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 :
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 :
