Formulaire OEM

~a ~
b
~
∇
~
∇.(~ ∇U) = ~ ∇2Udiv(
→
grad U) = ∆U
~
∇ ∧ (~ ∇U) = ~ 0
→
rot(
→
grad U) = ~ 0
~
∇.(~ ∇ ∧ ~a) = ~ 0div(
→
rot ~a) = ~ 0
~
∇ ∧ (~ ∇ ∧ ~a) = ~ ∇(~ ∇.~a)−~ ∇2~a
→
rot(
→
rot ~a) =
→
grad(div~a)−∆~a
~
∇(UV ) = V~ ∇(U) + U~ ∇(V)
→
grad(UV ) = V
→
grad U +U
→
grad V
~
∇.(U~a) = ~a(~ ∇U) + U(~ ∇.~a)div(U~a) =
→
grad U.~a +U div~a
~
∇ ∧ (U~a) = (~ ∇U)∧~a +U(~ ∇ ∧ ~a)
→
rot(U~a) =
→
grad U ∧~a +U
→
rot ~a
~
∇.(~a ∧~ b) = ~ b.(~ ∇ ∧ ~a)−~a.(~ ∇ ∧ ~ b)div(~a ∧~
b) = ~ b.
→
rot ~a −~a.
→
rot~ b
~
∇ ∧ (~a ∧~ b) = (~ ∇.~ b)~a −(~ ∇.~a)~ b+ (~ b.~ ∇)~a −(~a.~ ∇)~ b
→
rot(~a ∧~ b) = (div~ b)~a −(div~a)~ b+ (~ b.
→
grad)~a −(~a.
→
grad)~ b
~
∇(~a.~ b) = ~a ∧(~ ∇ ∧ ~ b) +~ b∧(~ ∇ ∧ ~a) + (~ b.~ ∇)~a + (~a.~ ∇)~ b
→
grad(~a.~ b) = ~a ∧(
→
rot~ b) +~ b∧(
→
rot ~a) + (~ b.
→
grad)~a + (~a.
→
grad)~ b
→
grad U =∂U
∂x ~ex+∂U ∂y ~ey+∂U ∂z ~ez
→
grad U =∂U
∂r ~er+1 r∂U ∂θ ~eθ+∂U ∂z ~ez
→
grad U =∂U
∂r ~er+1 r∂U ∂θ ~eθ+1 rsin θ∂U ∂ϕ ~eϕ
div~a =∂ax
∂x +∂ay ∂y +∂az ∂z 
div~a =1
r∂rar ∂r +1 r∂aθ ∂θ +∂az ∂z ~a =ar(r)~erdiv~a =1 r∂rar ∂r 
div~a =1
r2∂r2ar ∂r +1 rsin θ∂aθsin θ ∂θ +1 rsin θ∂aϕ ∂ϕ ~a =ar(r)~erdiv~a =1 r2∂r2ar ∂r 

→
rot ~a =∂az
∂y −∂ay ∂z ~ex+∂ax ∂z −∂az ∂x ~ey+∂ay ∂x −∂ax ∂y ~ez
→
rot ~a =1
r
∂az
∂θ −∂aθ ∂z ~er+∂ar ∂z −∂az ∂r ~eθ+1 r∂(raθ) ∂r −∂ar ∂θ ~ez
→
rot ~a =1
rsin θ∂(aϕsin θ) ∂θ −∂aθ ∂ϕ ~er+1 r1 sin θ
∂ar
∂ϕ −∂(raϕ) ∂r ~eθ+1 r∂(raθ) ∂r −∂ar ∂θ ~eϕ
∆U=∂2U
∂x2+∂2U ∂y2+∂2U ∂z2
∆U=∂2U
∂r2+1 r∂U ∂r +1 r2∂2U ∂θ2+∂2U ∂z2U=U(r) ∆U=1 r∂ ∂r r∂U ∂r 
∆U=∂2U
∂r2+2 r∂U ∂r +1
r2sin2θ∂2U
∂ϕ2+1 r2sin θ∂ ∂θ sin θ∂U ∂θ 
U=U(r) ∆U=1
r2∂ ∂r r2∂U ∂r 
~
rot~er=~ 0~ rot~eθ=~ez
r~
rot~ez=~ 0
div~er=1
rdiv~eθ= 0 div~ez= 0
~
rot~er=~ 0~ rot~eθ=~eϕ
r~
rot~eϕ=cos θ
rsin θ~er−~eθ r
div~er=2
rdiv~eθ=1 rtan θdiv~eϕ= 0