# EKF SLAM : SLAM specific Jacobians for new landmarks

I am currently trying to understand the books SLAM for dummies and Simulataneous localization and mapping with the extended Kalman filter to implement slam. I have understood steps 1 and 2 SLAM for dummies. However I am having difficulty understanding Step 3: Add new landmarks to the current state (Page 40). The corresponding section in Simulataneous localization and mapping with the extended Kalman filter is 2.3.4 Landmark initialization for full observations.

Specifically, I do not understand how the SLAM specific Jacobian $$J_{xr}$$ (SLAM for dummies) defined below is actually derived:

I am confused about what transformation function the Jacobian $$J_{xr}$$ is linearizing. I notice that $$J_{xr}$$ is the same as Jacobian $$G_R$$ that linearizes the Inverse Observation model as described in section 2.3.4 of Simulataneous localization and mapping with the extended Kalman filter. Why does $$J_{xr}$$, contain $$\Delta t$$, which is the thrust applied to the robot? $$\Delta t$$ which belongs to the State Prediction model shouldn't be in the Inverse Observation model, no? Am I right to say that the Inverse Observation model is a function of the robot pose : [$$x_{r}$$, $$y_{r}$$, $$\theta_{r}$$] and the new landmark observation: [$$d_{l1}$$, $$\phi_{l1}$$] and the output is the updated state vector : [$$x_{r}$$, $$y_{r}$$, $$\theta_{r}$$, $$x_{l1}$$, $$y_{l1}$$]?