MECANIQUE RATIONNELLE

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UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 47 A.KADI 5) → −−−→ −−→ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂ ∂ − ∂ ∂ ∂ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ ∧ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ = 0 0 0 0 ) ( 2 2 2 2 2 2 y x f y x f x z f x z f z y f z y f x f y y f x z f x x f z y f z z f y z f y f x f z y x f grad rot D’une autre manière : → → → → → −−−→ −−→ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∇∧ ∇ = ∇∧ ∇ = 0 ) ( f f f grad rot 6) ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ = ∇∧ ∇ = • ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ → → • → → → y A x A x A z A z A y A z y x A A rot div( x y z x y z ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ − ∂ ∂ ∂ ∂ = y A x A z x A z A y z A y A x x y z x y z 0 2 2 2 2 2 2 = ∂ ∂ ∂ − ∂ ∂ ∂ + ∂ ∂ ∂ − ∂ ∂ ∂ + ∂ ∂ ∂ − ∂ ∂ ∂ = y z A x z A x y A z y A z x A y x A x y z x y z D’une autre manière : ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ → → • → → → ∇∧ ∇ = A A rot div( ) soit les vecteurs sont perpendiculaires au vecteur résultat → ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ → → = ∧ ∇ B A → → ∇ A et → B . Nous avons alors : → • → → → ∇ = B A rot div( ) Comme d’où : → → ⊥ ∇ B ⇒ 0 = ∇ → • → B 0 ) = → → A rot div( 7) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − − − ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∧ • → → x y y x z x x z y z z y B A B A B A B A B A B A z y x B A div ( ) ( ) ( ) x y y x z x x z y z z y B A B A z B A B A y B A B A x − ∂ ∂ + − ∂ ∂ + − ∂ ∂ =
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI = B − A x x ⎛ ∂Az ⎜ ⎝ ∂y ⎛ ∂Bz ⎜ ⎝ ∂y ∂Ay − ∂z ∂B y − ∂z ⎞ ⎟ + B ⎠ ⎞ ⎟ − A ⎠ y y ⎛ ∂A ⎜ ⎝ ∂z ⎛ ∂B ⎜ ⎝ ∂z x x ∂Az − ∂x ∂Bz − ∂x ⎞ ⎟ + B ⎠ z ⎛ ∂Ay ⎜ ⎝ ∂x ⎞ ⎛ ∂B ⎟ − A ⎜ z ⎠ ⎝ ∂x y ∂Ax − ∂y ∂Bx − ∂y ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ → → div ( A∧ B) = → −−→ B • rotA → −−→ − ArotB Exercice 20 : → → → → Soit un vecteur r = x i + y j+ z k exprimé dans un repère orthonormé R( O, i , j, k) . → → → −−−→ () 1) Calculer grad r et −−−→ ⎛ ⎞ grad⎜ 1 ⎟ ; ⎝ r ⎠ 2) Si U(r) est un champ scalaire à symétrie sphérique, montrer que grad − −−→ ( U (r)) est un vecteur radial ; 3) Calculer div( → r ) et en déduire que pour un champ électrique Coulombien : → → r E = k on a r → → div E = 0 ; ⎛ 1 ⎞ 4) Montrer que Δ⎜ ⎟ = 0 avec r ≠ 0 ; ⎝ r ⎠ 5) Calculer −−→ ⎛ ⎞ rot → ⎜ r ⎟ ⎝ ⎠ Solution : 1 − 1 1 2 2 2 2 2 2 1) Nous avons : r = x + y + z = ( x + y + z ) 2 2 2 2 et = ( x + y + z ) 2 −−−→ → → → 1 → 1 ∂r ∂r ∂r − − → 2 2 2 () r i + j+ k = x( x + y + z ) 2 2 2 2 i + y( x + y + z ) 2 2 2 2 j+ z( x + y + z ) grad = 2 k ∂x ∂y ∂z r 1 − → = → → → x i + y j+ z k = → 2 2 2 r ( x + y + z ) 1 2 r 48
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MECANIQUE RATIONNELLE

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