Reynolds transport theoremLet Ω(t) be a region in Euclidean space with boundary . Let be the positions of points in the region and let be the velocity field in the region. Let be the outward unit normal to the boundary. Let be a vector field in the region (it may also be a scalar field). Show that This relation is also known as the Reynold's Transport Theorem and is a generalization of the Leibniz rule. Content of example. 
Proof:
Let Ω_{0} be reference configuration of the region Ω(t). Let the motion and the deformation gradient be given by
Let . Then, integrals in the current and the reference configurations are related by
The time derivative of an integral over a volume is defined as
Converting into integrals over the reference configuration, we get
Since Ω_{0} is independent of time, we have
Now, the time derivative of is given by (see Gurtin: 1981, p. 77)
Therefore,
where is the material time derivative of . Now, the material derivative is given by
Therefore,
or,
Using the identity
we then have
Using the divergence theorem and the identity we have
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