please consider this paper: A Primer on the Differential Calculus of 3D Orientations - Bloesch 2016. Equations 27, 29 and 30, for example, give nice results about differentiating the rotation of a vector via a rotation matrix with respect to the orientation represented by said matrix, and differentiating the concatenation of two orientations with respect to the involved rotations:
(27): $\frac{\partial}{\partial \Phi} (\Phi(r)) = -(\Phi(r))^\times$
(29): $\frac{\partial}{\partial \Phi_1} (\Phi_1 \circ \Phi_2) = I$
(30): $\frac{\partial}{\partial \Phi_2} (\Phi_1 \circ \Phi_2) = C(\Phi_1)$
Now consider being given a set of Euler angles to represent an orientation. As far as I understand, these equations are not directly "compatible" with a set of Euler angles. Equations 21, 31, and 33, for example, describe the construction of a rotation matrix from an orientation and the corresponding Jacobian. Here, the the orientation is described by $\varphi$ in the form of a rotation vector (as mentioned directly after equation 13 in section III):
(21): $C(\varphi):= C(exp(\varphi)) = I + \frac{sin(||\varphi||)\varphi^\times}{||\varphi||} + \frac{(1 – cos(||\varphi||))\varphi^{\times^2}}{||\varphi||^2}$
(31): $\frac{\partial}{\partial \varphi}(exp(\varphi)) = \Gamma(\varphi)$
(33): $\Gamma (\varphi) = I + \frac{(1 – cos(||\varphi||))\varphi^{\times}}{||\varphi||^2} + \frac{||\varphi|| - sin(||\varphi||))\varphi^{\times^2}}{||\varphi||^3}$
Thus, in order to apply these Jacobians listed above given my set of Euler angles, I first need the Jacobian for converting my angles to a rotation vector. This is what I am looking for.
I have not been able to find a nice expression relating my Euler angles to a rotation vector, and therefore I am having trouble coming up with the corresponding Jacobian.
Obtaining the Jacobian for converting from three Euler angles to a rotation matrix would also be great. However, as far as I can tell, doing this the classical way would yield a 3x3x3 tensor, and I am not sure how to further use that with the Jacobian given by equation 27, for example. If I understand correctly, this does not need to be a 3x3x3 tensor, however, since I am working on SO(3) and rotation matrices can be expressed equivalently as vectors of three elements.