Ce document est mis à disposition selon les termes de la licence Creative Commons “Attribution – Partage dans les mêmes conditions 4.0 International”. https://www.immae.eu/cours/ ρ ! " $ # ! ρ =% =% ρ ρ " ∂ =% ∂ & ∇⋅ = % " ' ( ! = $ = " ) # µ% π τ∧ * $+ µ% = π " % −- , ( ) ./ ε % µ% ! " ! µ% # ( 0 = " ) $ = ( & µ% π ∧ . * = µ% π ∧ * )) ) # " 1 2 • 3 • 4 2 )5 4 )) 4 1 , ! 6 • # µ% π ) )5 #' . = 7 ! # 9 # 9 : 7! " " 2# ) ; • • # : # # ) # ∧ ! ! * 8 ) " 0 " < ! . 6 ! 3. / = &7 7 µ% π ∧ 1 8 * , ! , 0 µ% π µ ≈ % π = $ = µ% π ∧ * ∧ * ∧ * = * 2 7 ) ! = ! # 0 ) 4 ) 7 1 = / " , ! ! 0 ! ) # ! ; / & 2! $ µ% π µ% = π µ = % π ∇ ⋅ = ∇ ⋅ ∧ * τ ∇ ⋅ ∧ * τ * ⋅∇ ∧ − ⋅∇ ∧ * τ =% µ = % π ⋅∇ ∧ − * τ= µ% π τ =% ⋅∇ ∧ ∇ =% > ∇ ⋅ 747 ) =% ! 0 " 1 ? ' $ # # / ⋅ =% & 9 0 / @ ) ) $ " A ! ! 4 # B ) :" ! ' $ ! ∇⋅ = % ⇔ ∃ & =∇∧ ' ! =∇∧ " " ; >= , + ∇φ ) +φ > > # ) # , ; 0 0 ; )) ; " " ∇⋅ = % # ) # ! )5 " ; # ! ! " * * ' " = $ µ% π ∧ τ= * µ% π ∇ τ ∧ = −∇ $ ∇ ∧ =∇ ∧ − ∇ ∧ =% ∇∧ )5 = $ = "∇ ∧ + ∇ ∧ µ% ∇ ∧ π µ = % π τ τ " • • $ ∇ ⋅ ! ! ! ) )) = ; µ% π 0 τ ∇ ⋅ $ ∇ ⋅ = ∇ ⋅ + ∇ ⋅ = −∇ ⋅ =% = ∇ ⋅ −∇ ⋅ = −∇ ⋅ =% & # # ∇ ⋅ =− $ ∇ ⋅ = −∇ ⋅ ! µ% π τ =− ∇ ⋅ µ% π ⋅ # )) ) ; 4 ) " ! # " ! #4 − ! + 4 #' ! #' ) #4 " $ )) ! =% ) $ 2# + ∇ ⋅ ) =% # =% ! ⋅ =% 4 " ! ) " < • 7 # # ( µ = % π $ 2 ! 0 ) # ! µ% π µ% = π ∇ 4 = τ τ ∇ τ ∇ − πδ = − µ% , ∇ $ ∇ = − µ% = −µ% ) ! ) ) # D 7 ( $ ∇ C ! ./ ; 0" 0 D # ! ! =% ;4 0 0 4 ) ) + # ! • 7 0 4 ) " ) ' / = ) $+ = )) $ = ∧ 0 ! " ' − % % ∧ " " = % 2 ∇∧ 7 . $ = 2! = ∂ ∂ ∂ ∂ ∂ ∂ − ∧ % % + = % = θ ∧ + 4 #) 4 = "" θ ) " C ( 0 0 = ! ! ; 0 ! ) # )) ∇⋅ = ∂ ∂ + ∂ ∂ =% =− = ! & ( " ( ! $ ( $ ∇∧ =∇∧ ∇∧ 2 ∇∧ ∇ = ∇ ∇⋅ −∇ ∧ ∇∧ = ∇ ∇⋅ −∇ =% ; , ( 0 = µ% ∇∧ ; ! 0 # ( 1 F 0 # " ! 0 " ) $ ∇∧ =% / #& / $ & ) Γ / 0 ! + Σ ) # " Γ " # / / , A ⋅ = ∇∧ ⋅ Σ = µ% ⋅ Σ 0 • 2) $ = Σ ⋅ E & # = µ% " ⋅ ! 4 / Γ 0 • ? @ " " ' > G , ! = # > # ⋅ > 0 > Γ = $ &! = > =− # =− > * =− + $ $ = −* $ =% + "# * $ & ! = $ " µ% π τ ! = ρ πε % τ # = " ( ! , < % ) " - ( ) < ) % * ( ! 