Formelsammlung Koordinatensysteme: Unterschied zwischen den Versionen
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[[Kartesische Koordinaten]] | [[Kartesische Koordinaten]] | ||
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[[Zylinderkoordinaten]] | [[Zylinderkoordinaten]] | ||
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[[Kugelkoordinaten]] | [[Kugelkoordinaten]] | ||
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+ | |||
+ | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | [[Datei:Kartesische Koordinaten.png|200px|miniatur|center]] | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | [[Datei:Zylinderkoordinaten.png|300px|miniatur|center]] | ||
+ | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | [[Datei:Kugelkoordinaten.png|180px|miniatur|center]] | ||
+ | |- | ||
+ | |style="background-color:#dde6f3;"| | ||
+ | Wertebereiche der Koordinaten | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>P=P(x,y,z)</math><br> | |
− | |style="background-color:#dde6f3"| | + | <math>-\infty\leq x\leq\infty</math><br> |
− | + | <math>-\infty\leq y\leq\infty</math><br> | |
+ | <math>-\infty\leq z\leq\infty</math> | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>P=P(\rho,\varphi,z)</math><br> | ||
+ | <math>0 \leq \rho \leq\infty</math><br> | ||
+ | <math>0 \leq \varphi < 2\pi</math><br> | ||
+ | <math>-\infty \leq z \leq\infty</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>P=P(r,\vartheta,\varphi)</math><br> | ||
+ | <math>0\leq r\leq\infty</math><br> | ||
+ | <math>0\leq \vartheta\leq\pi</math><br> | ||
+ | <math>0\leq \varphi < 2\pi</math> | ||
|- | |- | ||
− | |style="background-color:#dde6f3"|[[ | + | |style="background-color:#dde6f3;"| |
− | + | [[Einheitsvektoren]] | |
+ | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>\vec{\mathbf{e}}_x,\vec{\mathbf{e}}_y,\vec{\mathbf{e}}_z</math> | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\vec{\mathbf{e}}_{\rho},\vec{\mathbf{e}}_{\varphi},\vec{\mathbf{e}}_z</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>\vec{\mathbf{e}}_r, \vec{\mathbf{e}}_{\vartheta},\vec{\mathbf{e}}_{\varphi}</math> | |
+ | |- | ||
|style="background-color:#dde6f3"| | |style="background-color:#dde6f3"| | ||
− | + | [[Vektorprodukt|Kreuzprodukt]] | |
+ | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>\begin{align} | ||
+ | \vec{\mathbf{e}}_x\times\vec{\mathbf{e}}_y &=\vec{\mathbf{e}}_z\\ | ||
+ | \vec{\mathbf{e}}_y\times\vec{\mathbf{e}}_z &=\vec{\mathbf{e}}_x\\ | ||
+ | \vec{\mathbf{e}}_z\times\vec{\mathbf{e}}_x &=\vec{\mathbf{e}}_y | ||
+ | \end{align}</math> | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <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> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>\begin{align} | |
+ | \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> | ||
|- | |- | ||
− | |style="background-color:#dde6f3"|Zusammenhang | + | |style="background-color:#dde6f3"| |
− | + | Zusammenhang zu kartesischen Koordinaten | |
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | |style="background-color:#dde6f3;text-align:center;"| |
− | + | <math>\begin{align} | |
+ | x &=\rho\cos{\varphi} && && 0\leq\rho\leq\infty\\ | ||
+ | y &=\rho\sin{\varphi} &&\text{mit}&& 0\leq\varphi\leq2\pi\\ | ||
+ | z &=z | ||
+ | \end{align} </math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>\begin{align}x &=r\sin{\vartheta}\cos{\varphi} && && 0\leq r \leq\infty\\ | |
+ | y &=r\sin{\vartheta}\sin{\varphi} &&\text{mit}&& 0\leq\vartheta\leq \pi\\ | ||
+ | z &=r\cos{\vartheta} && && 0\leq\varphi\leq 2\pi | ||
+ | \end{align} </math> | ||
|- | |- | ||
|style="background-color:#dde6f3"|Umrechnungen | |style="background-color:#dde6f3"|Umrechnungen | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <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> | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\begin{align}\vec{\mathbf{e}}_{\rho} &= \vec{\mathbf{e}}_x\cos{\varphi}+\vec{\mathbf{e}}_y\sin{\varphi}\\ | ||
+ | \\ | ||
+ | \vec{\mathbf{e}}_{\varphi} &= -\vec{\mathbf{e}}_x\sin{\varphi}+\vec{\mathbf{e}}_y\cos{\varphi}\\ | ||
+ | \\ | ||
+ | \vec{\mathbf{e}}_z &= \vec{\mathbf{e}}_z | ||
+ | \end{align}</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>\begin{align}\vec{\mathbf{e}}_r &= \vec{\mathbf{e}}_x\sin{\vartheta}\cos{\varphi}+\vec{\mathbf{e}}_y\sin{\vartheta} \sin{\varphi}+\vec{\mathbf{e}}_z\cos{\vartheta}\\ | |
− | + | \\ | |
+ | \vec{\mathbf{e}}_{\vartheta} &= \vec{\mathbf{e}}_x\cos{\vartheta}\cos{\varphi}+\vec{\mathbf{e}}_y\cos{\vartheta}\sin{\varphi}-\vec{\mathbf{e}}_z\sin{\vartheta}\\ | ||
+ | \\ | ||
+ | \vec{\mathbf{e}}_{\varphi} &= -\vec{\mathbf{e}}_x\sin{\varphi}+\vec{\mathbf{e}}_y\cos{\varphi} | ||
