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Reynolds transport theorem
3D generalization of the Leibniz integral rule
Reynolds transport theorem
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: d d t ∫ Ω ( t ) f d V . {\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...
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