1 ! " " ( ! = $ < , µ% π τ∧ * = % ) " ( ) • • # ) " δ = δ ∇⋅ $ δφ + δφ + δφ = − ⋅ δ + > 747 $ δφ → % $ & − > 747 δφ = % =% ⋅ δ + δφ 90 H : )) ) ⋅δ + / ⋅δ = % − − ⋅ =% =% • 2 $ δ $ δ $ $ , − $ ) δ = ⋅δ + ⋅ 4δ = µ %δ ⋅ δ = µ% )) ⋅ δ ' ! % δ = ∧δ $ 2 # δ "# ∧ ⋅δ ∧ = µ% − ⋅ ∧ δ = µ% ) " = µ% − 2 ⋅ δ = µ% − ! ∧ % < = ∇⋅ ρ τ πε % = ∇∧ = * ρ ε% ∇⋅ / τ∧ * =% ∇ ∧ = µ% 0 F / #& / ? ' ! ) =% ! ! F µ% π @ ! & / = −∇ = ∇ = τ ε% = / =∇∧ µ = % π ∇ = −µ% −ρ − " ρ πε % & τ ( σ ε% − = µ% # . ∧ & & . ,. • π π . = > ∀ I • π ! . 4π . > ∀ " ( ⊂π ∈π & . • $ & • =− ∀ > ∈π =%" ! . π − 4π ∈π , 4π > * $ ! • , $ , ' = ! G . # D ⋅ # . " . ! ! G D • $ G D G D D G ! ∧ G D G D D G G D % $ • @ ! • ∇ 2! ∇⋅ • ( ∇∧ G G D D G G D D G D D G 1 & J = $ $ ρ τ = −∇ " πε % 0 G G G ρ ! D D D < $ = µ% π 0 τ =∇∧ G G D D D G " & < $ = µ% π ∧ θ * "2 . " = = , µ% π , µ% π + θ θ ∧ − * θ H # 0 0 K 4 / &! ⋅ * #& )) = µ% / . / ! =∇∧ ' ! ! ) " 1 ? H ./ ) Γ ,. θ = 2 F / #& θ / π " " = µ% = µ% µ 2 = % θ π" ( → +∞ →% . #& / * " • ( # µ% π ;4 ! +∞ = $ $ # µ% π = −∞ ;4 +∞ L " −∞ # ! ! " • $ $ • < F =∇∧ " 9 : 0 4 # " . " F 2 ( . " = ) . $ ⋅ = + ∇φ / , θ ! ) " # / ∇∧ ⋅ = = A ⋅ θ $ & − , = + − µ% π" = M0 . % ) . µ% π " % " = = − θ ) $ & µ = % π" ) " θ * 2# / × π " = µ% / #& / . >" = <" " µ% π" µ% π "" = & θ >" θ <" 2 , • & = • 7 , µ% π #' θ = µ% π "" ( θ) = µ% " θ θ θ= $ − 7 2 F # = "& 7 $ # = # µ% π ∧ * # . . = 2 & 2 = 2 7 µ% π = θ # θ " = $ µ% π 9 : 4 µ π" = % π θ µ% * " θ" = N ! $ θ= $ ) " " * θ ! " = " + 0 ( • . "J " * #' << " #' 4 ! = = = & " ! $ $ # 4 ! β =% " ! λ " 0/ 2 $ # #)' #' " # . #' " θ + + "α = "α α = % ∂ ∂ + β" % ! =% = 0 . . ≈ % ≈− )' "& ∂ ∂ =% C ( / =% ! = +ρ =% )) " • , O 2) # 0 = = = $ . " " 0 " << " $ #' θ= $ " = . = , θ = "" µ% * " << " / θ 0 = 2 × = µ% = µ% " × * θ" θ − −" θ θ θ E * 4 # #' " , O • / 0 ) $ )) 0/ • 2 F # . 