Formelsammlung Koordinatensysteme: Unterschied zwischen den Versionen
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[[Vektorprodukt|Kreuzprodukt]] | [[Vektorprodukt|Kreuzprodukt]] | ||
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− | <math>\vec{\mathbf{e}}_x\times\vec{\mathbf{e}}_y =\vec{\mathbf{e}}_z,\ | + | <math>\begin{align} |
− | \vec{\mathbf{e}}_y\times \vec{\mathbf{e}}_z =\vec{\mathbf{e}}_x,\ | + | \vec{\mathbf{e}}_x\times\vec{\mathbf{e}}_y &=\vec{\mathbf{e}}_z,\\ |
− | \vec{\mathbf{e}}_z \times\vec{\mathbf{e}}_x =\vec{\mathbf{e}}_y</math> | + | \vec{\mathbf{e}}_y\times\vec{\mathbf{e}}_z &=\vec{\mathbf{e}}_x,\\ |
− | |style="background-color:#dde6f3;text-align:center;"| | + | \vec{\mathbf{e}}_z\times\vec{\mathbf{e}}_x &=\vec{\mathbf{e}}_y |
− | <math>\vec{\mathbf{e}}_{\rho}\times\vec{\mathbf{e}}_{\varphi}=\vec{\mathbf{e}}_z,\ | + | \end{align}</math> |
− | \vec{\mathbf{e}}_{\varphi}\times \vec{\mathbf{e}}_z=\vec{\mathbf{e}}_{\rho},\ | + | |style="background-color:#dde6f3;text-align:center;"|\quad |
− | \vec{\mathbf{e}}_z \times\vec{\mathbf{e}}_{\rho}=\vec{\mathbf{e}}_{\varphi}</math> | + | <math>\begin{align} |
+ | \vec{\mathbf{e}}_{\rho}\times\vec{\mathbf{e}}_{\varphi} &=\vec{\mathbf{e}}_z,\\ | ||
+ | \vec{\mathbf{e}}_{\varphi}\times\vec{\mathbf{e}}_z &=\vec{\mathbf{e}}_{\rho},\\ | ||
+ | \vec{\mathbf{e}}_z \times\vec{\mathbf{e}}_{\rho} &=\vec{\mathbf{e}}_{\varphi} | ||
+ | \end{align}</math> | ||
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− | <math>\vec{\mathbf{e}}_r\times\vec{\mathbf{e}}_{\vartheta}=\vec{\mathbf{e}}_{\varphi},\ | + | <math>\begin{align} |
− | \vec{\mathbf{e}}_{\varphi} \times\vec{\mathbf{e}}_r=\vec{\mathbf{e}}_{\vartheta}</math> | + | \vec{\mathbf{e}}_r\times\vec{\mathbf{e}}_{\vartheta} &= \vec{\mathbf{e}}_{\varphi},\\ |
+ | \vec{\mathbf{e}}_{\vartheta}\times\vec{\mathbf{e}}_{\varphi} &= \vec{\mathbf{e}}_r,\\ | ||
+ | \vec{\mathbf{e}}_{\varphi}\times\vec{\mathbf{e}}_r &= \vec{\mathbf{e}}_{\vartheta} | ||
+ | \end{align}</math> | ||
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|style="background-color:#dde6f3"|Umrechnungen | |style="background-color:#dde6f3"|Umrechnungen | ||
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+ | <math>\begin{align} | ||
+ | \vec{\mathbf{e}}_x &= \vec{\mathbf{e}}_{\rho}\cos{\varphi}-\vec{\mathbf{e}}_{\varphi}\sin{\varphi}\\ | ||
+ | &= \vec{\mathbf{e}}_r\sin{\vartheta}\cos{\varphi}+\vec{\mathbf{e}}_{\vartheta} \cos{vartheta} \cos{\varphi}-\vec{\mathbf{e}}_{\varphi}\sin{\varphi}\\ | ||
+ | \vec{\mathbf{e}}_y &= \vec{\mathbf{e}}_{\rho}\sin{\varphi}+\vec{\mathbf{e}}_{\varphi}\cos{\varphi}\\ | ||
+ | &= \vec{\mathbf{e}}_r\sin{\vartheta}\sin{\varphi}+\vec{\mathbf{e}}_{\vartheta}\cos{vartheta}\sin{\varphi}+\vec{\mathbf{e}}_{\varphi}\cos{\varphi}\\ | ||
+ | |||
+ | \vec{\mathbf{e}}_z &= \vec{\mathbf{e}}_r\cos{\vartheta}-\vec{\mathbf{e}}_{\vartheta}\sin{\vartheta}\\ | ||
+ | \end{align}</math> | ||
+ | |||
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Version vom 28. August 2012, 10:44 Uhr
Einheitsvektoren |
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Zusammenhang mit den kartesischen Koordinaten |
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Umrechnungen |
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Ortsvektor | |||
Betrag des Ortsvektors | |||
vektorielles Wegelement | |||
Volumenelement | |||
vektorielles Flächenelement |