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| skalární součin vektorů: [{Math fontsize='14' |
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| \nabla\times\vec{v}=\left(\sum\limits_j\vec{\delta}_j\frac{\partial}{\partial |
| \begin{array}{l} |
| {\nabla\times\vec{v}=\left(\sum\limits_j\vec{\delta}_j\frac{\partial}{\partial |
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| x_j}v_k=\nonumber\\ |
| =\sum\limits_i\sum\limits_j\sum\limits_k\varepsilon_{ijk}\vec{\delta}_i\frac{\partial |
| x_j}v_k=\nonumber}\\ \\ |
| {=\sum\limits_i\sum\limits_j\sum\limits_k\varepsilon_{ijk}\vec{\delta}_i\frac{\partial |
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| =rot\,\vec{v} |
| =rot\,\vec{v}} |
| \end{array} |
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|
| Laplaceův operátor nabla%%sup 2%% působící na: |
|
| * skalární pole: |
|
| [{LTMath fontsize='12' |
|
| \nabla^2s=\nabla.(\nabla |
| s)=\left(\sum\limits_i\vec{\delta}_i\frac{\partial s}{\partial x_i}\right). |
| \left(\sum\limits_j\vec{\delta}_j\frac{\partial s}{\partial x_j}\right)= |
| \sum\limits_i\sum\limits_j\vec{\delta}_i\vec{\delta}_j\frac{\partial^2s}{\partial |
| x_j\partial x_i}= \sum\limits_i\sum\limits_j\delta_{ij}\frac{\partial^2s}{\partial |
| x_j\partial x_i}=\sum\limits_i\frac{\partial^2s} {\partial x_i^2} |
| }] |
|
| * vektorové pole: |
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| [{LTMath fontsize='12' |
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| \nabla^2\vec{v}=\nabla^2\sum\limits_i\vec{\delta}_iv_i=\sum\limits_i\vec{\delta}_i\nabla^2v_i }] |
|
| což neplatí v křivočarých souřadnicích; zde užijeme dentitu: |
|
| [{LTMath fontsize='12' |
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| \nabla^2\vec{v}=\nabla(\nabla.\vec{v})-\nabla\times(\nabla\times\vec{v}) }] |
|
| dále platí například: |
|
| [{LTMath fontsize='12' |
|
| \begin{array}{l} |
| \nabla rs = r\nabla s+s\nabla r\\ |
| \nabla.s\vec{v} = ...\\ |
| \nabla\times s\vec{v} = \nabla s\times\vec{v}+s\nabla\times\vec{v} |
| \end{array} |
| }] |