The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the
Leibniz integral rule. For a continuously
differentiable scalar
field 
and a continuously differentiable velocity vector field 
,
suppose the bounded material volume 
is smoothly transported by the fluid flow. The classical form of the Reynolds transport
theorem says
![D/(Dt)[int_(V_m(t))B(x,t)dV]=int_(V_m(t))[(partialB)/(partialt)+del ·(Bu)]dV.](/images/equations/ReynoldsTransportTheorem/NumberedEquation1.svg) |
Here, 
on the integral denotes its rate of change following
the material volume, corresponding to the convective
derivative for a transported scalar field; 
is the usual gradient, 
denotes the material volume
at time 
, and 
denotes the velocity vector.
The theorem also applies to a vector field or tensor field by applying the scalar formula to each component in a fixed Cartesian vector basis.
Because of its relation to the Leibniz integral rule, the Reynolds transport theorem is sometimes called the Leibniz-Reynolds transport theorem.
Variants of the Reynolds transport theorem appear in different contexts, with equations that may differ in appearance and complexity.
