formulaire mmc
DESCRIPTION
MMC FORMULAIRETRANSCRIPT
Formulaire
~OM = x ~ex + y ~ey + z ~ez
~gradf =
∂f
∂x
∂f
∂y
∂f
∂z
=∂f
∂x~ex +
∂f
∂y~ey +
∂f
∂z~ez (1)
¯̄grad (~u) = ¯̄grad
ux
uy
uz
=
∂ux∂x
∂ux∂y
∂ux∂z
∂uy∂x
∂uy∂y
∂uy∂z
∂uz∂x
∂uz∂y
∂uz∂z
⇔ ui,j (2)
div (~u) =∂ux∂x
+∂uy∂y
+∂uz∂z⇔ ui,i (3)
~∇ =
∂∂x
∂∂y
∂∂z
(4)
~rot (~u) = ~∇∧ ~u =
∂∂x
∂∂y
∂∂z
∧
ux
uy
uz
(5)
~div(
¯̄U)
= ~div
Uxx Uxy Uxz
Uyx Uyy Uyz
Uzx Uzy Uzz
(6)
~div(
¯̄U)
=
∂Uxx∂x
+∂Uxy∂y
+∂Uxz∂z
∂Uyx∂x
+∂Uyy∂y
+∂Uyz∂z
∂Uzx∂x
+∂Uzy∂y
+∂Uzz∂z
⇔ Uij,j (7)
~OM = r ~er + z ~ez
~gradf =
∂f
∂r
1
r
∂f
∂θ
∂f
∂z
=∂f
∂r~er +
1
r
∂f
∂θ~eθ +
∂f
∂z~ez (8)
¯̄grad (~u) = ¯̄grad
ur
uθ
uz
=
∂ur∂r
1
r
∂ur∂θ− uθ
r
∂ur∂z
∂uθ∂r
1
r
∂uθ∂θ
+urr
∂uθ∂z
∂uz∂r
1
r
∂uz∂θ
∂uz∂z
⇔ ui,j (9)
div (~u) = Tr(
¯̄grad (~u))
(10)
~∇ =
∂∂r
1r∂∂θ
∂∂z
(11)
~rot (~u) =1
r
(∂uz∂θ− ∂
∂z(r uθ)
)~er +
(∂ur∂z− ∂uz
∂r
)~eθ +
1
r
(∂
∂r(r uθ)−
∂ur∂θ
)~ez (12)
~div(
¯̄U)
= ~div
Urr Urθ Urz
Uθr Uθθ Uθz
Uzr Uzθ Uzz
(13)
~div(
¯̄U)
=
∂Urr∂r
+1
r
∂Urθ∂θ
+∂Urz∂z
+Urr − Uθθ
r
∂Uθr∂r
+1
r
∂Uθθ∂θ
+∂Uθz∂z
+2 Uθrr
∂Uzr∂r
+1
r
∂Uzθ∂θ
+∂Uzz∂z
+Uzrr
⇔ Uij,j (14)
~OM = r ~er
~gradf =
∂f
∂r
1
r
∂f
∂φ
1
r sin(φ)
∂f
∂θ
=∂f
∂r~er +
1
r
∂f
∂φ~eφ +
1
r sin(φ)
∂f
∂θ~eθ (15)
¯̄grad (~u) = ¯̄grad
ur
uφ
uθ
=
∂ur∂r
1
r
∂ur∂φ−uφr
1
r sin(φ)
∂ur∂θ− uθ
r
∂uφ∂r
1
r
∂uφ∂φ
+urr
1
r sin(φ)
∂uφ∂θ− uθ
rcot(φ)
∂uθ∂r
1
r
∂uθ∂φ
1
r sin(φ)
∂uθ∂θ
+urr
+uφr
cot(φ)
⇔ ui,j (16)
cot(x) = 1/ tan(x)
div (~u) = Tr(
¯̄grad (~u))
(17)
~rot (~u) =
(∂
∂φ(uθ r sin(φ))− ∂
∂θ(r uφ)
)1
r2 sin(φ)(∂ur∂θ− ∂
∂r(r uθ sin(φ))
)1
r sin(φ)(∂
∂r(r uφ)− ∂ur
∂φ
)1
r
(18)
~div(
¯̄U)
= ~div
Urr Urφ Urθ
Uφr Uφφ Uφθ
Uθr Uθφ Uθθ
(19)
~div(
¯̄U)
=
∂Urr∂r
+1
r
∂Urφ∂φ
+1
r sin(φ)
∂Urθ∂θ
+1
r(2 Urr − Uφφ − Uθθ + Urφ cot(φ))
∂Urφ∂r
+1
r
∂Uφφ∂φ
+1
r sin(φ)
∂Uφθ∂θ
+1
r((Uφφ − Uθθ) cot(φ) + 3 Urφ)
∂Urθ∂r
+1
r
∂Uθφ∂φ
+1
r sin(φ)
∂Uθθ∂θ
+1
r(3 Urθ − 2 Uθφ cot(φ))
⇔ Uij,j (20)