Take a look at the below picture, I would like to derive the Jacobian matrix without differentiation. In this Modern Robotics book, the screw theory is used. I've derived the forward kinematic using PoE formula which stated here:
$$ \begin{align} M &=\begin{bmatrix} 1&0&0& L_1+L_2\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix} \\ \mathcal{S}_2 &= [0,0,1,L_1,0,0]^T \\ \mathcal{S}_1 &= [0,0,1,0,0,0]^T \\ e^{[\mathcal{S}_2]\theta_2} &= \begin{bmatrix} c_{\theta_2} &-s_{\theta_2} &0&-L_1(c_{\theta_2}-1) \\s_{\theta_2} &c_{\theta_2} &0&-L_1s_{\theta_2}\\0&0&1&0\\0&0&0&1 \end{bmatrix} \\ e^{[\mathcal{S}_1]\theta_1} &= \begin{bmatrix} c_{\theta_1} &-s_{\theta_1} &0&0 \\s_{\theta_1} &c_{\theta_1} &0&0\\0&0&1&0\\0&0&0&1 \end{bmatrix} \\ T_2^0 &= e^{[\mathcal{S}_1]\theta_1} e^{[\mathcal{S}_2]\theta_2}M \\ &= \begin{bmatrix} c_{\theta_1+\theta_2}&-s_{\theta_1+\theta_2}&0&L_2c_{\theta_1+\theta_2}+L_1c_{\theta_1} \\ s_{\theta_1+\theta_2}&c_{\theta_1+\theta_2}&0&L_2s_{\theta_1+\theta_2}+L_1s_{\theta_1} \\0&0&1&0\\0&0&0&1\end{bmatrix} \end{align} $$ In the book, the authors differentiate the last column of $T_2^0$ to obtain this: $$ \begin{align} \dot{x} &= -L_1\dot{\theta}_1 s_{\theta_1} - L_2(\dot{\theta}_1+\dot{\theta}_2)s_{\theta_1+\theta_2} \\ \dot{y} &= L_1\dot{\theta}_1 c_{\theta_1} + L_2(\dot{\theta}_1+\dot{\theta}_2)c_{\theta_1+\theta_2} \end{align} $$ Or more compactly as $$ \begin{align} \begin{bmatrix} \dot{x}\\\dot{y}\end{bmatrix} &= \begin{bmatrix} (-L_1s_{\theta_1}-L_2s_{\theta_1+\theta_2})&(-L_2s_{\theta_1+\theta_2})\\(L_1c_{\theta_1}+L_2c_{\theta_1+\theta_2})&(L_2c_{\theta_1+\theta_2})\end{bmatrix} \begin{bmatrix} \dot{\theta}_1\\\dot{\theta}_2\end{bmatrix} \\ \dot{\mathbb{x}} &= [J_1(\theta) \ J_2(\theta)]\dot{\theta} \\ \dot{\mathbb{x}} &= J(\theta) \dot{\theta} \end{align} $$ where $J(\theta)$ the Jacobian matrix. The authors claim the Jacobian matrix can be obtained without differentiation using this formula:
I couldn't reach to the same result with the above formula. For the first column, $J_{1s}=\mathcal{S}_1$ which is not the same result obtained by differentiation. Any suggestions?