# Formelsammlung Mathematik: Koordinatensysteme

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## Polarkoordinaten

Umrechnung von Polarkoordinaten in kartesische Koordinaten Umrechnung von kartesischen Koordinaten in Polarkoordinaten
${\displaystyle x=r\cos \varphi ,}$
${\displaystyle y=r\sin \varphi .}$
${\displaystyle r={\sqrt {x^{2}+y^{2}}},}$
${\displaystyle \varphi =s(y)\arccos \left({\frac {x}{r}}\right)}$

mit ${\displaystyle s(y):={\begin{cases}+1&{\text{wenn}}\;y\geq 0,\\-1&{\text{wenn}}\;y<0.\end{cases}}}$

Jacobi-Matrix:

${\displaystyle J={\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{bmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{bmatrix}}.}$

Metrischer Tensor:

${\displaystyle g={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}.}$

Jacobi-Determinante:

${\displaystyle \det J=r.}$

Orthogonalbasis:

${\displaystyle \mathbf {e} _{r}={\begin{bmatrix}\cos \varphi \\\sin \varphi \end{bmatrix}},\quad \mathbf {e} _{\varphi }={\begin{bmatrix}-r\sin \varphi \\r\cos \varphi \end{bmatrix}}.}$

Orthonormalbasis:

${\displaystyle {\hat {\mathbf {e} }}_{r}=\mathbf {e} _{r},\quad {\hat {\mathbf {e} }}_{\varphi }={\frac {1}{r}}\mathbf {e} _{\varphi }.}$

${\displaystyle \operatorname {grad} f={\frac {\partial f}{\partial r}}{\hat {\mathbf {e} }}_{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \varphi }}{\hat {\mathbf {e} }}_{\varphi }.}$

## Zylinderkoordinaten

Umrechnung von Zylinderkoordinaten in kartesische Koordinaten Umrechnung von kartesischen Koordinaten in Zylinderkoordinaten
${\displaystyle x=\rho \cos \varphi ,}$
${\displaystyle y=\rho \sin \varphi ,}$
${\displaystyle z=z.}$
${\displaystyle \rho ={\sqrt {x^{2}+y^{2}}},}$
${\displaystyle \varphi =s(y)\arccos \left({\frac {x}{\rho }}\right)}$

mit ${\displaystyle s(y):={\begin{cases}+1&{\text{wenn}}\;y\geq 0,\\-1&{\text{wenn}}\;y<0.\end{cases}}}$

Jacobi-Matrix:

${\displaystyle g={\frac {\partial (x,y,z)}{\partial (\rho ,\varphi ,z)}}={\begin{bmatrix}\cos \varphi &-\rho \sin \varphi &0\\\sin \varphi &\rho \cos \varphi &0\\0&0&1\end{bmatrix}}.}$

Metrischer Tensor:

${\displaystyle g={\begin{bmatrix}1&0&0\\0&\rho ^{2}&0\\0&0&1\end{bmatrix}}.}$

Jacobi-Determinante:

${\displaystyle \det J=\rho .}$

Orthogonalbasis:

${\displaystyle \mathbf {e} _{\rho }={\begin{bmatrix}\cos \varphi \\\sin \varphi \\0\end{bmatrix}},\quad \mathbf {e} _{\varphi }={\begin{bmatrix}-\rho \sin \varphi \\\rho \cos \varphi \\0\end{bmatrix}},\quad \mathbf {e} _{z}={\begin{bmatrix}0\\0\\1\end{bmatrix}}.}$

Orthonormalbasis:

${\displaystyle {\hat {\mathbf {e} }}_{\rho }=\mathbf {e} _{\rho },\quad {\hat {\mathbf {e} }}_{\varphi }={\frac {1}{\rho }}\mathbf {e} _{\varphi },\quad {\hat {\mathbf {e} }}_{z}=\mathbf {e} _{z}.}$

