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Integral_Curviligne_Licence_3_Mathematics
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  • Paris VI - Université Pierre Et Marie Curie

Lecture notes and summary on curvilinear transformations, all the rules you need to solve your exercises. The words are in basic English but the math part is easy to grasp.

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Document information

  • November 25, 2023
  • 3
  • 2022/2023
  • Class notes
  • Patrick
  • All classes

Subjects

  • maths
  • mathematics
  • integrals
  • curvilinear integral
  • license 3
  • sorbonne
  • sorbonne university
  • analysis
  • 2022

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  • Paris VI - Université Pierre et Marie Curie
  • Mathématiques
  • Unknown

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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.

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