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Mathematics-Online course: Vector Calculus - Coordinates

Spherical Coordinates

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A point $ P=(x,y,z)$ can be represented by the following 3 values: the point's distance from the origin $ r=\vert\overline{OP}\vert$ , the angle $ \varphi\in[0,2\pi)$ formed by the $ x$ -axis and the projection of $ \overline{OP}$ onto the $ xy$ -plane, and the angle $ \vartheta\in[0,\pi]$ formed by $ \overline{OP}$ and the $ z$ -axis.

\includegraphics[width=.6\linewidth]{kugelkoordinaten_en}

We have

$\displaystyle x = r\cos(\varphi)\sin(\vartheta),\quad y = r\sin(\varphi)\sin(\vartheta),\quad z = r\cos(\vartheta) $

and

$\displaystyle r = \sqrt{x^2+y^2+z^2},\quad \varphi = \arctan(y/x),\quad \vartheta = \arctan(\sqrt{x^2+y^2}/z)\,, $

but you have got to select the appropriate branch of $ \arctan$ depending on the signs of $ x$ and $ y$ :

\begin{displaymath} \begin{array}{\vert l\vert l\vert} \hline x\ge 0 \wedge y... ... \le 0 & \vartheta \in [\pi/2,\pi] \\ \hline \end{array} \end{displaymath}

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  automatically generated 10/30/2007