${\displaystyle \operatorname {grad} f={\frac {\partial f}{\partial \rho }}{\hat {\mathbf {e} }}_{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\hat {\mathbf {e} }}_{\varphi }+{\frac {\partial f}{\partial z}}{\hat {\mathbf {e} }}_{z}.}$

## Kugelkoordinaten

Umrechnung von Kugelkoordinaten in kartesische Koordinaten Umrechnung von kartesischen Koordinaten in Kugelkoordinaten
${\displaystyle x=r\sin \theta \cos \varphi ,}$
${\displaystyle y=r\sin \theta \sin \varphi ,}$
${\displaystyle z=r\cos \theta .}$

Über ${\displaystyle \theta =\pi /2-\beta }$ ergibt sich ${\displaystyle \sin \theta =\cos \beta }$ und ${\displaystyle \cos \theta =\sin \beta }$, und somit

${\displaystyle x=r\cos \beta \cos \varphi ,}$
${\displaystyle y=r\cos \beta \sin \varphi ,}$
${\displaystyle z=r\sin \beta .}$

Der Winkel ${\displaystyle \beta }$ geht wie die geografische Breite vom Äquator aus nach Norden oder Süden, der Polarwinkel ${\displaystyle \theta }$ kommt dagegen vom Nordpol herab. Der Azimutwinkel ${\displaystyle \varphi }$ fährt wie die geografische Länge auf dem Äquator entlang.

${\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}},}$
${\displaystyle \theta =\arccos \left({\frac {z}{r}}\right),}$
${\displaystyle \varphi =s(y)\arccos \left({\frac {x}{\sqrt {x^{2}+y^{2}}}}\right)}$

mit ${\displaystyle s(y):={\begin{cases}+1&{\text{wenn}}\;y\geq 0,\\-1&{\text{wenn}}\;y<0.\end{cases}}}$

Jacobi-Matrix:

${\displaystyle J={\frac {\partial (x,y,z)}{\partial (r,\varphi ,\theta )}}={\begin{bmatrix}\sin \theta \cos \varphi &-r\sin \theta \sin \varphi &r\cos \theta \cos \varphi \\\sin \theta \sin \varphi &r\sin \theta \cos \varphi &r\cos \theta \sin \varphi \\\cos \theta &0&-r\sin \theta \end{bmatrix}}.}$

Metrischer Tensor:

${\displaystyle g={\begin{bmatrix}1&0&0\\0&(r\sin \theta )^{2}&0\\0&0&r^{2}\end{bmatrix}}.}$

Jacobi-Determinante:

${\displaystyle \det J=r^{2}\sin \theta .}$

Orthogonalbasis:

${\displaystyle \mathbf {e} _{\rho }={\begin{bmatrix}\sin \theta \cos \varphi \\\sin \theta \sin \varphi \\\cos \theta \end{bmatrix}},\quad \mathbf {e} _{\varphi }={\begin{bmatrix}-r\sin \theta \sin \varphi \\r\sin \theta \cos \varphi \\0\end{bmatrix}},\quad \mathbf {e} _{\theta }={\begin{bmatrix}r\cos \theta \cos \varphi \\r\cos \theta \sin \varphi \\-r\sin \theta \end{bmatrix}}.}$

Orthonormalbasis:

${\displaystyle {\hat {\mathbf {e} }}_{r}=\mathbf {e} _{r},\quad {\hat {\mathbf {e} }}_{\varphi }={\frac {1}{r\sin \theta }}\mathbf {e} _{\varphi },\quad {\hat {\mathbf {e} }}_{\theta }={\frac {1}{r}}\mathbf {e} _{\theta }.}$

${\displaystyle \operatorname {grad} f={\frac {\partial f}{\partial r}}{\hat {\mathbf {e} }}_{r}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}{\hat {\mathbf {e} }}_{\varphi }+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {\mathbf {e} }}_{\theta }.}$