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  over the time-dependent region  that has boundary , then taking the derivative with respect to time:
If we wish to move the derivative into the integral, there are two issues: the time dependence of , and the introduction of and removal of space from  due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.