71.225.101.109: /* Solution in terms of object heat capacity */

2014-02-23T17:23:26Z

Solution in terms of object heat capacity

en>Walrus62 at 04:15, 14 December 2013

2013-12-14T04:15:27Z

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+{{multiple issues|
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+In many applications, one needs to calculate the [[derivative|rate of change]] of a [[volume integral|volume]] or [[surface integral]] whose domain of [[integral|integration]], as well as the [[integrand]], are [[Function (mathematics)|functions]] of a particular parameter. In physical applications, that parameter is frequently [[time]] ''t''.
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+==Introduction==
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+:<math>\frac{d}{dt}\int_{a\left( t\right) }^{b\left( t\right) }f\left( t,x\right) dx=  \int_{a\left( t\right) }^{b\left( t\right) }\frac{\partial f\left( t,x\right) }{\partial t}dx+f\left( t,b\left( t\right) \right) b^{\prime }\left( t\right) -f\left( t,a\left( t\right) \right) a^{\prime }\left( t\right)</math>
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+The [[calculus of moving surfaces]]<ref name="Grinfeld1" >Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. {{doi|10.1111/j.1467-9590.2010.00485.x}}. ISSN 00222526.</ref> provides analogous [[formulas]] for volume integrals over [[Euclidean space|Euclidean domains]], and surface integrals over [[differential geometry of surfaces]], curved surfaces, including integrals over curved surfaces with moving contour [[Boundary (topology)|boundaries]].
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+==Volume integrals==
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+Let ''t'' be a time-like [[parameter]] and consider a time-dependent [[Domain of a function|domain]] &Omega; with a smooth [[surface]] boundary ''S''. Let ''F'' be a time-dependent [[Invariant (mathematics)|invariant]] field defined in the interior of &Omega;. Then the rate of change of the [[integral]] <math>\int_\Omega F \, d\Omega </math>

en>Dhollm: /* Mathematical statement */ improve grammar

2012-07-14T23:22:18Z

Mathematical statement: improve grammar

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Lumped capacitance model - Revision history

71.225.101.109: /* Solution in terms of object heat capacity */

en>Walrus62 at 04:15, 14 December 2013

en>Dhollm: /* Mathematical statement */ improve grammar