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Mathematics LibreTexts

7.1: Polar Coordinates

  • Page ID
    8341
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    Polar coordinates in two dimensions are defined by \[\begin{aligned} x = \rho\cos\phi, y= \rho\sin\phi,\\ \rho = \sqrt{x^2+y^2}, \phi = \arctan(y/x),\end{aligned}\] as indicated schematically in Fig. \(\PageIndex{1}\).

    polar.png
    Figure \(\PageIndex{1}\): Polar coordinates

    Using the chain rule we find

    \[\begin{aligned} \frac{\partial}{\partial x}{~} &= \frac{\partial \rho}{\partial x}\frac{\partial}{\partial \rho}{~}+\frac{\partial \phi}{\partial x}\frac{\partial}{\partial \phi}{~}\nonumber\\ &= \frac{x}{\rho}\frac{\partial}{\partial \rho}{~}-\frac{y}{\rho^2}\frac{\partial}{\partial \phi}{~}\nonumber\\ &= \cos\phi\frac{\partial}{\partial \rho}{~}-\frac{\sin\phi}{\rho}\frac{\partial}{\partial \phi}{~},\\ \frac{\partial}{\partial y}{~} &= \frac{\partial \rho}{\partial y}\frac{\partial}{\partial \rho}{~}+\frac{\partial \phi}{\partial y}\frac{\partial}{\partial \phi}{~}\nonumber\\ &= \frac{y}{\rho}\frac{\partial}{\partial \rho} {~}+\frac{x}{\rho^2}\frac{\partial}{\partial \phi}{~}\nonumber\\ &= \sin\phi\frac{\partial}{\partial \rho}{~}+\frac{\cos\phi}{\rho}\frac{\partial}{\partial \phi}{~},\end{aligned}\] We can write \[\begin{aligned} {\nabla} &= {\hat{e}}_\rho \frac{\partial}{\partial \rho}{~}+\hat{e}_\phi \frac{1}{\rho} \frac{\partial}{\partial \phi}{~}\end{aligned}\]

    where the unit vectors

    \[\begin{aligned} \hat{e}_\rho &= (\cos\phi,\sin\phi), \nonumber\\ \hat{e}_\phi &= (-\sin\phi,\cos\phi), \end{aligned}\] are an orthonormal set. We say that circular coordinates are orthogonal.

    We can now use this to evaluate \(\nabla^2\),

    \[\begin{aligned} \nabla^2 &= \cos^2\phi\frac{\partial^2}{\partial \rho^2}{~} + \frac{\sin\phi\cos\phi}{\rho^2}\frac{\partial}{\partial \phi} {~} +\frac{\sin^2\phi}{\rho}\frac{\partial}{\partial \rho} {~} + \frac{\sin^2\phi}{\rho^2}\frac{\partial^2}{\partial \phi^2} {~} + \frac{\sin\phi\cos\phi}{\rho^2} \frac{\partial}{\partial \phi} {~}\nonumber\\ &\space\space\space+\sin^2\phi\frac{\partial^2}{\partial \rho^2} {~}-\frac{\sin\phi\cos\phi}{\rho^2}\frac{\partial}{\partial \phi} {~} +\frac{\cos^2\phi}{\rho}\frac{\partial}{\partial \rho} {~} + \frac{\cos^2\phi}{\rho^2}\frac{\partial^2}{\partial \phi^2} {~} - \frac{\sin\phi\cos\phi}{\rho^2} \frac{\partial}{\partial \phi}{~}\\ &=\frac{\partial^2}{\partial \rho^2} {~} +\frac{1}{\rho}\frac{\partial}{\partial \rho}{~} + \frac{1}{\rho^2}\frac{\partial^2}{\partial \phi^2} {~}\nonumber\\ &=\frac{1}{\rho}\frac{\partial}{\partial \rho} {~} \left(\rho\frac{\partial}{\partial \rho} {~}\right) +\frac{1}{\rho^2}\frac{\partial^2}{\partial \phi^2} {~}.\nonumber\\\end{aligned}\]

    A final useful relation is the integration over these coordinates.

    imageedit_2_5805175308.png
    Figure \(\PageIndex{2}\): Integration in polar coordinates

    As indicated schematically in Fig. \(\PageIndex{2}\), the surface related to a change \(\rho \rightarrow \rho + \delta \rho\), \(\phi \rightarrow \phi+\delta\phi\) is \(\rho \delta \rho \delta\phi\). This leads us to the conclusion that an integral over \(x,y\) can be rewritten as \[\int_V f(x,y) dx dy = \int_V f(\rho\cos\phi,\rho\sin\phi) \rho \,d\rho\, d\phi\]