# 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.

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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:

If we wish to move the derivative within 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.