Proof for a material element
Let Ω0 be reference configuration of the region Ω(t). Let
the motion and the deformation gradient be given by

Let J(X,t) = det F(X,t). Define
Then the integrals in the current and the reference configurations are related by

That this derivation is for a material element is implicit in the time constancy of the reference configuration: it is constant in material coordinates. The time derivative of an integral over a volume is defined as

Converting into integrals over the reference configuration, we get

Since Ω0 is independent of time, we have

The time derivative of J is given by:[6]

Therefore,
where
is the material time derivative of f. The material derivative is given by

Therefore,
or,

Using the identity
we then have

Using the divergence theorem and the identity (a ⊗ b) · n = (b · n)a, we have
Q.E.D.