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| Integral form | Differential form |
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| LaTeX Math Block |
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| anchor | ContinuityIntegral |
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| alignment | left |
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| \frac{d}{dt} \iiint_\Omega \rho \, dV = \frac{dm_\Omega}{dt} |
| | LaTeX Math Block |
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| anchor | ContinuityDifferential |
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| alignment | left |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \frac{dmd\rho (t, {\bf r})}{dt} |
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where
| time | | LaTeX Math Inline |
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| body | --uriencoded--\rho(t, %7B\bf r%7D) |
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| continuum body spatial density distribution |
| LaTeX Math Inline |
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| body | --uriencoded--%7B\bf r %7D |
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| position vector | | LaTeX Math Inline |
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| body | --uriencoded--%7B\bf u%7D(t, %7B\bf r) |
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| continuum body spatial velocity distribution |
| space volume (could be finite or infinite) | | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7Bdm_\Omega%7D%7Bdt%7D |
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| mass generation rate with the space volume |
| gradient operator | | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \frac%7Bdm(t, %7B\bf r%7D)%7D%7Bdt%7D |
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| mass generation rate at a given point in space | LaTeX Math Inline |
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| body | --uriencoded--%7B\bf r %7D |
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For the specific case of finite number of mass generation locations the differential equation
| LaTeX Math Block Reference |
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| anchor | ContinuityDifferential |
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takes form:
| LaTeX Math Block |
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| anchor | ContinuityDifferential |
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| alignment | left |
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| \frac{\partial \rho}{\partial t} + \nabla (\rho \, {\bf u}) = \sum_k \dot m_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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where
| LaTeX Math Inline |
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| body | --uriencoded--%7B\bf r %7D_k |
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| position vector of the -th source/sink |
| mass generation rate at -th source/sink: | LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \dot m_k(t) = \frac%7Bdm_k%7D%7Bdt%7D |
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| Dirac delta function |
See also
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Natural Science / Physics / Mechanics / Continuum mechanics
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