Classical Mechanics by Herbert Goldstein is one of the most used textbooks on this subject, perhaps the most used one. However, I found a couple of errors in Section 4.9 (in 3rd ed, written with Charles Poole and John Safko) about rotations. First, at p. 172, the angular velocity vector ω is defined by the relation ωdt = dΩ. The authors write in a footnote: "Note that ω is not the derivative of any vector." (by which they mean a time derivative). But this is wrong. If ω is a (vector valued) continuous function of time (even weaker conditions suffice), then it follows from the Fundamental Theorem of Calculus, applied to each of its components, that it has a (vector) antiderivative w.r.t time. Presumably, the authors meant that ω is not the derivative of any physically interesting vector. Second, at p. 173, the authors try to express the angular velocity vector in terms of the Euler angles and their time derivatives, and they write the following about infinitesimal rotations: "The general infinitesimal rotation associated with ω can be considered as consisting of three successive infintesimal rotations [associated with the Euler angles] ... " But these rotations associated with the Euler angles will not always be all infinitesimal, even if the resulting rotation is. For example, if the new z-axis (the ς-axis), after the resulting infinitesimal rotation, lies in the old xz-plane, but differs slightly from the old z-axis, then the first Euler angle, φ, will be 90° (or -90°), since the node line must be the old y-axis, and the third Euler angle, ψ, will be approximately -φ. These Euler angles, and the roatations associated with them, are certainly not infinitesimal, although they almost cancel each other out, making the resulting rotation infintesimal. Only if a unit vector ez along the z-axis is moved by the resulting infinitesimal rotation by a vector dez, whose angle to the old yz-plane is infinitesimal, all three Euler angles are infinitesimal. In general, I think Goldstein et. al. talk too much about infinitesimal roatations. I think it is simpler to look at time derivatives of rotation matrices instead.