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09 fonctions vectorielles à valeurs dans un espace euclidien

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09 fonctions vectorielles à valeurs dans un espace euclidien

  1. 1. !!" #= $ ! → % ! & !!!# % % = =∈∀ "" "& % $ ' % ( ∈ ∈ !!!# ! [ ] )" "")) =−⇔ =∈∀⇔ <−<−∈∀>∃>∀⇔= εααε ' % * ∞+ ∞− + , ! ' % + ] [+∞∩ ] [∞−∩ ! & $ + % → ∈λ →ϕ % ! + λ ϕ ⋅ + + λ !ϕ ⋅ ! - . / ∧ + ∧ & 0 ( →ϕ " ⊂ → " ⊂"ϕ ∈α ϕ ∈ α ϕ α + ! ' % ( ∈ ! ⇔ " * " [ ]∈∀⇔ ! "' % * 1 touscours.net
  2. 2. # ' % ∈∀⇔ ! [ ]∈∀⇔ ! & , % + % 2 & % 0 ( → $ " 3 + * 4 ∈ + ≤∈∀ " & %% *+ $ [ ]∈∀⇔ $ ! 5 % $ = " % ' % ( ∈ ! $ ⇔ * { } − − → "" 6 ! [ ]∈∀⇔ $ ! "3 " = = "3"3 ! ' % * $ $ ! "7 ( $ )3 =⇔= 8 % 4 % % 2 9 & % $ ( → $ ∈λ →ϕ $ + λ ϕ ⋅ $ "3"""3"3" "3!""!"3"3!" "3!"3!" "3"3"3" ⋅+⋅=⋅ += = +=+ ϕϕϕ λλ - . / ∧ $ "3"""3"3" ∧+∧=∧ ! touscours.net
  3. 3. / 7 + & $ % : $ ; "" "!" ⋅ϕ " " ∧ + ϕ ! < 0 ( →ϕ → ⊂"ϕ ∈α ϕ $ α $ "αϕ ϕ $ α ""3"3"3" αϕαϕαϕ = ( → $ )" ≠ $ " "3" "3" ⋅ = - %% ⋅= + 0 % ⋅ ( )"" ≠⋅ " 3 )" ≠ ⋅ $ " "3" "" "3"# # ⋅ = ⋅ ⋅ ! ( $ = 3⊥ " 3 "3" ⊥∈∀ - %% ( )== * + )== *. ! ( " "3" "3") ⋅ ==∈∀ -4 * ! " # $ 4 $ " & , % * " ∈ % ! $ ! ! : + * = % ! "! # π+ = $ ! % ( →=> ! = " 4 $ !!!" #= ! % " ! : % * 4% 42 touscours.net
  4. 4. ? 2 < 0 " ( →=> " 4 ≤ ≤ "" < 0 ( → % " * * + ! 5 + → """ −= ! + 0 * $ 2 7 + : % % " = 4 ! ! 5 % ( => $ >= 4 + ∈ + ≤∈∀ "3>= −≤− "" ! :* % % " = 5 < @: * − ( " ≥ => "" =>∈ = "+ 4 * 0 A " A" " !!!"3"""" " − ≤ − − ++−+− − − :* < @: % " < " * − →=> " ≥ − − + − − ++−+= − − − " A" " A" " !!!"3""" "" " % * $ B * < @: B 0 , < @C " * ( → ) 4 →ε + )" ) =ε ")"!!!!)"33!)"3!)"" " A# # ε+++++=∈∀ touscours.net
  5. 5. ' % ( % ∈) = { })6 → ! & + ': * ) + * 4 % →ε !!!) + """!!!""" )" ))# # ))) ) ε ε −+−++−+−+=∈∀• =• D * ': * ) ! -4 * ': * ) 4 ': ) ≤ ! -4 * ': * ) ) ⇔ 4 * ) - + ∈) " 3 + = -4 ': * ) ⇔ $ ) "E * 4 ) ': * ) "' % < @C & ': ( ! + + ! touscours.net

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