“doc” — 2002/9/17 — 15:42 — page 247 — #245 i
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COORDONNÉES CARTÉSIENNES
COORDONNÉES CYLINDRIQUES
COORDONNÉES SPHÉRIQUES z
y x
O M
uy uz
ux
j k
i
z
z x
O
uθ uz
uρ
θ M ρ
z
x O
ur
uθ uϕ
ϕ θ
M
uϕ
x = ρ cosθ = rsinθ cosϕ
y = ρ sinθ = rsinθ sinϕ
z = z = rcosθ
0 ≤ θ ≤ 2π 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π
= icosθ+ jsinθ ux
y
z k
ρ
θ =−isinθ+ jcosθ z=k
r= isinθcosϕ + jsinθ + jcosθ
sinϕ +ksinθ q = icosθcosϕ sinϕ − sinθ ϕ =−isinϕ
u = j
+ jcosϕ
k
= i
= u
u u u
u u u
d O M= dl dx dy dz
d O M= dl dρ ρdθ dz
dO M= dl dr rdθ rsin q dϕ dl2= dx2+ dy2+ dz2 dl2= dρ2+ ρ 2dθ 2+ dz2 dl2= dr2+ r2dθ2+ r2sin2θ dϕ 2
N Q Q
P P
M M
dρ P? P'
N? N'
M? M?
Q? Q'
dS2= dρdz
dS3 = ρdθdρ dS
1= ρdθdz dz
dθ
dθ
dV = ρdθdρdz dV = r2sinθdθdrdϕ
dV = dxdydz
dS = dxdy = dxdz = dydz N
N Q
M dS
3= rdθdr
P
M?
N? P?
Q?
dS 1= r2sinθdθdϕ dS2= rsinθdϕdθ
dϕ
∇? =
∂ ?
∂x +
∂ ?
∂y +
∂ ?
∂z k
∂ ?
∂ρ +
1 ρ
∂ ?
∂θ +
∂ ?
∂z
∂ ?
∂r + 1 r
∂ ?
∂θ + 1 rsinθ
∂ ?
∇? = ∇? = ∂ ϕ
i u
ρ u
θ uz ur u q uϕ
∂x ∂y ∂z
∇.A · =
∂Ax
∂ +
∂Ay +
∂Az 1
ρ
∂(ρ Aρ )
∂ρ + 1 ρ
∂Aθ
∂θ +
∂Az
∂z
1 r2
∂ (r2Ar)
∂r +
1 rsinθ
∂(A θ sinθ)
∂θ +
∂Aϕ
∂ ϕ
∂
∂x
∂
∂y
∂
∂z Ax Ay Az
1 ρ
ρ
∂
∂ρ
∂
∂θ
∂
∂z Aρ ρ Aθ Az
1 r2sinθ
r rsinθ
∂
∂ r
∂
∂θ
∂
∂ ϕ Ar r Aθ rsinθ Aϕ
∇.A = ∇.A =
∇∧A = ∇∧ A = ∇∧ A =
k
i uρ uθ uθ u
uz ur ϕ
??=
∂ 2
∂x2+
∂ 2
∂y2
+
∂ 2
∂z2
?
??= 1 ρ
∂
∂ ρ
∂
∂
+ 1 ρ 2
∂ 2
∂ θ 2+
∂2
∂z2
?
??= 1 r2
∂
∂ r r2
∂
∂ r + 1 r2sinθ
∂
∂ θ sinθ
∂
∂ θ
+ 1 r2sin2θ ∂ ϕ 2
∂ 2
? r
Fig. A.4.
A. ÉLÉMENTS D’ANALYSE VECTORIELLE
