Chapitre 3 :Théorème de Gauss
Transcription
Chapitre 3 :Théorème de Gauss
T his work is licensed under a Creative Commons “Attribution-ShareAlike 4.0 International” license. θ ϕ ρ ! θ ϕ = θ =! " ϕ =! " ! # θ θ= ϕ ϕ= $ ! θ ! ϕ % ϕ & ' ( = !& θ & ϕ & * + & θ ρ π , ) " θ −θ θ ρ θ θ = θ" + = θ" ρ − θ" ρ θ" Ismaël Bouya http://melusine.eu.org/syracuse/immae/ = = + " " = !− θ" θ + = !− " θ+ = θ" θ ρ θ θ+ ρ + θ " ϕ" = " - θ" θ + ! θ" θ " ϕ" ρ ϕ ϕ + " θ" θ θ " ϕ" + " ϕ ) ∂ ∂ = ∂ ∂θ ∂ θ ∂ϕ θ θ ϕ . + . + ! - ϕ+ ϕ ) ,π × ) ' θ " ϕ" = = π× !/ θ , θ " ϕ" θ = , θ ! θ !θ + θ " ϕ = θ" ϕ θ" ϕ 0 = × θ" " θ" ϕ 1 2 = ( = π ,π , θ =3 ϕ =3 θ" θ" ϕ = π ,π θ =3 , θ " θ = 4π , 2 - 5 6 1 7 5 5 ) ' ) " , 5 & 5 8 5 & 5 0 9 :" 5 = ! " 8 & 6 1 φ= ! ⋅ = ⋅ " , 6 18 & 5 5 & ! ) 5 Γ - 5 φ= ! φ ! ⋅ = " " 5 ! ⋅ ! 5 ! = ⋅ ! ! ! 5 5 5 & & " φ ! = ⋅ ! ! $ 5 2 φ ! - = ε3 5 ! 5 ( " & 1 "+ & ! 2 5 ( 5 σ3 " 5 =! ; ) " ! ! $ 7( 7 ) ! + ! - " ( ! " 8 " = ! ! <& - ∂ ∂ "∂ ∂ % - =3 = ! =3 ! " ! , 7 5 5 5 5 5 5 = φ ! = 5 ( ) = + 8 > − " + 5 φ= = φ+ >3 ) 5 φ+ " φ 5 5 ! =− φ= ! ⋅ "! − = ! ⋅ !− 4 ! = φ =− - φ - !− ! = ! 5 =− ! ! = ! ⋅ " =3 5 φ % = ! ! ⊥ φ= ! ⋅ " = φ ! =3 φ ! = , ! - 2 =, ! =σ3 ∩ ) ! = σ3 ε3 ! = ! σ3 " ,ε 3 σ3 ,ε 3 = −σ3 ,ε 3 >3 <3 3 ∆ = σ3 ε3 +σ −σ - 1 ) 5 5 5 σ >3 −σ < 3 " = σ" − @ 55 ' A 5 " ? 2 8 )1 1 ! =3 1 ! = −σ ε3 =3 =3 + &" ! - 8) 5 & & =− − ∂∂ = − ∂∂ − ∂∂ −σ ε3 - ! = 8 )1 ! σ" + ε % = 8) " ! − C, C, − = C = - − − = ! , − ! −, = σ" σ" " = = ε3 ε3 !ε 3 C = ε3 C B " "# ! $! & 6 →, , = = , 4πε 3 ! , = ! , , , 4πε 3 ↔ ↔− →, , , = 4πε 3 = , − , = ϕ! ! ! , ! E ϕ" ϕ = , 5 ' − & " ! = ϕ! ! = , , , =− = φ ! , , =− ! , , , − = ! 4πε 3 − = φ ! ε3 = × !−4π 2 ( @ & = B433F " 4 = µ3 π ' ! & µ3 5 " = B 3" 3 ,4 F 7( 7( ϕ ! =− ϕ= ! − ϕ = ! D , 5 ( 5 = ! ! φ= ! ⋅ φ= = " ! = ! " ⋅ = ! ! ! = ! 4π " , 8 ! < > > 4 = π 7 " = ! ! @ 7 ∈ ! 5 µ3 = < 4 = π " µ3 = φ ! = −4π A4π " ≤ -) H ! - ! = −4π ! =− ! A4π " ≥ , , =− , ! 2 8 )1 ) " = −4π ! , =− =− , & 8 " ∝ !∝ ∝ 5 9 −, 8: G " & ϕ⇔ =− ϕ! = ϕ! = , - − ϕ =− + , 3≤ ≤ + , ≥ + , , , + − = → +∞ ϕ! = ϕ! = - ϕ =3 − , − + , , , =3" 2 = − , " 3≤ ≤ ≥ 4 2 $ ( 4 = π ρ3 & ρ3 " ρ3 ! = ! = " = 4πε 3 4πε 3 ! = ! = 3≤ ≤ ε3 ≥ , − , 4πε 3 4πε 3 ρ3 + ≥ " , 4πε 3 , 3≤ ≤ 5 "