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raccourcis clavier

Reynolds transport theorem

Also known as Leibniz-Reynolds transport therem

A three-dimensional generalization of Leibniz integral rule

theorem

Consider integrating $\mathbf{f} = \mathbf{f}(x, t)$ over time-dependent region $\Omega (t)$ that has boundary $\partial \Omega (t)$ then take derivative w.r.t time:
$\frac{d}{dt} \int_{\Omega (t)} \mathbf{f} dV$

general form

\frac{d}{dt} \int_{\Omega(t)} \mathbf{f} dV = \int_{\Omega (t)} \frac{\partial{\mathbf{f}}}{\partial{t}} dV + \int_{\partial{\Omega (t)}}(\mathbf{v}_b \cdot \mathbf{n}) \mathbf{f} dA

where:

$\mathbf{n}(\mathbf{x},t)$ is the outward-pointing unit normal vector
$\mathbf{x}$ is the variable of integrations
$dV$ and $dA$ are volume and surface elements at $\mathbf{x}$
$\mathbf{v}_b(\mathbf{x},t)$ is the velocity of the area element.