Reynolds transport theorem

In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or in short Reynolds theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign.
The theorem is named after Osborne Reynolds (1842–1912). It is used to recast 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 within 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.
General form.
Reynolds transport theorem can be expressed as follows:
in which is the outward-pointing unit normal vector, is a point in the region and is the variable of integration, and are volume and surface elements at , and is the velocity of the area element ("not" the flow velocity). The function may be tensor-, vector- or scalar-valued. Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.
Form for a material element.
In continuum mechanics, this theorem is often used for material elements. These are parcels of fluids or solids which no material enters or leaves. If is a material element then there is a velocity function , and the boundary elements obey
This condition may be substituted to obtain:
A special case.
If we take to be constant with respect to time, then and the identity reduces to
as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.)
Interpretation and reduction to one dimension.
The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose is independent of and , and that is a unit square in the -plane and has limits and . Then Reynolds transport theorem reduces to
which, up to swapping and , is the standard expression for differentiation under the integral sign.