Laplaceoperator in Kugelkoordinaten:

\[\vec{\nabla}^{2}F\left(\vec{r}\right)=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\,\frac{\partial F\left(\vec{r}\right)}{\partial r}\right)+\frac{1}{r^{2}\sin\vartheta}\frac{\partial}{\partial\vartheta}\left(\sin\vartheta\,\frac{\partial F\left(\vec{r}\right)}{\partial\vartheta}\right)+\frac{1}{r^{2}\sin^{2}\vartheta}\frac{\partial^{2}F\left(\vec{r}\right)}{\partial\phi^{2}} \]

Der Radialteil kann auch so geschrieben werden:

\[\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\,\frac{\partial F\left(\vec{r}\right)}{\partial r}\right)=\frac{\partial^{2}F\left(\vec{r}\right)}{\partial r^{2}}+\frac{2}{r}\frac{\partial F\left(\vec{r}\right)}{\partial r}=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}\Big(r^{2}F\left(\vec{r}\right)\Big) \]

Laplaceoperator in Zylinderkoordinaten:

\[\vec{\nabla}^{2}F\left(\vec{r}\right)=\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\,\frac{\partial F\left(\vec{r}\right)}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}F\left(\vec{r}\right)}{\partial\phi^{2}}+\frac{\partial^{2}F\left(\vec{r}\right)}{\partial z^{2}} \]

Herleitung:

Kugelkoordinaten:

\[\left(x,y,z\right)\rightarrow\left(r,\theta,\phi\right)\]

\[x=r\sin\theta\cos\phi \]


\[y=r\sin\theta\sin\phi \]


\[z=r\cos\theta \]


\[r=\sqrt{x^{2}+y^{2}+z^{2}} \]


\[\theta=\arccos\left(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right) \]


\[\phi=\arctan\left(\frac{y}{x}\right) \]


Ableitungen mit Kettenregel:

\[\frac{\partial}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial x}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial x}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial y}=\frac{\partial r}{\partial y}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial y}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial y}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial z}=\frac{\partial r}{\partial z}\frac{\partial}{\partial r}+\frac{\partial\theta}{\partial z}\frac{\partial}{\partial\theta}+\frac{\partial\phi}{\partial z}\frac{\partial}{\partial\phi} \]


Partielle Ableitungen:

\[\frac{\partial r}{\partial x}=\frac{x}{r}=\sin\theta\cos\phi \]


\[\frac{\partial\theta}{\partial x}=\frac{\cos\theta\cos\phi}{r} \frac{\partial\phi}{\partial x}=-\frac{\sin\phi}{r\sin\theta} \]


\[\frac{\partial r}{\partial y}=\frac{y}{r}=\sin\theta\sin\phi \]


\[\frac{\partial\theta}{\partial y}=\frac{\cos\theta\sin\phi}{r} \frac{\partial\phi}{\partial y}=\frac{\cos\phi}{r\sin\theta} \]


\[\frac{\partial r}{\partial z}=\frac{z}{r}=\cos\theta \]


\[\frac{\partial\theta}{\partial z}=-\frac{\sin\theta}{r} \]


\[\frac{\partial\phi}{\partial z}=0 \]


Führt zu Ableitungen:

\[\frac{\partial}{\partial x}=\sin\theta\cos\phi\frac{\partial}{\partial r}+\frac{\cos\theta\cos\phi}{r}\frac{\partial}{\partial\theta}-\frac{\sin\phi}{r\sin\theta}\frac{\partial}{\partial\phi} \]

\[\frac{\partial}{\partial y}=\sin\theta\sin\phi\frac{\partial}{\partial r}+\frac{\cos\theta\sin\phi}{r}\frac{\partial}{\partial\theta}+\frac{\cos\phi}{r\sin\theta}\frac{\partial}{\partial\phi} \]


\[\frac{\partial}{\partial z}=\cos\theta\frac{\partial}{\partial r}-\frac{\sin\theta}{r}\frac{\partial}{\partial\theta} \]


Laplace-Operator:

\[\vec{\nabla}^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial}{\partial r}\right)+\frac{1}{r^{2}\sin^{2}\theta}\left[\sin\theta\frac{\partial}{\partial}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{\partial^{2}}{\partial\phi^{2}}\right]\]

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Eine kurze Vorstellung der Delta-Funktion, ihrer Eigenschaften und Beispiele, wie man damit rechnet.

PDF-Version

 

Eigenschaften der Deltafunktion:

 

  • \[\int_{-\infty}^{\infty}f(x)\,\delta(x-a)\,\mathrm{d}x=\int_{-\infty}^{\infty}f(x)\,\delta(a-x)\,\mathrm{d}x=f(a)\]
  • \[\delta(g(x))=\sum_{i}\frac{1}{|\partial_{x}g(x_{i})|}\delta(x-x_{i})\]

Beispiele zum Rechnen:

  • \[\int_{1}^{7}(2x^{2}-18x+1)\,\delta(x-3)\,\mathrm{d}x=2\cdot3^{2}-18\cdot3+1=-35\]
  • \[\int_{1}^{7}(2x^{3}-18x^{2}+x)\,\delta(x-8)\,\mathrm{d}x=0\]
  • \[\int_{-4}^{4}(7x-1)\,\delta(bx)\,\mathrm{d}x=-\frac{1}{|b|}\]
  • \[\int_{a}^{\infty}\delta(x+b)\,\mathrm{d}x=\begin{cases}
    0 & \mathrm{f\ddot{u}r}\ -b<a\\
    \frac{1}{2} & \mathrm{f\ddot{u}r}\ -b=a\\
    1 & \mathrm{f\ddot{u}r}\ -b>a\end{cases}\]
  • \[\int_{-37}^{14}(x^{3}-3x+2)\,\delta(1-x)\,\mathrm{d}x=0\]
  • \[\int_{-\infty}^{\infty}f(x)\,\delta(g(x))\,\mathrm{d}x=\begin{cases}
    \sum_{i}\frac{1}{|\partial_{x}g(x_{i})|}\cdot f(x_{i}) & \mathrm{f\ddot{u}r}\, x_{i}\, Nullstellen\, von\, g\\
    0 & \mathrm{f\ddot{u}r}\ g\, hat\, keine\, Nullstellen\end{cases}\]

