ϕ) ;

thus the metric tensor and the first fundamental form associated with the parametrization

tV, ψu are

g(θ, ϕ) = [Dψ]T[Dψ](θ, ϕ) = [ sin2 ϕ ]
R2 0 0
R2

and g = det(g) = R4 sin2 ϕ.

What does the first fundamental form do for us?
(

Let p = ψ)(u0, v0) be a point in Σ. Then the surface area of the region ψ [u0, u0 + h] ˆ
[v0, v0 + k] , where h, k are very small, can be approximated by the sum of the area of two
triangles, one with vertices ψ(u0, v0), ψ(u0 + h, v0), ψ(u0, v0 + k) and the other with vertices
ψ(u0 + h, v0), ψ(u0, v0 + k), ψ(u0 + h, v0 + k).

ψ(u0, v0 + k)

ψ(u0 + h, v0 + k)

ψ(u0, v0)

ψ(u0 + h, v0)

Here we remark that the approximation of the surface area of a regular C 1-surface obeys

()
the surface area of ψ [u0, u0 + h] ˆ [v0, v0 + k]
lim the sum of area of the two triangles given in the context = 1. (4.10)

(h,k)Ñ(0,0)