NFYB21003U-Fluid-Mechanics/Lecture1BSlide3.html at main · team-ocean/NFYB21003U-Fluid-Mechanics · GitHub

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    <title>From Leibniz's Theorem to the Reynolds Transport Theorem</title>
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        <div class="title">From Leibniz's Theorem to the Reynolds Transport Theorem</div>

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                <p><span class="sql">Leibniz's Theorem:</span>

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                    \[
                    \frac{d}{dt} \int_{a(t)}^{b(t)} F(x,t) dx = \int_{a}^{b} \frac{\partial F}{\partial t} dx + \frac{db}{dt} F(b,t) - \frac{da}{dt} F(a,t)
                    \]
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                A geometrical generalization of Leibniz's rule is presented using control volumes.</p>
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                    \[
                    \frac{d}{dt} \int_{V^*(t)} F(x,t) dV = \lim_{\Delta t \to 0} \frac{1}{\Delta t} \left\{ \int_{V^*(t+\Delta t)} F(x,t+\Delta t) dV - \int_{V^*(t)} F(x,t) dV \right\}
                    \]
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