Die Flächenelemente bei verschiedenen Parametrisierungen
Anuncio
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)