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In diesem Artikel wird beschrieben, wie der Epsilon-Tensor definiert ist und welche Eigenschaften er hat. Außerdem werden Identitäten für Vektorfelder angeschrieben und mithilfe des Epsilon Tensors bewiesen.

Der Epsilon-(Pseudo-)Tensor:

  • \[\varepsilon_{ijk}=\begin{cases}+1, & \mbox{falls }(i,j,k) \mbox{ eine gerade Permutation von } (1,2,3) \mbox{ ist,} \\-1, & \mbox{falls }(i,j,k) \mbox{ eine ungerade Permutation von } (1,2,3) \mbox{ ist,} \\0, & \mbox{wenn mindestens zwei Indizes gleich sind.}\end{cases} \]

Eigenschaften des Epsilon-Tensors (Zusammenhang mit Kronecker-Delta) ohne Beweis:

  • \[\varepsilon_{ikl}\varepsilon_{jml}=\delta_{ij}\delta_{km}-\delta_{im}\delta_{jk}\]
  • \[\varepsilon_{ikl}\varepsilon_{jkl}=2\delta_{ij}\]
  • \[\varepsilon_{ikl}\varepsilon_{ikl}=6\]

Identitäten für Vektorfelder:

  1. \[a\cdot(b\times c)=b\cdot(c\times a)=c\cdot(a\times b)\]
  2. \[a\times(b\times c)=b(a\cdot c)-c(a\cdot b)\]
  3. \[(a\times b)\cdot(c\times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)\]
  4. \[\nabla\times(a\times b)=(b\cdot\nabla)a-b(\nabla\cdot a)+a(\nabla\cdot b)-(a\cdot\nabla)b\]
    mit
    \[\nabla_{i}=\partial_{i}=\frac{\partial}{\partial x_{i}}\]

 

Beweis der Identitäten:

 

  1. \[a\cdot(b\times c)=a_{i}(e_{i}\varepsilon_{ijk}b_{j}c_{k})=a_{i}b_{j}c_{k}\varepsilon_{ijk}=a_{i}b_{j}c_{k}\varepsilon_{jki}=b_{j}(e_{j}\varepsilon_{jki}c_{k}a_{i})=\]

    \[b\cdot(c\times a)=a_{i}b_{j}c_{k}\varepsilon_{kij}=c_{k}(e_{k}\varepsilon_{kij}a_{i}b_{j})=c\cdot(a\times b)\]
  2. \[a\times(b\times c)=\varepsilon_{ijk}e_{i}a_{j}(b\times c)_{k}=\varepsilon_{ijk}e_{i}a_{j}(\varepsilon_{klm}b_{l}c_{m})=e_{i}a_{j}b_{l}c_{m}\varepsilon_{ijk}\varepsilon_{lmk}=\]

    \[e_{i}a_{j}b_{l}c_{m}(\delta_{il}\delta_{jm}-\delta_{im}\delta_{lj})=e_{i}a_{j}b_{l}c_{m}\delta_{il}\delta_{jm}-e_{i}a_{j}b_{l}c_{m}\delta_{im}\delta_{lj}=b(a\cdot c)-c(a\cdot b)\]
  3. \[(a\times b)\cdot(c\times d)=(\varepsilon_{ijk}a_{j}b_{k})_{l}(\varepsilon_{imn}c_{m}d_{n})=a_{j}b_{k}c_{m}d_{n}(\varepsilon_{jki}\varepsilon_{mni})=\]

    \[a_{j}b_{k}c_{m}d_{n}(\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{mk})=a_{j}b_{k}c_{m}d_{n}\delta_{jm}\delta_{kn}-a_{j}b_{k}c_{m}d_{n}\delta_{jn}\delta_{mk}\]

    \[=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)\]
  4. \[\nabla\times(a\times b)=e_{i}\varepsilon_{ijk}\partial_{j}(\varepsilon_{klm}a_{l}b_{m})=e_{i}\varepsilon_{ijk}\varepsilon_{lmk}b_{m}\partial_{j}a_{l}+e_{i}\varepsilon_{ijk}\varepsilon_{lmk}a_{l}\partial_{j}b_{m}\]

    \[=e_{i}(\delta_{il}\delta_{jm}-\delta_{im}\delta_{lj})b_{m}\partial_{j}a_{l}+e_{i}(\delta_{il}\delta_{jm}-\delta_{im}\delta_{lj})a_{l}\partial_{j}b_{m}\]

    \[=e_{l}\delta_{jm}b_{m}\partial_{j}a_{l}-e_{m}\delta_{lj}b_{m}\partial_{j}a_{l}+e_{l}\delta_{jm}a_{l}\partial_{j}b_{m}-e_{m}\delta_{lj}a_{l}\partial_{j}b_{m}\]

    \[=(b\cdot\nabla)a-b(\nabla\cdot a)+a(\nabla\cdot b)-(a\cdot\nabla)b\]