University of Toronto – MAT237Y1 – LEC5201
Multivariable calculus
Poincaré lemma: potentials and vector potentials
Jean-Baptiste Campesato April 2
nd, 2020
Contents
1 Star-shaped sets 1
2 Conservative vector fields 1
2.1 𝑛 = 2 . . . 2 2.2 𝑛 = 3 . . . 2 2.3 General case . . . 2
3 Vector potentials 3
4 The general Poincaré lemma 3
1 Star-shaped sets
Definition 1. We say that𝑆 ⊂ ℝ𝑛isstar-shapedif there exists𝑝0 ∈ 𝑆 such that for any𝑞 ∈ 𝑆the line segment from𝑝0to𝑞is included in𝑆.
Proposition 2. A non-empty convex set is star-shaped.
Beware:the empty set is convex but not star-shaped (because “∀𝑝 ∈ ∅, 𝑃 (𝑝)” is vacuously true whereas
“∃𝑝 ∈ ∅, 𝑃 (𝑝)” is always false).
2 Conservative vector fields
Theorem 3. Let𝑈 ⊂ ℝ𝑛be open (𝑛 ≥ 2), andF∶ 𝑈 → ℝ𝑛be a continuous vector field.
The following are equivalent:
1. There exists𝑓 ∶ 𝑈 → ℝ 𝐶1such thatF= ∇𝑓.
2. For any piecewise smooth closed oriented curve𝐶in𝑈, we have
∫𝐶F⋅dx= 0.
3. Fsatisfies thepath-independence property: for any two piecewise smooth oriented curves𝐶1and𝐶2with same startpoint and same endpoint, we have
∫𝐶
1
F⋅dx=
∫𝐶
2
F⋅dx.
Then we say thatFisconservativeand that𝑓 is apotentialofF(beware: in physics we usually say that−𝑓 is a potential ofF).
2 Poincaré lemma: potentials and vector potentials
2.1 𝑛 = 2
Proposition 4. Let𝑈 ⊂ ℝ2be open andF∶ 𝑈 → ℝ2be a𝐶1vector field.
IfFis conservative then 𝜕𝐹2
𝜕𝑥 = 𝜕𝐹1
𝜕𝑦 on𝑈.
When𝑈is star-shaped, the converse is true:
Theorem 5(Poincaré lemma). Let𝑈 ⊂ ℝ2be astar-shapedopen set andF∶ 𝑈 → ℝ2be a𝐶1vector field.
If 𝜕𝐹2
𝜕𝑥 = 𝜕𝐹1
𝜕𝑦 on𝑈then there exists𝑓 ∶ 𝑈 → ℝ 𝐶2such thatF= ∇𝑓. Beware:the above theorem is false with no assumption on the domain.
Indeed, defineF(𝑥, 𝑦) = (𝑥2+𝑦𝑦 2,𝑥2−𝑥+𝑦2)on𝑈 = ℝ2⧵ {0}. Then𝜕𝐹2
𝜕𝑥 = 𝜕𝐹1
𝜕𝑦 but
∫𝐶F⋅dx= −2𝜋 ≠ 0where 𝐶is the unit circle with the counterclockwise orientation, soFis not conservative.
Alexis Clairaut proved a first version of this Poincaré lemma in 1739 but there was a flaw since he didn’t real- ize that he needed some assumptions on the domain. Then Jean Le Rond d’Alembert gave this counter-example in 1768. Notice that the notation
∫𝐶𝑃 (𝑥, 𝑦)d𝑥 + 𝑄(𝑥, 𝑦)d𝑦was introduced by A. Clairaut in the above cited paper.
Proposition 6. Let𝑈 ⊂ ℝ2be an open rectangle andF∶ 𝑈 → ℝ2be a𝐶1conservative vector field.
Let𝑝 = (𝑎, 𝑏) ∈ 𝑈and define𝑓 ∶ 𝑈 → ℝby𝑓 (𝑥, 𝑦) =
∫
𝑥 𝑎
𝐹1(𝑡, 𝑏)d𝑡 +
∫
𝑦 𝑏
𝐹2(𝑥, 𝑡)d𝑡.
Then𝑓 is𝐶2andF= ∇𝑓.
2.2 𝑛 = 3
Proposition 7. Let𝑈 ⊂ ℝ3be open andF∶ 𝑈 → ℝ3be a𝐶1vector field.
IfFis conservative thencurlF=0on𝑈.
When𝑈is star-shaped, the converse is true:
Theorem 8(Poincaré lemma). Let𝑈 ⊂ ℝ3be astar-shapedopen set andF∶ 𝑈 → ℝ3be a𝐶1vector field.
IfcurlF=0on𝑈then there exists𝑓 ∶ 𝑈 → ℝ 𝐶2such thatF= ∇𝑓.
Proposition 9. Let𝑈 ⊂ ℝ3be an open rectangle andF∶ 𝑈 → ℝ3be a𝐶1conservative vector field.
