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Mathematics LibreTexts

7.1: Polar Coordinates

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    8341
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    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Polar and spherical coordinate systems

    \( \newcommand{\pdr}[1]{\dfrac{\partial}{\partial r}}\) \( \newcommand{\pdtheta}[1]{\dfrac{\partial}{\partial \theta}}\) \( \newcommand{\pdrho}[1]{\dfrac{\partial}{\partial \rho}}\) \( \newcommand{\pdphi}[1]{\dfrac{\partial}{\partial \phi}}\) \( \newcommand{\pdw}[1]{\dfrac{\partial}{\partial w}}\) \( \newcommand{\pdx}[1]{\dfrac{\partial}{\partial x}}\) \( \newcommand{\pdy}[1]{\dfrac{\partial}{\partial y}}\) \( \newcommand{\pdz}[1]{\dfrac{\partial}{\partial z}}\) \( \newcommand{\pdt}[1]{\dfrac{\partial}{\partial t}}\) \( \newcommand{\pdtt}[1]{\dfrac{\partial^2}{\partial t^2}}\) \( \newcommand{\pdyy}[1]{\dfrac{\partial^2}{\partial y^2}}\) \( \newcommand{\pdww}[1]{\dfrac{\partial^2}{\partial w^2}}\) \( \newcommand{\pdwz}[1]{\dfrac{\partial^2}{\partial w \partial z}}\) \( \newcommand{\half}[1]{\dfrac{1}{2}}\) \( \newcommand{\pdxx}[1]{\dfrac{\partial^2}{\partial x^2}}\)

    Polar coordinates

    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. [fig:polar].

     
    Polar coordinates

    Using the chain rule we find

    \[\begin{aligned} \pdx{~} &=& \pdx{\rho}\pdrho{~}+\pdx{\phi}\pdphi{~}\nonumber\\ &=& \frac{x}{\rho}\pdrho{~}-\frac{y}{\rho^2}\pdphi{~}\nonumber\\ &=& \cos\phi\pdrho{~}-\frac{\sin\phi}{\rho}\pdphi{~},\\ \pdy{~} &=& \pdy{\rho}\pdrho{~}+\pdy{\phi}\pdphi{~}\nonumber\\ &=& \frac{y}{\rho}\pdrho{~}+\frac{x}{\rho^2}\pdphi{~}\nonumber\\ &=& \sin\phi\pdrho{~}+\frac{\cos\phi}{\rho}\pdphi{~},\end{aligned}\] We can write \[\begin{aligned} {\vect{\nabla}} &=& {\unitvec{e}}_\rho \pdrho{~}+\unitvec{e}_\phi \frac{1}{\rho} \pdphi{~}\end{aligned}\]

    where the unit vectors

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

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

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

    A final useful relation is the integration over these coordinates.

     
    Integration in polar coordinates

    As indicated schematically in Fig. [fig:polar2], 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 \[\int_V f(x,y) dx dy = \int_V f(\rho\cos\phi,\rho\sin\phi) \rho d\rho d\phi\]

    spherical coordinates

    Spherical coordinates are defined as \[\begin{aligned} x = r \cos\phi\sin\theta,\; y= r \sin\phi\sin\theta,\; z =r \cos\theta,\\ r = \sqrt{x^2+y^2+z^2},\; \phi = \arctan(y/x), \; \theta=\arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right),\end{aligned}\] as indicated schematically in Fig. [fig:spherical].

    Spherical coordinates
    Spherical coordinates

    Using the chain rule we find

    \[\begin{aligned} \pdx{~} &=& \pdx{r}\pdr{~}+\pdx{\phi}\pdphi{~}+\pdx{\theta}\pdtheta{~}\nonumber\\ &=& \frac{x}{r} \pdr{~}-\frac{y}{x^2+y^2}\pdphi{~} +\frac{xz}{r^2\sqrt{x^2+y^2}}\pdtheta{~} \nonumber\\ &=& \sin\theta\cos\phi\pdr{~}-\frac{\sin\phi}{r\sin\theta} \pdphi{~} +\frac{\cos\phi\cos\theta}{r} \pdtheta{~},\\ \pdy{~} &=& \pdy{r}\pdr{~}+\pdy{\phi}\pdphi{~}+\pdy{\theta}\pdtheta{~}\nonumber\\ &=& \frac{y}{r} \pdr{~}+\frac{x}{x^2+y^2}\pdphi{~} +\frac{yz}{r^2\sqrt{x^2+y^2}}\pdtheta{~} \nonumber\\ &=& \sin\theta\sin\phi\pdr{~}+\frac{\cos\phi}{r\sin\theta} \pdphi{~} +\frac{\sin\phi\cos\theta}{r} \pdtheta{~},\\ \pdz{~} &=& \pdz{r}\pdr{~}+\pdz{\phi}\pdphi{~}+\pdz{\theta}\pdtheta{~}\nonumber\\ &=& \frac{z}{r} \pdr{~} -\frac{\sqrt{x^2+y^2}}{r^2}\pdtheta{~} \nonumber\\ &=& \sin\theta\sin\phi\pdr{~}-\frac{\sin\theta}{r} \pdtheta{~}.\\\end{aligned}\]

    once again we can write \(\vect{\nabla}\) in terms of these coordinates.

    \[\begin{aligned} {\vect{\nabla}} &=& \unitvec{e}_r \pdr{~}+\unitvec{e}_\phi \frac{1}{r\sin\theta}\pdphi{~} + \unitvec{e}_\theta \frac{1}{r}\pdtheta{~}\end{aligned}\] where the unit vectors \[\begin{aligned} \unitvec{e}_r &=& (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta), \nonumber\\ \unitvec{e}_\phi &=& (-\sin\phi,\cos\phi,0), \nonumber\\ \unitvec{e}_\theta &=& (\cos\phi\cos\theta,\sin\phi\cos\theta,-\sin\theta).\end{aligned}\]

    are an orthonormal set. We say that spherical coordinates are orthogonal.

    We can use this to evaluate \(\Delta={\vect{\nabla}}^2\),

    \[\Delta = \frac{1}{r^2}\pdr{~}\left(r^2 \pdr{~}\right) +\frac{1}{r^2} \frac{1}{\sin\theta} \pdtheta{~} \left( \sin\theta\pdtheta{~} \right) + \frac{1}{r^2}\pdphip{~}\]

    Integration in spherical coordinates
    Integration in spherical coordinates

    Finally, for integration over these variables we need to know the volume of the small cuboid contained between \(r\) and \(r+\delta r\), \(\theta\) and \(\theta + \delta\theta\) and \(\phi\) and \(\phi+\delta\phi\). The length of the sides due to each of these changes is \(\delta r\), \(r \delta \theta\) and \(r \sin \theta \delta \theta\), respectively. We thus conclude that

    \[\int_V f(x,y,z) dx dy dz = \int_V f(r,\theta,\phi) r^2\sin\theta dr d\theta d\phi.\]

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