Applying (10), the D-NURBS motion equations are
The two vectors involving 
on the right side of
(36)
may be be combined into a single vector:
Using the product rule of differentiation, we have 
. For (37) to hold, we must have
It is obvious from (11) that the two sides of
(38) are integrals of the two vectors, respectively. The two
vectors in (38) are equal when, for 
,
We now prove (39). The right side is represented as R. Based on the product rule of differentiation and the property of the Jacobian matrix, we obtain the simpler expression
Furthermore, according to the property of the Jacobian matrix and the observation that we can interchange the order of the cross derivatives
Combining the above two expressions we obtain
Since R is a scalar, (39) is proved.
Next, we derive another mathematical identity:
The left side of (40) is the integral of the summation of the
five terms of (16). Each of these five vectors
is the zero vector. To see this, note that for , we
have
According to the definition of the Jacobian matrix, the left hand side
is 
, . Thus, we have
The order of the second cross derivative with respect to the variables
and u is irrelevant, so we further have
Now, (41) is the ith component of the first vector on the left side of (40). Similarly, the other four vectors inside the integral operator on the left hand side of (40) are zero.