Let𝑝 = (𝑎, 𝑏, 𝑐) ∈ 𝑈and define𝑓 ∶ 𝑈 → ℝby𝑓 (𝑥, 𝑦, 𝑧) =
∫
𝑥 𝑎
𝐹1(𝑡, 𝑏, 𝑐)d𝑡 +
∫
𝑦 𝑏
𝐹2(𝑥, 𝑡, 𝑐)d𝑡 +
∫
𝑧 𝑐
𝐹2(𝑥, 𝑦, 𝑡)d𝑡.
Then𝑓 is𝐶2andF= ∇𝑓.
2.3 General case
This subsection is not part of MAT237.
The proof of the above Poincaré Lemma for𝑛 = 2relied on the Gradient Theorem, and the proof of the above Poincaré Lemma for𝑛 = 3relied on Kelvin–Stokes theorem. They admit a generalization to any𝑛 from which we may derive the following general results.
Proposition 10. Let𝑈 ⊂ ℝ𝑛be open andF∶ 𝑈 → ℝ𝑛be a𝐶1vector field.
IfFis conservative then∀𝑖, 𝑗 = 1, … , 𝑛, 𝜕𝐹𝑖
𝜕𝑥𝑗 = 𝜕𝐹𝑗
𝜕𝑥𝑖 on𝑈.
When𝑈is star-shaped, the converse is true:
Theorem 11(Poincaré lemma). Let𝑈 ⊂ ℝ𝑛be astar-shapedopen set andF∶ 𝑈 → ℝ𝑛be a𝐶1vector field.
If∀𝑖, 𝑗 = 1, … , 𝑛, 𝜕𝐹𝑖
𝜕𝑥𝑗 = 𝜕𝐹𝑗
𝜕𝑥𝑖 on𝑈then there exists𝑓 ∶ 𝑈 → ℝ 𝐶2such thatF=∇𝑓.
MAT237Y1 – LEC5201 – J.-B. Campesato 3 Proposition 12. Let𝑈 ⊂ ℝ𝑛be an open rectangle andF∶ 𝑈 → ℝ𝑛be a𝐶1conservative vector field.
Let𝑝 = (𝑎1, … , 𝑎𝑛) ∈ 𝑈and define𝑓 ∶ 𝑈 → ℝby𝑓 (𝑥1, … , 𝑥𝑛) =
𝑛
∑𝑖=1∫
𝑥𝑖 𝑎𝑖
𝐹𝑖(𝑥1, … , 𝑥𝑖−1, 𝑡, 𝑎𝑖+1, … , 𝑎𝑛)d𝑡.
Then𝑓 is𝐶2andF= ∇𝑓.
3 Vector potentials
Proposition 13. Let𝑈 ⊂ ℝ3be open andF∶ 𝑈 → ℝ3be a𝐶1vector field.
IfF=curlGforG∶ 𝑈 → ℝ3𝐶2thendivF= 0on𝑈.
Then we say thatGis avector potentialofF.
When𝑈is star-shaped, the converse is true:
Theorem 14(Poincaré lemma). Let𝑈 ⊂ ℝ3be astar-shapedopen set andF∶ 𝑈 → ℝ3be a𝐶1vector field.
IfdivF= 0on𝑈then there existsG∶ 𝑈 → ℝ3𝐶2such thatF=curlG.
Beware:the above theorem is false with no assumption on the domain.
Indeed, defineF= ‖r‖r3 onℝ3⧵ {0}wherer(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧).
Then divF= 0but there is noG∶ ℝ3⧵ {0} → ℝ3such thatF=curlG.
Proposition 15. Let𝑈 ⊂ ℝ3be astar-shapedopen set andF∶ 𝑈 → ℝ3be a𝐶1vector field such thatdivF= 0.
Let𝑝0= (𝑥0, 𝑦0, 𝑧0) ∈ 𝑈such that for any𝑞 ∈ 𝑈the line segment from𝑝0to𝑞is included in𝑈.
DefineG∶ 𝑈 → ℝ3by
G(𝑥, 𝑦, 𝑧) =
∫
1
0
F
⎛⎜
⎜⎝
𝑥0+ 𝑡(𝑥 − 𝑥0) 𝑦0+ 𝑡(𝑦 − 𝑦0) 𝑧0+ 𝑡(𝑧 − 𝑧0)
⎞⎟
⎟⎠
×
⎛⎜
⎜⎝ 𝑥 − 𝑥0 𝑦 − 𝑦0 𝑧 − 𝑧0
⎞⎟
⎟⎠ 𝑡d𝑡
ThenGis𝐶2andF=curlG.
Quite often (but not always),𝑈iscenteredat0so that we can take𝑝0= 0, then the above formula may rewritten
G=
∫
1 0
F(𝑡r) × (𝑡r)d𝑡 wherer(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, 𝑧).
4 The general Poincaré lemma
In all the above results namedPoincaré Lemma, we may relax the assumption on the domain by assuming that𝑈 is open and contractible (which is weaker than open and star-shaped since a star-shaped set is contractible) but this notion is not part of MAT237.
Then, they are all special cases of the following very general result:
Theorem 16(Poincaré Lemma).
If𝑀is a contractible manifold then𝐻dR𝑛 (𝑀) = {
ℝ if𝑛 = 0 0 otherwise , i.e. the closed differential forms on𝑀are exact.
You should come back here after you learn differential forms and de Rham cohomology.