2 • 7 , ! " # 4 . " = 4 # #' θ =% θ =π " = µ% " #' 2 7 3 $ 4 # ⋅ $ = µ% =% ' • $ C P #' " 4 #' µ% − $ = µ% $ O # 4 0 > = µ% " 747 =% ) ! ) = θ ) - $ ! $ O = µ% & #' =% ; O " " = ⋅ θ = " O & " " 2 = µ% $ $ "+ − ! " " − = % − µ% = − µ% 4 = µ% ! ∧ # ) " " & ' R R/ / 9 : (Q λ ) 0 (Q / '/ > 47 ! ! B " $ ) ω σ $ #' $ / = 0 µ% # * / θ" σ = 2 $ ⋅ % " θ 5 4 = σ " "ω" δ ) = σ " = σ " "ω" ! ! ) = π "2 ω = δ δ = σ × π" " = σ " "ω" θ 7 < 5 µ%σ " "ω" = * θ" = µ%σ "ω" * θ" θ θ $ =− 2 = & = θ − µ %σ "ω" µ% * σω" − > %" = & / µ% θ θ= θ θ θ θ = <% µ% σω" − θ % α θ − " " + 1 2 $ =% α →% π α= #. = σ" ω $ 7 σ & S %C σ= $ 7 7 π #' %" S πε % " ε% S " µ%σω µ %ε % "S ωS ω ω 4 # (Q L % $ S ) 0 #' # − 4 # S % −C F " F ! ) ! " %I S %− F %- F L ) ) . " # ! ! ! ) 0 ! # α π +ε = α α= " + θ − α θ − +ε σω"" − α= θ θ µ %σ "ω" =− " I ! )) . % −C F C # # / # # / ) 0 ! ! ! ! 0 4 # 4) 0 " . # ) * = − J" % = " " % %< < " θ >" θ " $ 2 ;4 ! ∇⋅ ) =% #' 6 $ ∇∧ ! ) = µ% # T " ,. θ ) 2 ;4 & 2 = #' " & = * ) . ! " ) = ! $ ⋅ = , $ 2 µ% ⋅ . π" = ⋅ + − = µ% × π " = µ% × π " " % 2# + = * = , µ% − % µ% " * − J" <" " >" =% ) " ) "+ − $ "− = % − % θ " 2 "+ − "− = # . # "" = % − ∧ . = µ% ) ∧ = , = − % µ% C ) ,# . #' " ; →%" * < 2 10 π" 3 2 − J" % " →% #. K ) 0 0 $ = = + " • , #' ! ! % # = % %% + ∂ ∂ ∂ ∂ × + =% × + =% ∂* ∂ * =% ∂ 2 = ∂ #' #)' & • $ =% ∂ ! ! ! ! # ! * 10 ! # #' " #' " 3 K $ $ ! 0 0 ) 7 ! ! ) # ω " . ρ $ . 2 ! $ " "J #' = ρ " = ρ " ω" θ * ' =% . 4 =% * =% ∂ =% × ! + $ = ρ "ω" × " 4 O ) 0 , >< , >> > = µ % ρω , >" > =% 0 =% , <" = " " . ρωµ % > > = µ% ρω " − ? ! 2) $ ! τ ! • "! ) = ρ& = τ & & ! =ρ − N = +ρ 0 . ' ! ' − + + 0 ρ& = $ & − τ # ρ& = + & & ρ& & +ρ & ρ • ! ! J ! =σ =ρ J • ρ& = & =λ − & ' • ! ) 5 4 = µ% π" θ λ ) ! = µ %λ " π" 4 θ & K (' = = ) λ πε % " 5 ) 4 ! ) K( = * $ )) ) #)) • 7 ? $ ! + =% ! (' S = µ %ε % ( )) 4 )' ) 0 . = # = ! ' + + + '− ' + +' − − = '− ' + +ρ =% − =% =ρ + + +ρ − − − =ρ − − − $ = 7 µ% π" θ =% , $ ρ = ρP + ρ + = ρP ρ& = $ & 2 & & = µ& + = P ρ& + =% ! +ρ − − = =% − =% = µ& = & − =% & & + = − = ρP + P ) & & & µ& = & & λ*& + +ρ − − +' − + 0 " = ρ ! " # 0 $ << − << −