Calculus on Manifolds Calculations-with the Metric Tensor, guaranteed and verified 100% Pass

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Calculations with the Metric Tensor


Now let’s apply these tensor concepts to the metric tensor for a surface in ℝ3 .
Recall that if a surface, 𝑆, in ℝ3 is parameterized by:
⃗Φ
⃗ : 𝑈 ⊆ ℝ2 → 𝑆 ⊆ ℝ 3
⃗Φ
⃗ (𝑢, 𝑣 ) = (𝑥 (𝑢, 𝑣 ), 𝑦(𝑢, 𝑣 ), 𝑧(𝑢, 𝑣 ))

then, the first fundamental form, or metric tensor, is given by:
𝑔11 𝑔12
𝑔 = (𝑔 𝑔22 )
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where 𝑔11 = ⃗Φ
⃗⃗ 𝑢 ∙ ⃗Φ
⃗𝑢
𝑔12 = 𝑔21 = ⃗Φ
⃗ 𝑢 ∙ ⃗Φ⃗𝑣
𝑔22 = ⃗Φ⃗ 𝑣 ∙ ⃗Φ
⃗⃗ 𝑣 .


At each point, 𝑝 ∈ 𝑆, 𝑔 ∈ 𝒯 2 (𝑇𝑝 𝑆).

Thus if given 𝑤
⃗⃗ 1 , 𝑤
⃗⃗ 2 ∈ 𝑇𝑝 𝑆, then:

𝑔 𝑔12 𝑎21
𝑔(𝑤 ⃗⃗ 2 ) = (𝑎11
⃗⃗ 1 , 𝑤 𝑎12 ) ( 11
𝑔21 𝑔22 ) (𝑎22 )
where:
⃗⃗ 1 = 𝑎11 ⃗Φ
𝑤 ⃗⃗ 𝑢 + 𝑎12 ⃗Φ
⃗⃗ 𝑣
𝑤
⃗⃗ 2 = 𝑎21 Φ⃗⃗⃗ 𝑢 + 𝑎22 Φ⃗⃗⃗ 𝑣 .

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Ex. Let 𝑆 be the surface parameterized by ⃗Φ
⃗⃗ : ℝ2 → 𝑆 ⊆ ℝ3 and
⃗Φ
⃗ (𝑢, 𝑣 ) = (𝑢, 𝑣, 𝑢2 + 𝑣 2 ).

a) Find the metric tensor, 𝑔, at (𝑢, 𝑣 ) = (1, 2).


b) If 𝑤 ⃗⃗⃗ 𝑢 (1, 2) − 3Φ
⃗⃗ 1 = 2Φ ⃗⃗ 𝑣 (1, 2) and 𝑤 ⃗⃗⃗ 𝑢 (1, 2) + 2Φ
⃗⃗ 2 = −Φ ⃗⃗⃗ 𝑣 (1, 2)
⃗⃗ 𝑢 (1,2), ⃗Φ
then find 𝑔(Φ ⃗ 𝑣 (1,2)), 𝑔(𝑤 ⃗⃗ 2 ).
⃗⃗ 1 , 𝑤


⃗⃗⃗ (1, 2): 𝑇(1,2)ℝ2 → 𝑇(1,2,5)𝑆 is a linear transformation.
c) We know that 𝐷Φ
If 𝑈 ∈ 𝒯 𝑘 (𝑇(1,2,5) 𝑆), then
∗
⃗⃗ (1, 2)) 𝑈 ∈ 𝒯 𝑘 (𝑇(1,2) ℝ2 ).
(𝐷Φ
∗
⃗⃗ (1, 2)) 𝑔(𝑣1 , 𝑣2 ) where 𝑣1 = (−3, 2), 𝑣2 = (1, −1)
Find (𝐷Φ
and 𝑣1 , 𝑣2 ∈ 𝑇(1,2) ℝ2.
⃗⃗⃗ (𝑢, 𝑣) = (𝑢, 𝑣, 𝑢2 + 𝑣 2 )
Φ


𝑇(1,2) ℝ2
⃗⃗⃗ (1, 2): 𝑇(1,2) ℝ2 → 𝑇(1,2,5) 𝑆
𝐷Φ
(1,2,5)
(1,2)


⃗Φ
⃗⃗
𝑇(1,2,5) 𝑆