I can't seem to follow the chain of reasoning that leads to equation 4-100). Can anybody please help me to understand that why under infinitesimal rotation x1 transforms in the way as shown ?
This is from Goldstein's Classical Mechanics page chapter 5 and page 168 on the Kinematics of Rigid body motion.
Suppose a finite rotation around x axis. The rotation matrix is:
\begin{vmatrix} 1 & 0 & 0\\ 0 & cos(\theta_1) & sin(\theta_1)\\ 0 & -sin(\theta_1) & cos(\theta_1) \end{vmatrix}
Now a rotation around y axis:
\begin{vmatrix} cos(\theta_2) & 0 & -sin(\theta_2)\\ 0 & 1 & 0\\ sin(\theta_2) & 0 & cos(\theta_2) \end{vmatrix}
Finally a rotation around z axis:
\begin{vmatrix} cos(\theta_3) & sin(\theta_3) & 0\\ -sin(\theta_3) & cos(\theta_3) & 0\\ 0 & 0 & 1 \end{vmatrix}
If the rotations are infinitesimal, $cos(\theta_i)\sim 1$ and $sin(\theta_i) \sim \theta_i$.
If we want now a composite infinitesimal rotation, we multiply that matrices, following that approximations, and disregarding the products of $\theta_i*\theta_j$ that are second order infinitesimals.
We get:
\begin{vmatrix} 1 & \theta_3 & -\theta_2\\ -\theta_3 & 1 & -\theta_1\\ \theta_2 & -\theta_1 & 1 \end{vmatrix}
If we use that matrix to transform a vector:
$$
\begin{vmatrix} x^{'}_1 \\ x^{'}_2 \\x^{'}_3 \end{vmatrix} = \begin{vmatrix}
1 & \theta_3 & -\theta_2\\
-\theta_3 & 1 & -\theta_1\\
\theta_2 & -\theta_1 & 1
\end{vmatrix} \times \begin{vmatrix} x_1 \\ x_2 \\ x_3 \end{vmatrix}
$$
We have the same expression from the book, except that $\epsilon_{ij} = 0$ if $i = j$
Imagine you have your point $(x_1,x_2,x_3)$. If you do a traslation over the x axis, the $x'_1$ would be $x'_1=x_1+a$ but if the traslation is infinitesimal you will have $x'_1=x_1+\varepsilon x_1$.
If you have a rotation, you can extrapolate from the traslational movement, now you have 3 traslations in space, each one infinitesimal for $x_1,x_2,x_3$:
$$x'_1=x_1+\varepsilon_{11} x_1+\varepsilon_{12} x_2+\varepsilon_{13} x_3$$
