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Cylindrical Coordinates

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A point $ P=(x,y,z)$ can be represented by the following 3 values: the angle $ \varphi\in[0,2\pi)$ formed by the $ x$ -axis and the projection of $ \overline{OP}$ onto the $ xy$ -plane, the length $ \varrho$ of the projection of $ \overline{OP}$ onto the $ xy$ -plane, and the $ z$ -coordinate.

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

We have

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

and

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

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

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  automatically generated 3/28/2008