Die Flächenelemente bei verschiedenen Parametrisierungen

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Die Flächenelemente bei verschiedenen Parametrisierungen
Kartesische Koordinaten
Zylinderkoordinaten
Kugelkoordinaten∗)
z = z(x, y)
z = z(ρ, ϕ)
ρ = ρ(ϑ, ϕ)
Parametrisierung
Ortsvektor



ρ cos ϕ
~r =  ρ sin ϕ 
z(ρ, ϕ)


ρ cos ϑ cos ϕ
~r =  ρ cos ϑ sin ϕ 
ρ sin ϑ


1
~rx =  0 
zx
1. Tangential
vektor


cos ϕ
~rρ =  sin ϕ 
zρ

ρϑ cos ϑ cos ϕ − ρ sin ϑ cos ϕ
~rϑ =  ρϑ cos ϑ sin ϕ − ρ sin ϑ sin ϕ 
ρϑ sin ϑ + ρ cos ϑ
2. Tangential
vektor


0
~ry =  1 
zy

−ρ sin ϕ
~rϕ =  ρ cos ϕ 
zϕ

ρϕ cos ϑ cos ϕ − ρ cos ϑ sin ϕ
~rϕ =  ρϕ cos ϑ sin ϕ + ρ cos ϑ cos ϕ 
ρϕ sin ϑ
E
|~rx |2 = 1 + zx2
|~rρ |2 = 1 + zρ2
|~rϑ |2 = ρ2 + ρ2ϑ
F
(~rx , ~ry ) = zx zy
(~rρ , ~rϕ ) = zρ zϕ
(~rϑ , ~rϕ ) = ρϑ ρϕ
G
|~ry |2 = 1 + zx2
|~rϕ |2 = ρ2 + zϕ2
|~rϕ |2 = ρ2 cos2 ϑ + ρ2ϕ
√
EG − F 2
vektorielles
Flächenelement
dF~ = ~r_ × ~r_ d_d_
∗) Für


x

y
~r = 
z(x, y)
dF =
q
1 + zx2 + zy2 dxdy

dF =
dF~ = ~rx × ~ry dxdy =


−zx
 −zy  dxdy
1
konstantes ρ vereinfachen sich die Formeln wesentlich.
q

q
dF = ρ cos2 ϑ(1 + ρ2ϑ ) + ρ2ϕ dϑdϕ
ρ2 (1 + zρ2 ) + zϕ2 dρdϕ
dF~ = ~rρ × ~rϕ dρdϕ =


zϕ sin ϕ − ρzρ cos ϕ
 −zϕ cos ϕ − ρzρ sin ϕ  dρdϕ
ρ
dF~ = ~rϑ × ~rϕ dϑdϕ =


−ρ (ρϕ sin ϕ + ρϑ sin ϑ cos ϑ cos ϕ + ρ cos2 ϑ cos ϕ)
 r ρϑ sin ϑ cos ϑ sin ϕ + ρ cos2 ϑ sin ϕ − ρϕ cos ϕ  dϑdϕ
ρ cos ϑ (−ρϑ cos ϑ + ρ sin θ)
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