Kugelkoordinaten
Kugelkoordinaten x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ r = x 2 + y 2 + z ∂ r ∂ x = x r ∂ r ∂ y = y r ∂ r ∂ z = z r ∂ θ ∂ x = cos θ cos ϕ r ∂ θ ∂ y = cos θ sin ϕ r ∂ θ ∂ z = − sin θ r ∂ ϕ ∂ x = − sin ϕ r sin θ ∂ ϕ ∂ y = cos ϕ r sin θ ∂ ϕ ∂ z = 0 \begin{array}{|c|c|c|}
\hline \text { Kugelkoordinaten } & \begin{array}{l}
x=r \sin \theta \cos \phi \\
y=r \sin \theta \sin \phi \\
z=r \cos \theta
\end{array} & r=\sqrt{x^2+y^2+z} \\
\hline \frac{\partial r}{\partial x}=\frac{x}{r} & \frac{\partial r}{\partial y}=\frac{y}{r} & \frac{\partial r}{\partial z}=\frac{z}{r} \\
\hline \frac{\partial \theta}{\partial x}=\frac{\cos \theta \cos \phi}{r} & \frac{\partial \theta}{\partial y}=\frac{\cos \theta \sin \phi}{r} & \frac{\partial \theta}{\partial z}=\frac{-\sin \theta}{r} \\
\hline \frac{\partial \phi}{\partial x}=\frac{-\sin \phi}{r \sin \theta} & \frac{\partial \phi}{\partial y}=\frac{\cos \phi}{r \sin \theta} & \frac{\partial \phi}{\partial z}=0 \\
\hline
\end{array} Kugelkoordinaten ∂ x ∂ r = r x ∂ x ∂ θ = r c o s θ c o s ϕ ∂ x ∂ ϕ = r s i n θ − s i n ϕ x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ ∂ y ∂ r = r y ∂ y ∂ θ = r c o s θ s i n ϕ ∂ y ∂ ϕ = r s i n θ c o s ϕ r = x 2 + y 2 + z ∂ z ∂ r = r z ∂ z ∂ θ = r − s i n θ ∂ z ∂ ϕ = 0 Kugelkoordinaten funktionieren auf der Eben wie Polarkoordinaten , die Höhe beschreiben wir aber
zusätzlich auch durch einen Winkel θ \theta θ
x = r sin θ cos ϕ r = x 2 + y 2 + z 2 y = r sin θ sin ϕ z = r cos θ . \begin{aligned}
& x=r \sin \theta \cos \phi \quad r=\sqrt{x^2+y^2+z^2} \\
& y=r \sin \theta \sin \phi \\
& z=r \cos \theta .
\end{aligned} x = r sin θ cos ϕ r = x 2 + y 2 + z 2 y = r sin θ sin ϕ z = r cos θ .