Leibniz's Theorem & Reynolds Transport Theorem

Leibniz's Theorem: Consider a function \( F \) that depends on one independent spatial variable, \( x \), and time \( t \). The time derivative of the integral of \( F(x,t) \) between \( x = a(t) \) and \( x = b(t) \) is given by:

\[ \frac{d}{dt} \int_{a(t)}^{b(t)} F(x,t) dx = \int_{a}^{b} \frac{\partial F}{\partial t} dx + \frac{db}{dt} F(b,t) - \frac{da}{dt} F(a,t) \]

where \( a, b, F \), and their derivatives appearing on the right side of the equation are all evaluated at time \( t \).

Reynolds Transport Theorem is a generalization of the Leibniz rule. The key difference is that the velocity has three components, and only the perpendicular component enters the calculations.

A geometrical generalization of Leibniz’s rule is presented using control volumes.

Consider a moving volume \( V^*(t) \) with a closed surface \( A^*(t) \) having an outward normal \( \mathbf{n} \) and a local velocity \( \mathbf{b} \).

\[ \frac{d}{dt} \int_{V^*(t)} F(x,t) dV = \lim_{\Delta t \to 0} \frac{1}{\Delta t} \left\{ \int_{V^*(t+\Delta t)} F(x,t+\Delta t) dV - \int_{V^*(t)} F(x,t) dV \right\} \]