Calculus on Manifolds Closed and Exact Differential-Forms, guaranteed and verified 100% Pass

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Closed and Exact Differential Forms

Def. A differential 𝑘-form 𝜔 is called closed if 𝑑𝜔 = 0.

Ex. Let 𝜔 = (𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦. Show that 𝜔 is closed.

𝑑𝜔 = 𝑑[(𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦]
= 𝑑 [(𝑥 2 + 𝑦 2 )𝑑𝑥] + 𝑑[2𝑥𝑦𝑑𝑦]
= 𝑑 (𝑥 2 + 𝑦 2 ) ∧ 𝑑𝑥 + 𝑑(2𝑥𝑦) ∧ 𝑑𝑦
𝜕 𝜕
= (𝜕𝑥 (𝑥 2 + 𝑦 2 )𝑑𝑥 + 𝜕𝑦 (𝑥 2 + 𝑦 2 )𝑑𝑦) ∧ 𝑑𝑥
𝜕 𝜕
+(𝜕𝑥 (2𝑥𝑦)𝑑𝑥 + 𝜕𝑦 (2𝑥𝑦)𝑑𝑦) ∧ 𝑑𝑦
= (2𝑥𝑑𝑥 + 2𝑦𝑑𝑦) ∧ 𝑑𝑥 + (2𝑦𝑑𝑥 + 2𝑥𝑑𝑦) ∧ 𝑑𝑦
= 2𝑦𝑑𝑦 ∧ 𝑑𝑥 + 2𝑦𝑑𝑥 ∧ 𝑑𝑦 = 0.


Ex. Show that any 2 form on ℝ2 is closed.

Any 2 form on ℝ2 , 𝜔, can be written as 𝜔 = 𝑓 (𝑥, 𝑦)𝑑𝑥 ∧ 𝑑𝑦.

𝑑𝜔 = 𝑑(𝑓(𝑥, 𝑦)𝑑𝑥 ∧ 𝑑𝑦)
= 𝑑𝑓 ∧ 𝑑𝑥 ∧ 𝑑𝑦
𝜕𝑓 𝜕𝑓
= ( 𝑑𝑥 + 𝑑𝑦) ∧ 𝑑𝑥 ∧ 𝑑𝑦
𝜕𝑥 𝜕𝑦
𝜕𝑓 𝜕𝑓
= 𝑑𝑥 ∧ 𝑑𝑥 ∧ 𝑑𝑦 + 𝜕𝑦 𝑑𝑦 ∧ 𝑑𝑥 ∧ 𝑑𝑦 = 0.
𝜕𝑥



Ex. Show that 𝜔 = 𝑑𝑥𝑖 ∧ 𝑑𝑥𝑗 is closed as a 2 form on ℝ𝑛 .

𝑑𝜔 = 𝑑(𝑑𝑥𝑖 ∧ 𝑑𝑥𝑗 ) = 𝑑 (𝑑𝑥𝑖 ) ∧ 𝑑𝑥𝑗 + (−1)1 𝑑𝑥𝑖 ∧ 𝑑(𝑑𝑥𝑗 ) = 0.

By induction one can show that 𝜔 = 𝑑𝑥𝑖1 ∧ 𝑑𝑥𝑖2 ∧ … ∧ 𝑑𝑥𝑖 𝑘 is closed on
ℝ𝑛 .

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Def. A differential 𝑘-form 𝜔 is called exact if 𝜔 = 𝑑𝜂 for some
(𝑘 − 1)-form 𝜂.

Ex. Show that 𝜔 = (𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 is exact on ℝ2 .



So we have to show we can find a real valued function 𝑓 on ℝ2 such that
𝑑𝑓 = 𝜔 = (𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦.
However, we know that:
𝜕𝑓 𝜕𝑓
𝑑𝑓 = 𝜕𝑥 𝑑𝑥 + 𝜕𝑦 𝑑𝑦.

So we have to find a function 𝑓 such that:
𝜕𝑓 𝜕𝑓
𝑑𝑓 = 𝜕𝑥 𝑑𝑥 + 𝜕𝑦 𝑑𝑦 = (𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦 .

Thus we need to have:
𝜕𝑓
= 𝑥2 + 𝑦2
𝜕𝑥
𝜕𝑓
= 2𝑥𝑦.
𝜕𝑦

We solve these 2 equations as was done in second year calculus.
𝑥3
𝑓(𝑥, 𝑦) = ∫(𝑥 2 + 𝑦 2 )𝑑𝑥 = + 𝑥𝑦 2 + 𝑔(𝑦).
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Now differentiate this equation with respect to 𝑦.
𝜕𝑓
𝜕𝑦
= 2𝑥𝑦 + 𝑔′(𝑦).

𝜕𝑓
But we also know that = 2𝑥𝑦, so
𝜕𝑦
2𝑥𝑦 + 𝑔′ (𝑦) = 2𝑥𝑦.

Thus 𝑔′ (𝑦) = 0 and 𝑔(𝑦) = 𝑐.

𝑥3
Thus if 𝑓 (𝑥, 𝑦) = + 𝑥𝑦 2 + 𝑐, then 𝑑𝑓 = 𝜔 = (𝑥 2 + 𝑦 2 )𝑑𝑥 + 2𝑥𝑦𝑑𝑦.
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