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 F is given by:

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

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