Review Note
Last Update: 01/12/2025 04:37 PM
Current Deck: Physikalische Rechenmethoden::Integration und Differentiation
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Front
Dreifach Integral:
Koord. Transformation
Kartesisch \(\rightarrow\) Zylinder, Kugel
1. \(f(x,y,z) \rightarrow\)
2. \(dV = dx dy dz \rightarrow \)
3. Grenzen?
Koord. Transformation
Kartesisch \(\rightarrow\) Zylinder, Kugel
1. \(f(x,y,z) \rightarrow\)
2. \(dV = dx dy dz \rightarrow \)
3. Grenzen?
Back
Zylinder
1. \(f(\rho \cdot \cos \varphi, \rho \cdot \sin \varphi, z)\)
2. \(dV' = \rho \cdot d \rho \cdot d \varphi \cdot dz\)
\(\int_{0}^{r} \: \int_{0}^{2\pi} \:\int_{0}^{h} \rho \, d \rho \, d \varphi \, dz\)
Kugel
1. \(f(r \cdot \sin \vartheta \cdot \cos \varphi,\:\:\, r \cdot \sin \vartheta \cdot \sin \varphi, \)
\(r \cdot \cos \vartheta)\)
2. \(dV' = r^2 \cdot \sin \vartheta \cdot dr \cdot d \vartheta \cdot d \varphi \)
\(\int_{0}^{2\pi} \: \int_{0}^{\pi} \: \int_{0}^{R} r^2 \sin \vartheta \:dr \,d \vartheta \, d \varphi\)
1. \(f(\rho \cdot \cos \varphi, \rho \cdot \sin \varphi, z)\)
2. \(dV' = \rho \cdot d \rho \cdot d \varphi \cdot dz\)
\(\int_{0}^{r} \: \int_{0}^{2\pi} \:\int_{0}^{h} \rho \, d \rho \, d \varphi \, dz\)
Kugel
1. \(f(r \cdot \sin \vartheta \cdot \cos \varphi,\:\:\, r \cdot \sin \vartheta \cdot \sin \varphi, \)
\(r \cdot \cos \vartheta)\)
2. \(dV' = r^2 \cdot \sin \vartheta \cdot dr \cdot d \vartheta \cdot d \varphi \)
\(\int_{0}^{2\pi} \: \int_{0}^{\pi} \: \int_{0}^{R} r^2 \sin \vartheta \:dr \,d \vartheta \, d \varphi\)
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