Calculus-on Manifolds Greens Theorem Stokes Theorem the Divergence Theorem, guaranteed and verified 100% Pass

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Green’s Theorem, Stokes’ Theorem, the Divergence Theorem and the
Fundamental Theorem of Calculus

In this section we will show that The Fundamental Theorem of Calculus, Green’s
Theorem, Stokes’ Theorem (on surfaces in ℝ3 ), and the Divergence Theorem are
all special cases of Stokes’ Theorem on manifolds.

Def. A singular 0-cube, 𝑐, is a map of 𝑐: {0} → 𝐴 ⊆ ℝ𝑛 . If 𝜔 is a zero form
(i.e. a real valued function) we define:

∫ 𝝎 = 𝝎(𝒄(𝟎)).
𝒄




The Fundamental Theorem of Calculus:
If 𝑓 is a smooth function on [𝑎, 𝑏], then
𝑏
∫ 𝑓 ′ (𝑥 )𝑑𝑥 = 𝑓(𝑏) − 𝑓(𝑎 ).
𝑎
In this case, 𝑑𝑓 = 𝑓 ′ (𝑥 )𝑑𝑥.
If we call [𝑎, 𝑏] = 𝐼, then The Fundamental
Theorem of Calculus becomes:


∫ 𝑑𝑓 = ∫ 𝑓 = 𝑓(𝑏) − 𝑓(𝑎).
𝐼 𝜕𝐼

That is, The Fundamental Theorem of Calculus is Stokes’ Theorem
where 𝑀 = 𝐼.

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Green’s Theorem:
Let 𝐷 be a simple region and 𝐶 be its boundary. Suppose that 𝑃: 𝐷 → ℝ
and 𝑄: 𝐷 → ℝ are smooth function, then:

𝜕𝑄 𝜕𝑃
∫ 𝑃𝑑𝑥 + 𝑄𝑑𝑦 = ∬ ( − ) 𝑑𝑥 𝑑𝑦.
𝐶 𝐷 𝜕𝑥 𝜕𝑦


Notice that if 𝜔 = 𝑃(𝑥, 𝑦)𝑑𝑥 + 𝑄(𝑥, 𝑦)𝑑𝑦, then:

𝜕𝑃 𝜕𝑄 𝜕𝑄 𝜕𝑃
𝑑𝜔 = 𝜕𝑦 𝑑𝑦 ∧ 𝑑𝑥 + 𝜕𝑥 𝑑𝑥 ∧ 𝑑𝑦 = ( 𝜕𝑥 − 𝜕𝑦) 𝑑𝑥 ∧ 𝑑𝑦

Since 𝐶 = 𝜕𝐷, we can write Green’s Theorem as:

∫ 𝜔 = ∫ 𝑑𝜔.
𝜕𝐷 𝐷


This is Stokes’ Theorem where 𝑀 = 𝐷.




Stokes’ Theorem (for parameterized surfaces in ℝ3 ):
Let 𝑆 be an oriented surface in ℝ3 defined by a one-to-one
parameterization ⃗Φ
⃗ : 𝐷 ⊆ ℝ2 → ℝ3 , where 𝐷 is a simple region. Let
𝐹 (𝑥, 𝑦, 𝑧) be a smooth vector field on 𝑆, then:

∬ (∇ × 𝐹 ) ∙ 𝑑𝑆 = ∫ 𝐹 ∙ 𝑑𝑠.
𝑆 𝜕𝑆