Vector Analysis Integrating Differential Forms over Subsets, guaranteed and verified 100% Pass

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Integrating Differential Forms over Subsets of ℝ3



We will focus on three types of subsets of ℝ3 :

1. Oriented simple curves and oriented simple closed curves

2. Oriented surfaces

3. Elementary subregions.



Integrals of 1-Forms over Curves

Let 𝜔 be a 1-form on 𝐾 ⊆ ℝ3 , and let 𝑐 be any oriented simple curve. From
our study of line integrals, we are already familiar with integrating a 1-form
𝜔 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 + 𝑅𝑑𝑧 along a curve.



Ex. Let 𝜔 = 𝑥𝑦 2 𝑑𝑥 + 𝑧 3 𝑑𝑦 + 𝑑𝑧 be a 1-form on ℝ3 , and let 𝑐 be the oriented
simple curve: 𝑐⃗(𝑡 ) =< 1, 𝑡 3 , 𝑡 > 0 ≤ 𝑡 ≤ 1. Find ∫𝑐 𝜔.


𝑐⃗(𝑡)
∫𝑐 𝜔 = ∫𝑐 𝑥𝑦 2 𝑑𝑥 + 𝑧 3 𝑑𝑦 + 𝑑𝑧.


𝑐⃗′(𝑡 ) =< 0, 3𝑡 2 , 1 > ;

so 𝑑𝑥 = 0𝑑𝑡, 𝑑𝑦 = 3𝑡 2 𝑑𝑡, 𝑑𝑧 = 1𝑑𝑡 .


𝑡=1
∫𝑐 𝑥𝑦 2 𝑑𝑥 + 𝑧 3 𝑑𝑦 + 𝑑𝑧 = ∫𝑡=0 (1)(𝑡 3 )2 (0)𝑑𝑡 + 𝑡 3 (3𝑡 2 )𝑑𝑡 + 1𝑑𝑡
1 3
= ∫0 (3𝑡 5 + 1) 𝑑𝑡 = .
2

, 2


Integrals of 2-Forms over Surfaces

⃗⃗ (𝑢, 𝑣) =< 𝑥 (𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣) >; where 𝛷
Let 𝛷 ⃗⃗ : 𝐷 ⊂ ℝ2 → ℝ3 is a
parametriztion of a smooth oriented surface 𝑆 ⊆ ℝ3 and

𝜂 = 𝐹(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦 + 𝐺(𝑥, 𝑦, 𝑧)𝑑𝑦𝑑𝑧 + 𝐻(𝑥, 𝑦, 𝑧)𝑑𝑧𝑑𝑥 a 2-form on 𝐾 ⊆ ℝ3 ,
where 𝑆 ⊆ 𝐾 ⊆ ℝ3 .

How do we evaluate ∬𝑆 𝜂 ?



Definition: If 𝑆 is an oriented surface such that 𝑆 ⊆ 𝐾, an open set in ℝ3 , we
define ∬𝑆 𝜂 by the formula:

∬𝑺 𝜼 = ∬𝑺 𝑭𝒅𝒙𝒅𝒚 + 𝑮𝒅𝒚𝒅𝒛 + 𝑯𝒅𝒛𝒅𝒙
𝝏(𝒙,𝒚) 𝝏(𝒚,𝒛) 𝝏(𝒛,𝒙)
⃗⃗⃗⃗(𝒖, 𝒗))
= ∬𝑫 [𝑭( 𝜱 ⃗⃗⃗⃗(𝒖, 𝒗))
+ 𝑮 (𝜱 ⃗⃗⃗⃗(𝒖, 𝒗))
+ 𝑯(𝜱 ]𝒅𝒖𝒅𝒗
𝝏(𝒖,𝒗) 𝝏(𝒖,𝒗) 𝝏(𝒖,𝒗)

𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑧 𝜕𝑧
𝜕(𝑥,𝑦) 𝜕(𝑦,𝑧) 𝜕(𝑧,𝑥)
where: = |𝜕𝑢
𝜕𝑦
𝜕𝑣
𝜕𝑦
|, = |𝜕𝑢
𝜕𝑧
𝜕𝑣
𝜕𝑧
|, = |𝜕𝑢
𝜕𝑥
𝜕𝑣
𝜕𝑥
|.
𝜕(𝑢,𝑣) 𝜕(𝑢,𝑣) 𝜕(𝑢,𝑣)
𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣




Let’s see where this definition comes from. Recall that for surface integrals of
vector fields we had:


∬ 𝐹⃗ ∙ 𝑑𝑆⃗ = ∬ 𝐹⃗ (𝛷 ⃗⃗𝑢 × 𝑇
⃗⃗(𝑢, 𝑣)) ∙ (𝑇 ⃗⃗𝑣 )𝑑𝑢𝑣
𝑆 𝐷

⃗⃗: 𝐷 ⊆ 𝑅2 → 𝑅3 ; and 𝛷
where 𝛷 ⃗⃗(𝐷) = 𝑆.