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© 1999-2008 HEAT
JSPWiki v2.4.104
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| [{ALLOW view All}] |
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| [{Math fontsize='14' |
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| skalární součin vektorů: [{Math fontsize='14' |
| skalární součin vektorů: [{LTMath fontsize='14' |
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| | [{Math fontsize='14' |
| | [{LTMath fontsize='14' |
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| [{Math fontsize='12' latex='\vec{v}.\vec{w}=\left(\sum\limits_i \vec{\delta}_i v_i\right) |
| [{LTMath fontsize='12' |
| \vec{v}.\vec{w}=\left(\sum\limits_i \vec{\delta}_i v_i\right) |
| At line 26 changed 1 line. |
| \delta_{ij}v_iw_j=\sum\limits_iv_iw_j' }] |
| \delta_{ij}v_iw_j=\sum\limits_iv_iw_j }] |
| At line 30 changed 1 line. |
| [{Math fontsize='14' |
| [{LTMath fontsize='14' |
| At line 34 changed 1 line. |
| [{Math fontsize='12' latex='\vec{w}\times\vec{v}=\left(\sum\limits_j\vec{\delta}_jw_j\right) |
| [{LTMath fontsize='12' |
| \vec{w}\times\vec{v}=\left(\sum\limits_j\vec{\delta}_jw_j\right) |
| At line 43 changed 1 line. |
| \right|' }] |
| \right| }] |
| vektorové diferenciální operace: |
| df: |
| [{LTMath fontsize='14' |
| \nabla=\sum\limits_i\vec{\delta}_i\frac{\partial}{\partial x_i} }] |
| (čti nabla) |
| gradient skalárního pole: |
| [{LTMath fontsize='14' |
| \nabla |
| s=\sum\limits_i\vec{\delta}_i\frac{\partial s}{\partial x_i} = \textrm{grad}\,s }] |
| (čti nabla na s) |
| [{LTMath fontsize='14' |
| \nabla s\neq s\nabla;\qquad (\nabla r)s\neq\nabla(rs); \qquad \nabla(r+s)=\nabla r+\nabla s }] |
| divergence vektorového pole: |
| [{LTMath fontsize='14' maxwidth='700' |
| \nabla.\vec{v}=\sum\limits_i\nabla_i.v_i=\sum\limits_i\frac{\partial v_i}{\partial x_i}=div\,\vec{v} }] |
| rotace vektorového pole: |
| [{LTMath fontsize='12' maxwidth='900' |
| \begin{array}{l} |
| {\nabla\times\vec{v}=\left(\sum\limits_j\vec{\delta}_j\frac{\partial}{\partial |
| x_j}\right)\times\left(\sum\limits_k\vec{\delta}_k |
| v_k\right)=\sum\limits_j\sum\limits_k(\vec{\delta}_j\times\vec{\delta}_k)\frac{\partial}{\partial |
| x_j}v_k=\nonumber}\\ \\ |
| {=\sum\limits_i\sum\limits_j\sum\limits_k\varepsilon_{ijk}\vec{\delta}_i\frac{\partial |
| v_k}{\partial x_j} =\left| |
| \begin{array}{ccc} |
| \vec{\delta}_1 &\vec{\delta}_2 &\vec{\delta}_3\\ |
| \frac{\partial}{\partial x_1} &\frac{\partial}{\partial x_2} |
| &\frac{\partial}{\partial x_3}\\v_1 & v_2 & v_3 |
| \end{array}\right| |
| =rot\,\vec{v}} |
| \end{array} |
| }] |
| 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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uživatelem xkrumpha.