I am currently working through "An Introduction to Dynamic Meteorology (5th Ed.)" by Holton and Hakim, and am looking for more detail on material derivatives in rotating reference frames than what they provide.
On page 33 they begin a derivation of the material derivative of a vector field $\mathbf{A}$ in a frame rotating with angular velocity $\mathbf{\Omega}$. They express $\mathbf{A}$ in the inertial frame as $$\mathbf{A}=\mathbf{i}'A_x'+\mathbf{j}'A_y'+\mathbf{k}'A_z'$$ and in the rotating frame as $$\mathbf{A}=\mathbf{i}A_x+\mathbf{j}A_y+\mathbf{k}A_z\,.$$ They define $D_a\mathbf{A}/Dt$ to be the material derivative of $\mathbf{A}$ in the inertial frame, and $D\mathbf{A}/Dt$ be the material derivative in the rotating frame, and state \begin{align} \frac{D_a\mathbf{A}}{Dt}&=\mathbf{i}'\frac{DA_x'}{Dt}+\mathbf{j}'\frac{DA_y'}{Dt}+\mathbf{k}'\frac{DA_z'}{Dt}\\ &=\mathbf{i}\frac{DA_x}{Dt}+\mathbf{j}\frac{DA_y}{Dt}+\mathbf{k}\frac{DA_z}{Dt}+\frac{D_a\mathbf{i}}{Dt}A_x+\frac{D_a\mathbf{j}}{Dt}A_y +\frac{D_a\mathbf{k}}{Dt}A_z\, . \end{align}
The only justification they give for this last step is
The first three terms on the preceding line can be combined to give... the material derivative of $\mathbf{A}$ as viewed in the rotating coordinates (i.e., the rate in $\mathbf{A}$ following the relative motion.) The last three terms arise because the directions of the unit vectors ($\mathbf{i},$ $\mathbf{j},$ $\mathbf{k}$) change their orientation in space as Earth rotates. (Page 34)
Can anyone provide a more detailed derivation, or a reference where this is treated more formally? Thanks!