Applying (10), the D-NURBS motion equations with linear constraints are
To simplify (42) we first show that it reduces to the following
As in Appendix B, 
. Hence, (43) is also expressed as
Similar to (38), the two sides of (44) are integrals
of two vectors, respectively. Hence, (44) holds if
corresponding components of the two vectors are equal; i.e., for
,
We now prove (45). Denoting the right side as R, we further expand it using the product rule of differentiation
Furthermore, according to the property of the Jacobian matrix and the irrelevance of the order of differentiation, we have
Combining the above two equations, we have
Since R is a scalar, (45) follows.
The proof of
parallels that in Appendix B, with 
replacing 
and 
replacing 
.
