Calculus-on Manifolds Vector Fields and Differential Forms on Manifolds, guaranteed and verified 100% Pass

1


Vector Fields and Differential Forms on Manifolds


Def. Let 𝑀 ⊆ ℝ𝑛 be a 𝑘-dimensional manifold. A vector field on 𝑴 is a function
on 𝑀 that associates to each point 𝑝 ∈ 𝑀 a vector 𝐹(𝑝) ∈ 𝑇𝑝 𝑀.

⃗⃗ : 𝑈 → 𝑀 be a parametrization. Given a vector field 𝐹(𝑥) on
Let Φ
𝑀, there is a unique vector field 𝐺 on 𝑈 such that:
⃗⃗⃗ ∗ (𝐺 (𝑎 )) = 𝐹 (Φ
Φ ⃗⃗⃗ (𝑎 ))
for 𝑎 ∈ 𝑈, and where:

⃗Φ
⃗⃗ ∗ (𝐺 (𝑎 )) = (𝐷Φ
⃗⃗⃗ (𝑎 )(𝐺 (𝑎 ))) .
⃗⃗⃗ (𝑎)
Φ


We say 𝐹 is differentiable if 𝐺 is differentiable. Note that the definition of
differentiability of 𝐹 does not depend on which parametrization is used.



Ex. Let 𝑀 be parametrized by ⃗Φ
⃗⃗ (𝑢, 𝑣 ) = (𝑢, 𝑣, 𝑢2 + 𝑣 2 ). Then at each point
𝑝 = (𝑢, 𝑣, 𝑢2 + 𝑣 2 ) on 𝑀, the tangent space 𝑇𝑝 𝑀 has a basis of
⃗⃗⃗
𝜕Φ ⃗⃗⃗
𝜕Φ
{ 𝜕𝑢 , 𝜕𝑣 } = {(1,0,2𝑢), (0,1,2𝑣)}. Thus we can express any vector field
on 𝑀 as:
⃗⃗⃗
𝜕Φ ⃗⃗⃗
𝜕Φ
𝐹 (𝑝) = 𝑓1 (𝑝) 𝜕𝑢 + 𝑓2 (𝑝) 𝜕𝑣 ;

where 𝑓1 and 𝑓2 are real valued function on 𝑀.

⃗⃗⃗
𝜕Φ ⃗⃗⃗
𝜕Φ
For example, 𝐹 (𝑢, 𝑣 ) = 𝑢𝑣 + (𝑢 − 𝑣) 𝜕𝑣
𝜕𝑢
Is a vector field on 𝑀.

Def. A function 𝜔, which assigns 𝜔(𝑥 ) ∈ Ω𝑝 (𝑇𝑥 𝑀) for each 𝑥 ∈ 𝑀, is called a
𝒑-form on 𝑴.