fabian s. klinke

Differentiation in Polarkoordinaten

Um die Partielle Ableitung für Polarkoordinaten zu berechnen, betrachten wir zunächst

x=ρcos⁡ϕ,y=ρsin⁡ϕx=\rho \cos \phi, \quad y=\rho \sin \phi

Hieraus ergeben sich

∂x∂ρ=cos⁡ϕ,∂x∂ϕ=−ρsin⁡ϕ∂y∂ρ=sin⁡ϕ,∂y∂ϕ=ρcos⁡ϕ\begin{array}{ll} \frac{\partial x}{\partial \rho}=\cos \phi, & \frac{\partial x}{\partial \phi}=-\rho \sin \phi \\ \frac{\partial y}{\partial \rho}=\sin \phi, & \frac{\partial y}{\partial \phi}=\rho \cos \phi \end{array}

wodurch sich durch die Kettenregel ergibt, dass

∂f∂ρ=∂f∂x∂x∂ρ+∂f∂y∂y∂ρ=∂f∂xcos⁡ϕ+∂f∂ysin⁡ϕ∂f∂ϕ=∂f∂x∂x∂ϕ+∂f∂y∂y∂ϕ=−ρ∂f∂xsin⁡ϕ+ρ∂f∂ycos⁡ϕ\begin{aligned} & \frac{\partial f}{\partial \rho}=\frac{\partial f}{\partial x} \frac{\partial x}{\partial \rho}+\frac{\partial f}{\partial y} \frac{\partial y}{\partial \rho}=\frac{\partial f}{\partial x} \cos \phi+\frac{\partial f}{\partial y} \sin \phi \\ & \frac{\partial f}{\partial \phi}=\frac{\partial f}{\partial x} \frac{\partial x}{\partial \phi}+\frac{\partial f}{\partial y} \frac{\partial y}{\partial \phi}=-\rho \frac{\partial f}{\partial x} \sin \phi+\rho \frac{\partial f}{\partial y} \cos \phi \end{aligned}

oder veranschaulicht dargestellt durch Gradient einer Funktion

∂f∂ρ=gradf⋅(cos⁡ϕ,sin⁡ϕ)∂f∂ϕ=gradf⋅(−ρsin⁡ϕ,ρcos⁡ϕ).\begin{aligned} & \frac{\partial f}{\partial \rho}=\text{grad} f \cdot(\cos \phi, \sin \phi) \\ & \frac{\partial f}{\partial \phi}=\text{grad} f \cdot(-\rho \sin \phi, \rho \cos \phi). \end{aligned}

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