Zylinderkoordinaten
Zylinderkoordinaten x = ρ cos ϕ y = ρ sin ϕ z = z ρ = x 2 + y 2 ∂ ρ ∂ x = x ρ ∂ ρ ∂ y = y ρ ∂ ρ ∂ z = 0 ∂ ϕ ∂ x = − sin ϕ ρ ∂ ϕ ∂ y = cos ϕ ρ ∂ ϕ ∂ z = 0 ∂ z ∂ x = 0 ∂ z ∂ y = 0 ∂ z ∂ z = 1 \begin{array}{|c|c|c|}
\hline \text { Zylinderkoordinaten } & \begin{array}{l}
x=\rho \cos \phi \\
y=\rho \sin \phi \\
z=z
\end{array} & \rho=\sqrt{x^2+y^2} \\
\hline \frac{\partial \rho}{\partial x}=\frac{x}{\rho} & \frac{\partial \rho}{\partial y}=\frac{y}{\rho} & \frac{\partial \rho}{\partial z}=0 \\
\hline \frac{\partial \phi}{\partial x}=\frac{-\sin \phi}{\rho} & \frac{\partial \phi}{\partial y}=\frac{\cos \phi}{\rho} & \frac{\partial \phi}{\partial z}=0 \\
\hline \frac{\partial z}{\partial x}=0 & \frac{\partial z}{\partial y}=0 & \frac{\partial z}{\partial z}=1 \\
\hline
\end{array} Zylinderkoordinaten ∂ x ∂ ρ = ρ x ∂ x ∂ ϕ = ρ − s i n ϕ ∂ x ∂ z = 0 x = ρ cos ϕ y = ρ sin ϕ z = z ∂ y ∂ ρ = ρ y ∂ y ∂ ϕ = ρ c o s ϕ ∂ y ∂ z = 0 ρ = x 2 + y 2 ∂ z ∂ ρ = 0 ∂ z ∂ ϕ = 0 ∂ z ∂ z = 1 Zylinderkoordinaten sind in der Ebene Polarkoordinaten die durch eine z z z
x = ρ cos ϕ ρ = x 2 + y 2 y = ρ sin ϕ z = z \begin{aligned}
& x=\rho \cos \phi \quad \rho=\sqrt{x^2+y^2} \\
& y=\rho \sin \phi \\
& z=z
\end{aligned} x = ρ cos ϕ ρ = x 2 + y 2 y = ρ sin ϕ z = z Entsprechend sind die Ableitungen nach der
Differentiation in Polarkoordinaten ergänzt durch z z z
∂ f ∂ ρ = ∂ f ∂ x cos ϕ + ∂ f ∂ y sin ϕ ∂ f ∂ ϕ = − ρ ∂ f ∂ x sin ϕ + ρ ∂ f ∂ y cos ϕ ∂ f ∂ z = ∂ f ∂ z . \begin{aligned}
& \frac{\partial f}{\partial \rho}=\frac{\partial f}{\partial x} \cos \phi+\frac{\partial f}{\partial y} \sin \phi \\
& \frac{\partial f}{\partial \phi}=-\rho \frac{\partial f}{\partial x} \sin \phi+\rho \frac{\partial f}{\partial y} \cos \phi \\
& \frac{\partial f}{\partial z}=\frac{\partial f}{\partial z} .
\end{aligned} ∂ ρ ∂ f = ∂ x ∂ f cos ϕ + ∂ y ∂ f sin ϕ ∂ ϕ ∂ f = − ρ ∂ x ∂ f sin ϕ + ρ ∂ y ∂ f cos ϕ ∂ z ∂ f = ∂ z ∂ f .