+ | \end{align}</math> | ||
|- | |- | ||
− | |style="background-color:#dde6f3"|Ortsvektor | + | |style="background-color:#dde6f3"|[[Ortsvektor]] |
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <math>\vec{\mathbf{r}}=\vec{\mathbf{e}}_x x+\vec{\mathbf{e}}_y y+\vec{\mathbf{e}}_z z</math> |
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\vec{\mathbf{r}}=\vec{\mathbf{e}}_{\rho}\rho+\vec{\mathbf{e}}_z z</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>\vec{\mathbf{r}}=\vec{\mathbf{e}}_r r</math> | ||
|- | |- | ||
|style="background-color:#dde6f3"|Betrag des Ortsvektors | |style="background-color:#dde6f3"|Betrag des Ortsvektors | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <math>r=\sqrt{x^2+y^2+z^2}</math> |
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>r=\sqrt{\rho^2+z^2}</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>r=\sqrt{r^2}</math> | |
|- | |- | ||
|style="background-color:#dde6f3"| [[Wegelemente|vektorielles Wegelement]] | |style="background-color:#dde6f3"| [[Wegelemente|vektorielles Wegelement]] | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <math>\mathrm{d}\vec{\mathbf{r}}=\vec{\mathbf{e}}_x\mathrm{d}x+\vec{\mathbf{e}}_y\mathrm{d}y+\vec{\mathbf{e}}_z\mathrm{d}z</math> |
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\mathrm{d}\vec{\mathbf{r}}=\vec{\mathbf{e}}_{\rho}\mathrm{d}\rho+\vec{\mathbf{e}}_{\varphi}\rho\mathrm{d}\varphi+\vec{\mathbf{e}}_z\mathrm{d}z</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | + | <math>\mathrm{d}\vec{\mathbf{r}}=\vec{\mathbf{e}}_r\mathrm{d}r+\vec{\mathbf{e}}_{\vartheta}r\mathrm{d}\vartheta+\vec{\mathbf{e}}_{\varphi}r\sin\vartheta\mathrm{d}\varphi</math> | |
|- | |- | ||
|style="background-color:#dde6f3"| [[Volumenelemente| Volumenelement]] | |style="background-color:#dde6f3"| [[Volumenelemente| Volumenelement]] | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <math>\mathrm{d}V=\mathrm{d}x\mathrm{d}y\mathrm{d}z</math> |
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\mathrm{d}V=\mathrm{d}\rho\cdot\rho\mathrm{d}\varphi\cdot\mathrm{d}z=\rho\mathrm{d}\rho\mathrm{d}\varphi\mathrm{d}z</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>\mathrm{d}V=\mathrm{d}r\cdot r\mathrm{d}\vartheta\cdot r\sin\vartheta\mathrm{d}\varphi=r^2\sin\vartheta\mathrm{d}r\mathrm{d}\vartheta\mathrm{d}\varphi</math> | ||
|- | |- | ||
|style="background-color:#dde6f3"|[[Flächenelemente|vektorielles Flächenelement]] | |style="background-color:#dde6f3"|[[Flächenelemente|vektorielles Flächenelement]] | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
− | |style="background-color:#dde6f3"| | + | <math>\begin{align} |
+ | \mathrm{d}\vec{\mathbf{A}} &= \vec{\mathbf{e}}_z\mathrm{d}x\mathrm{d}y \text{ (x-y-Ebene)}\\ | ||
+ | \mathrm{d}\vec{\mathbf{A}} &= \vec{\mathbf{e}}_y\mathrm{d}x\mathrm{d}z \text{ (x-z-Ebene)}\\ | ||
+ | \mathrm{d}\vec{\mathbf{A}} &= \vec{\mathbf{e}}_x\mathrm{d}y\mathrm{d}z \text{ (y-z-Ebene)} | ||
+ | \end{align}</math> | ||
+ | |style="background-color:#dde6f3;text-align:center;"| | ||
+ | <math>\begin{align} | ||
+ | \mathrm{d}\vec{\mathbf{A}} &= \vec{\mathbf{e}}_z\rho\mathrm{d}\rho\mathrm{d}\varphi \text{ (Deckel)}\\ | ||
+ | \mathrm{d}\vec{\mathbf{A}} &= \vec{\mathbf{e}}_\rho\rho\mathrm{d}\varphi\mathrm{d}z \text{ (Mantel)} | ||
+ | \end{align}</math> | ||
|style="background-color:#c9d7ec;text-align:center;"| | |style="background-color:#c9d7ec;text-align:center;"| | ||
+ | <math>\mathrm{d}\vec{\mathbf{A}}=\vec{\mathbf{e}}_r r^2\sin\vartheta\mathrm{d}\vartheta\mathrm{d}\varphi</math> | ||
|} | |} | ||
+ | |||
+ | <noinclude>==Literatur== | ||
+ | * Manfred Albach, ''Grundlagen der Elektrotechnik 1: Erfahrungssätze, Bauelemente, Gleichstromschaltungen'', 3. Auflage (Pearson Studium, 2011) | ||
+ | </noinclude> | ||
+ | |||
+ | [[Kategorie:Artikel]] | ||
+ | [[Kategorie:Feedback]] |
Aktuelle Version vom 25. Oktober 2023, 16:34 Uhr
Wertebereiche der Koordinaten |
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Zusammenhang zu kartesischen Koordinaten |
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Umrechnungen |
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Ortsvektor |
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Betrag des Ortsvektors |
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vektorielles Wegelement |
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Volumenelement |
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vektorielles Flächenelement |
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Literatur
- Manfred Albach, Grundlagen der Elektrotechnik 1: Erfahrungssätze, Bauelemente, Gleichstromschaltungen, 3. Auflage (Pearson Studium, 2011)