I have a quadric $F(x,y,z) = 0$. It is convenient for me to represent this function in "standard" form (i.e. without the complicating $xy$, $xz$, $yz$ terms) by first rotating the world into the orientation of the quadric surface (so that my $x, y, z$ axes are along the axes of the surface). To accomplish this, I'm using an axis-angle approach.
However, I am now needing to take derivatives of $F$ w.r.t. the variables and also the axis and angle by which I am rotating everything. I can take a derivative of the rotation in quaternion space, and I can take derivatives of $F$ once it's in the canonical form, but I'm not entirely sure how to compose these derivatives. Ordinarily a "change of variables" requires multiplication by the derivative of the variable substitution (chain rule).... but what is the Jacobian of the map from Cartesian coordinates into Quaternion coordinates? The ordinary mapping suggests that it's just:
$$\begin{bmatrix} \bf{0} & I_3 \end{bmatrix}$$
But I'm 90% sure my reasoning here is entirely too simplistic.
How does the differential calculus work when composing quaternion rotations AND Cartesian space operations to a vector? I realize the long of this is probably 10 tons of textbooks on the matter... so any pointers to good resources are also appreciated.