Jack plug as logo for the webpageFABIAN S. KLINKE

Zylinderkoordinaten

 Zylinderkoordinaten x=ρcos⁡ϕy=ρsin⁡ϕz=zρ=x2+y2∂ρ∂x=xρ∂ρ∂y=yρ∂ρ∂z=0∂ϕ∂x=−sin⁡ϕρ∂ϕ∂y=cos⁡ϕρ∂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 sind in der Ebene Polarkoordinaten die durch eine zz Komponente, also einer höhe ergänzt werden. Wir definieren das System also durch

x=ρcos⁡ϕρ=x2+y2y=ρsin⁡ϕz=z\begin{aligned} & x=\rho \cos \phi \quad \rho=\sqrt{x^2+y^2} \\ & y=\rho \sin \phi \\ & z=z \end{aligned}

Entsprechend sind die Ableitungen nach der Differentiation in Polarkoordinaten ergänzt durch zz

∂f∂ρ=∂f∂xcos⁡ϕ+∂f∂ysin⁡ϕ∂f∂ϕ=−ρ∂f∂xsin⁡ϕ+ρ∂f∂ycos⁡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}

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