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Back to MatematickyAparatFyzikyKontinua1, or MatematickyAparatFyzikyKontinua1 version history

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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}}
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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:
[{LTMath fontsize='12'
\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'
\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}
}]
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Tato strana (revision-16) byla změněna 11:09 20.11.2007 uživatelem xkrumpha.