Reynolds transport theorem

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${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
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In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds (1842–1912), is a three-dimensional generalization of the Leibniz integral rule. It is used to recast time derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.

Consider integrating f = f(x,t) over the time-dependent region Ω(t) that has boundary ∂Ω(t), then taking the derivative with respect to time: $${\displaystyle {\frac {d}{dt}}\int _{\Omega (t)}\mathbf {f} \,dV.}$$ If we wish to move the derivative into the integral, there are two issues: the time dependence of f, and the introduction of and removal of space from Ω due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.