Departamento de Física Aplicada III Universidad de Sevilla

Resumen de fórmulas de coordenadas curvilı́neas ortogonales
=
r cos θ
z
ϕ
y
arctg
x
= z = r cos θ
Diferencial de longitud:
dr = h1 dq1 u1 + h2 dq2 u2 + h3 dq3 u3
dr = dxux + dy uy + dzuz
dr = dρ uρ + ρ dϕ uϕ + dzuz
dr = drur + r dθ uθ + r sen θ dϕ uϕ
∇·A=
1
h1 h2 h3
dSx = dy dz ux
2
dSρ = ρ dϕ dz uρ
h3 u3 ∂ ∂q3 h A 3
dSr = r2 sen θ dθ dϕ ur
dSz = dx dy uz
3
hx = 1 hy = 1
hρ = 1 hϕ = ρ
hr = 1 hθ = r
Diferencial de volumen:
dτ = h1 h2 h3 dq1 dq2 dq3
hz = 1
hz = 1
hϕ = r sen θ
dτ = dx dy dz
dτ = ρ dρ dϕ dz
dτ = r2 sen θ dr dθ dϕ
dSϕ = dρ dz uϕ
dSz = ρ dρ dϕ uz
dSθ = r sen θ dr dϕ uθ
dSϕ = r dr dθ uϕ
Gradiente:
ux = cos ϕuρ − sen ϕuϕ = sen θ cos ϕur + cos θ cos ϕuθ − sen ϕuϕ
uy = sen ϕuρ + cos ϕuϕ = sen θ sen ϕur + cos θ sen ϕuθ + cos ϕuϕ
uz =
uz
=
cos θur − sen θuθ
sen θ cos ϕux + sen θ sen ϕuy + cos θuz = sen θuρ + cos θuz = ur
cos θ cos ϕux + cos θ sen ϕuy − sen θuz = cos θuρ − sen θuz = uθ
− sen ϕux + cos ϕuy
=
uϕ
= uϕ
∇φ =
1 ∂φ
1 ∂φ
1 ∂φ
u1 +
u2 +
u3
h1 ∂q1
h2 ∂q2
h3 ∂q3
∇φ =
∇φ =
∂φ
∂φ
∂φ
ux +
uy +
uz
∂x
∂y
∂z
∂φ
1 ∂φ
∂φ
uρ +
uϕ +
uz
∂ρ
ρ ∂ϕ
∂z
∂φ
1 ∂φ
1 ∂φ
ur +
uθ +
uϕ
∂r
r ∂θ
r sen θ ∂ϕ
∂Ay
∂Az
∂Ax
∂Az
∂Ax
∂Ay
−
ux +
−
uy +
−
uz
∇×A=
∂y
∂z
∂z
∂x
∂x
∂y
∂Aϕ
∂Az
1 ∂(ρAϕ ) ∂Aρ
1 ∂Az
∂Aρ
−
uρ +
−
uϕ +
−
uz
∇×A =
ρ ∂ϕ
∂z
∂z
∂ρ
ρ
∂ρ
∂ϕ
∂(rAϕ )
1
1
1 ∂(rAθ ) ∂Ar
∂(sen θAϕ ) ∂Aθ
1 ∂Ar
−
ur +
−
uθ +
−
uϕ
∇×A=
r sen θ
∂θ
∂ϕ
r sen θ ∂ϕ
∂r
r
∂r
∂θ
∂(A1 h2 h3 ) ∂(h1 A2 h3 ) ∂(h1 h2 A3 )
+
+
∂q1
∂q2
∂q3
∇·A=
∇·A=
2
dSy = dx dz uy
Divergencia:
∇·A=
=ϕ
dS|q3 =cte = h1 h2 dq1 dq2 u3
Rotacional:
h1 u1 h2 u2
∂
1
∂
∇×A =
h1 h2 h3 ∂q1
∂q2
h A h A
1
ϕ
Diferencial de superficie coordenada:
Vectores unitarios:
1 ∂r
ui =
hi ∂qi
cos ϕux + sen ϕuy = uρ = sen θur + cos θuθ
− sen ϕux + cos ϕuy = uϕ =
uϕ
uz
= uz = cos θur − sen θuθ
1
=
Factores de
escala:
∂r hi = ∂qi III
z
y
=ϕ=
x
Vector de posición:
r = xux + y uy + zuz
r = ρ uρ + zuz
r = rur
rt
am
U e
n
VE ive nto
RS rs de
IÓ ida F
N d ísi
DE de ca
EX Se Ap
AM vil lic
EN la ada
z=
arctg
x2 + y 2 + z 2 = ρ2 + z 2 = r
ρ
x2 + y 2
= arctg
=θ
arctg
z
z
∇φ =
pa
y = ρ sen ϕ = r sen θ sen ϕ
x2 + y 2 = ρ = r sen θ
De
x = ρ cos ϕ = r sen θ cos ϕ
∂Ay
∂Az
∂Ax
+
+
∂x
∂y
∂z
∂Az
1 ∂(ρAρ ) 1 ∂Aϕ
+
+
ρ ∂ρ
ρ ∂ϕ
∂z
1 ∂(sen θAθ )
1 ∂Aϕ
1 ∂(r2 Ar )
+
+
r2 ∂r
r sen θ
∂θ
r sen θ ∂ϕ
∇2 φ =
1
h1 h2 h3
∂
∂q1
Laplaciano:
∂
∂
h2 h3 ∂φ
h1 h3 ∂φ
h1 h2 ∂φ
+
+
h1 ∂q1
∂q2
h2 ∂q2
∂q3
h3 ∂q3
∂ 2φ ∂ 2φ ∂ 2φ
+ 2 + 2
∂x2
∂y
∂z
∂φ
1 ∂ 2φ
1 ∂
∂ 2φ
ρ
+ 2
∇2 φ =
+
ρ ∂ρ
∂ρ
ρ ∂ϕ2
∂z 2
1
∂φ
1
1 ∂
∂φ
∂
∂ 2φ
r2
+ 2
sen θ
+ 2
∇2 φ = 2
2
r ∂r
∂r
r sen θ ∂θ
∂θ
r sen θ ∂ϕ2
∇2 φ =