Calculus on Manifolds The First Fundamental Form of a Surface in-R3, guaranteed and verified 100% Pass

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The First Fundamental Form of a Surface in ℝ3

We know from multivariable calculus that if 𝛾(𝑡 ) = (𝑥 (𝑡 ), 𝑦(𝑡 ), 𝑧(𝑡 )) is a
curve in ℝ3 , then it’s length is given by:


𝑏 𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2
𝐿= √
∫𝑎 ( 𝑑𝑡 ) + ( 𝑑𝑡 ) + (𝑑𝑡 ) 𝑑𝑡 .

𝑑𝑥 𝑑𝑦 𝑑𝑧 𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2
𝛾 ′ (𝑡) = ( 𝑑𝑡 , 𝑑𝑡 , 𝑑𝑡 ) and ‖𝛾 ′ (𝑡 )‖2 = 𝛾′ ⋅ 𝛾′ = ( 𝑑𝑡 ) + ( 𝑑𝑡 ) + (𝑑𝑡 )

so we can also write:
𝑏
𝐿 = ∫ ‖𝛾 ′ (𝑡)‖ 𝑑𝑡.
𝑎



Now we want to consider a formula for the length of a curve that lies on a smooth
surface 𝑆 ⊆ ℝ3 . Suppose 𝑆 is parameterized by:

⃗⃗ : 𝑈 ⊆ ℝ2 → 𝑆 ⊆ ℝ3
Φ
⃗Φ
⃗⃗ (𝑢, 𝑣 ) = (𝑥 (𝑢, 𝑣 ), 𝑦(𝑢, 𝑣 ), 𝑧(𝑢, 𝑣 )).

If 𝑆 is a regular surface (i.e. Φ ⃗⃗⃗ 𝑣 ≠ ⃗0), then Φ
⃗⃗ 𝑢 × Φ ⃗⃗⃗ 𝑢 and Φ
⃗⃗⃗ 𝑣 span the
tangent space of 𝑆 at 𝑝 ∈ 𝑆. Thus, any vector 𝑤 ⃗⃗ in 𝑇𝑝 𝑆 can be written as:

𝑤 ⃗⃗⃗ 𝑢 + 𝑏Φ
⃗⃗ = 𝑎Φ ⃗⃗⃗ 𝑣 ; 𝑎, 𝑏 ∈ ℝ.


Notice that:
𝑤
⃗⃗ ∙ 𝑤 ⃗⃗ 𝑢 + 𝑏Φ
⃗⃗ = (𝑎Φ ⃗⃗⃗ 𝑣 ) ∙ (𝑎Φ
⃗⃗ 𝑢 + 𝑏Φ
⃗⃗ 𝑣 )

= 𝑎 2 ⃗Φ
⃗⃗ 𝑢 ∙ ⃗Φ
⃗ 𝑢 + 2𝑎𝑏Φ
⃗⃗⃗ 𝑢 ∙ ⃗Φ
⃗⃗ 𝑣 + 𝑏 2 ⃗Φ
⃗⃗ 𝑣 ∙ ⃗Φ
⃗𝑣 .

, 2


If we let: 𝐸 = ⃗Φ
⃗ 𝑢 ∙ ⃗Φ⃗⃗ 𝑢
𝐹=Φ ⃗⃗ 𝑢 ∙ Φ⃗⃗ 𝑣 = Φ
⃗⃗⃗ 𝑣 ∙ Φ
⃗⃗ 𝑢
𝐺 = ⃗Φ⃗ 𝑣 ∙ ⃗Φ
⃗𝑣
then:
𝑤 ⃗⃗ = 𝐸𝑎 2 + 2𝐹𝑎𝑏 + 𝐺𝑏 2 .
⃗⃗ ∙ 𝑤

𝐸𝐹
In fact, the matrix: ( ) represents a bilinear form on 𝑇𝑝 𝑆. This bilinear form
𝐹𝐺
is called the first fundamental form on 𝑆. If 𝑣 = 𝑐Φ ⃗⃗⃗ 𝑢 + 𝑑Φ
⃗⃗⃗ 𝑣 then

𝐸 𝐹 𝑐
⃗⃗ ∙ 𝑣 = (𝑎
𝑤 𝑏) ( ) ( ) = 𝑎𝑐𝐸 + (𝑎𝑑 + 𝑏𝑐 )𝐹 + 𝑏𝑑𝐺.
𝐹 𝐺 𝑑

Suppose that 𝛾 is a curve on 𝑆 such that: 𝛾 (𝑡 ) = ⃗Φ
⃗ (𝑢(𝑡 ), 𝑣 (𝑡 )).




By the chain rule:
𝛾 ′ (𝑡 ) = ⃗Φ
⃗⃗ 𝑢 𝑢′(𝑡) + ⃗Φ
⃗ 𝑣 𝑣 ′ (𝑡 )
Thus we have:

‖𝛾 ′ (𝑡)‖2 = 𝛾 ′ (𝑡 ) ⋅ 𝛾 ′ (𝑡) = (𝑢′ ⃗Φ
⃗⃗ 𝑢 + 𝑣 ′ ⃗Φ
⃗⃗ 𝑣 ) ⋅ (𝑢′ ⃗Φ
⃗ 𝑢 + 𝑣 ′ ⃗Φ
⃗⃗ 𝑣 )
= (𝑢′ )2 ⃗Φ
⃗ 𝑢 ∙ ⃗Φ
⃗ 𝑢 + 2(𝑢′ )(𝑣 ′ )⃗Φ
⃗⃗ 𝑢 ∙ ⃗Φ
⃗⃗ 𝑣 + (𝑣 ′ )2 ⃗Φ
⃗ 𝑣 ∙ ⃗Φ
⃗𝑣
= 𝐸 (𝑢′ )2 + 2𝐹 (𝑢′ )(𝑣 ′ ) + 𝐺 (𝑣 ′ )2 .
And so we get:
𝑏 𝑏 1
𝐿 = ∫ ‖𝛾 𝑡 )‖ 𝑑𝑡 = ∫ (𝐸 (𝑢′ )2 + 2𝐹 (𝑢′ )(𝑣 ′ ) + 𝐺 (𝑣 ′ )2 )2 𝑑𝑡.
′(
𝑎 𝑎