How to transform a nonlinear velocity field to a new frame of reference?

Example:

Velocity Field:

v =
 
 - x*sin(C*t) - y*(w/2 - cos(C*t))
   y*sin(C*t) + x*(w/2 + cos(C*t))
                                 0

The linear velocity field can be written as:

X = [x,y,z];
A = [-sin(C*t), cos(C*t) - w/2,0; cos(C*t) + w/2 , sin(C*t),0;0,0,0;];
v = A*transpose(X);

Transformation (Observer change): where

Q = [cos(C/2*t),sin(C/2*t),0;-sin(C/2*t),cos(C/2*t),0;0,0,1];

Based on continuum mechanics, velocity field transforms as . Applying this formula as:

vy  = diff(Q,t,1)*transpose(X) + Q*v;

I do not get the correct answer. I believe that is because v and are still in the old frame. If I change them to the new frame it works. The procedure I used is as follows:

If is also linear, then it can be written as:

Then B is computed as:

Qtr = transpose(Q); assume(Qtr,'real')'; 
Q*(A*Qtr - diff(Qtr,t,1))

and it works. However, now the problem is that, I can not define A and B for nonlinear velocity fields, for numerical velocity fields not possible at all, so what do I do? How do I transform a velocity field with a given observer change?