I am currently working through definitions and examples of KJets and one-parameter point groups. The text I am working with describes kjets as vector fields defined by: \begin{equation} (j_k T_\epsilon)(x,j_k u) := (\tilde{x}, j_k \tilde{x}),\mbox{ whenever }(\tilde{x},\tilde{u}) = T_\epsilon(x,u), \epsilon \in I_{(x,u)}. \end{equation} But nowhere is there a concrete definition of $j_k \tilde{x}$. It is only described as a generic element of the kth jet space.
So, in one of the examples the concrete 1-jet of the point transformation group \begin{equation} T_\theta(x,u) =\big(x\cos(\theta) -u(x)\sin(\theta),\, x\sin(\theta),\, x\sin(\theta) + u(x)\cos(\theta)\big) \end{equation} is given by: \begin{equation} \begin{split} &(j_1 T_\theta)(x,u,u_x) =\\ =& \left(x\cos(\theta) -u_x\sin(\theta),\, x\sin(\theta),\, x\sin(\theta) + u\cos(\theta),\, \frac{\sin(\theta) + u_x \cos(\theta)}{\cos(\theta) -u_x \sin(\theta)}\right) \end{split} \end{equation}
How does one calculate the last term of this result? The textbook shows that it can be achieved by direct computation but it nowhere says how to achieve this computation.