Intégrale_Curviligne_Licence_3_Mathématiques

III. Line integrals :

Let 𝛾 be a curve arc in an open set 𝑈 of ℝ2 defined by its parametric representation :

𝑡 ⟼ 𝑥(𝑡)
{ ; 𝑡 ∈ [𝑎, 𝑏]
𝑡 ⟼ 𝑦(𝑡)

We call the line integral of the differential form 𝑤 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 defined on 𝑈, the number ∫𝛾 𝜔 such
that :

𝒃
∫𝜸 𝝎 = ∫𝒂 [𝑷(𝒙(𝒕), 𝒚(𝒕))𝒙′ (𝒕) + 𝑸(𝒙(𝒕), 𝒚(𝒕))𝒚′ (𝒕)]. 𝒅𝒕

𝜕𝑓 𝜕𝑓
We can write : 𝑑𝑓 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦 = 𝜕𝑥 𝑑𝑥 + 𝜕𝑦 𝑑𝑦

𝑏 𝜕𝑓 𝜕𝑓
Then : ∫𝛾 𝜔 = ∫𝛾 𝑑𝑓 = ∫𝑎 [𝜕𝑥 (𝑥(𝑡), 𝑦(𝑡))𝑥 ′ (𝑡) + 𝜕𝑦 (𝑥(𝑡), 𝑦(𝑡))𝑦 ′ (𝑡)] . 𝑑𝑡

We say that 𝜔 is an exact form if :

𝑏
∫ 𝜔 = ∫ [𝑓(𝑥(𝑡), 𝑦(𝑡))]𝑑𝑡 = 𝑓(𝑥(𝑏), 𝑦(𝑏)) − 𝑓(𝑥(𝑎), 𝑦(𝑎)) = 𝑓(𝐵) − 𝑓(𝐴)
𝛾 𝑎


Which means that ∫𝛾 𝜔 depends only on the origin and the endpoint.

❖ Properties :

𝐵 𝐴
▪ ∫𝛾 𝜔 = ∫𝐴 𝑃𝑑𝑥 + 𝑄𝑑𝑦 = − ∫𝐵 𝑃𝑑𝑥 + 𝑄𝑑𝑦
▪ ∫𝛾 (𝜔1 + 𝜔2 ) = ∫𝛾 𝜔1 + ∫𝛾 𝜔2 ; 𝜔1, 𝜔2 being differential forms.
▪ ∫𝛾 𝜆𝜔 = 𝜆 ∫𝛾 𝜔 ; (𝜆 ∈ ℝ).
▪ 𝛾 = 𝛾1 + 𝛾2 : ∫𝛾 𝜔 = ∫𝛾 𝜔 + ∫𝛾 𝜔
1 2


❖ Line integral of a vector field :

Let 𝑓(𝑥, 𝑦) = (𝑃(𝑥, 𝑦), 𝑄(𝑥, 𝑦)) be a vector field moving along a curve arc. The work of 𝑓 on 𝐴𝐵 is
𝐵
given by the line integral : ∫𝐴 𝑃𝑑𝑥 + 𝑄𝑑𝑦

Definition 1 : A closed curve is said to be simple if it does not intersect itself.




Simple Not Simple


Definition 2 : A connected open set 𝐷 in ℝ2 is said to ss
be connected if any two points in 𝐷
can be joined by a curve that lies entirely in 𝐷.