Class notes
Integrals_de_area_license_3_mathematics
- Course
- Institution
Notes and course summaries on surface integrals, all the rules you need to solve your exercises. The words are in basic English but the math part is easy to grasp.
[Show more]
Seller
Follow
scienceexplore
Content preview
IV. Surface integrals :Let 𝑆 be a surface defined by its parametric representation :
𝑥 = 𝑥(𝑢, 𝑣)
{𝑦 = 𝑦(𝑢, 𝑣) ; (𝑢, 𝑣) ∈ 𝐷
𝑧 = 𝑧(𝑢, 𝑣)
𝐹 is a continuous scalar function on 𝑆. The surface integral is given by : ∬𝑆 𝐹 = ∬𝑆 𝐹(𝑆)𝑑𝑆
It is defined by :
⃗⃗ (𝒖, 𝒗)‖. 𝒅𝒖𝒅𝒗
∬ 𝑭(𝑺)𝒅𝑺 = ∬ 𝑭(𝒙(𝒖, 𝒗), 𝒚(𝒖, 𝒗), 𝒛(𝒖, 𝒗))‖𝑵
𝑺 𝑫
⃗ (𝑢, 𝑣) is defined by : (𝑁
The vector 𝑁 ⃗ normal to the surface 𝑆)
⃗⃗
𝜕𝑀 ⃗⃗
𝜕𝑀
⃗ (𝑢, 𝑣) =
𝑁 (𝑢, 𝑣) × (𝑢, 𝑣)
𝜕𝑢 𝜕𝑣
If 𝑆 is the graphical representation of a function 𝜑, then (𝜑 = 𝜑(𝑥, 𝑦)):
𝜕𝜑 𝜕𝜑
⃗ : (−
𝑁 ,− , 1)
𝜕𝑥 𝜕𝑦
We have then :
𝜕𝜑 2 𝜕𝜑 2
√
∬ 𝐹(𝑆)𝑑𝑆 = ∬ 𝐹(𝑥, 𝑦, 𝜑(𝑥, 𝑦)) 1 + ( ) + ( ) . 𝑑𝑥𝑑𝑦
𝑆 𝐷 𝜕𝑥 𝜕𝑦
𝜕𝜑 𝜕𝜑
By denoting = 𝑝 and 𝜕𝑦 = 𝑞, then we have :
𝜕𝑥
∬ 𝐹(𝑆)𝑑𝑆 = ∬ 𝐹(𝑥, 𝑦, 𝜑(𝑥, 𝑦))√1 + 𝑝2 + 𝑞 2 . 𝑑𝑥𝑑𝑦
𝑆 𝐷
We define the area of a surface by : 𝑎(𝑆) = ∬𝑆 𝑑𝑆
𝑎(𝑆) = ∬ √1 + 𝑝2 + 𝑞 2 . 𝑑𝑥𝑑𝑦
𝐷
❖ Ampere-Stoke’s formula :
𝝏𝑹 𝝏𝑸 𝝏𝑷 𝝏𝑹 𝝏𝑸 𝝏𝑷
∫ 𝑷𝒅𝒙 + 𝑸𝒅𝒚 + 𝑹𝒅𝒛 = ∬ ( − ) 𝒅𝒚𝒅𝒛 + ( − ) 𝒅𝒛𝒅𝒙 + ( − ) 𝒅𝒙𝒅𝒚
𝜸 𝑺 𝝏𝒚 𝝏𝒛 𝝏𝒛 𝝏𝒙 𝝏𝒙 𝝏𝒚
𝛾 is the curve along which the line integral is taken, and 𝑆 is a surface resting on 𝛾.
Let 𝛼, 𝛽, 𝜆 be the direction cosines of the normal to 𝑆, then the formula above can be written as :
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller scienceexplore. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $3.26. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75759 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now