What does it mean for $g^{\epsilon}:(x, u) \mapsto(\tilde{x}, \tilde{u})$ to describe a 1-parameter group of transformations? Explain how to compute the vector field

$$V=\xi(x, u) \frac{\partial}{\partial x}+\eta(x, u) \frac{\partial}{\partial u}$$

that generates such a 1-parameter group of transformations.

Suppose now $u=u(x)$. Define the $n$th prolongation, $\mathrm{pr}^{(n)} g^{\epsilon}$, of $g^{\epsilon}$ and the vector field which generates it. If $V$ is defined by $(*)$ show that

$$\mathrm{pr}^{(n)} V=V+\sum_{k=1}^{n} \eta_{k} \frac{\partial}{\partial u^{(k)}}$$

where $u^{(k)}=\mathrm{d}^{k} u / \mathrm{d} x^{k}$ and $\eta_{k}$ are functions to be determined.

The curvature of the curve $u=u(x)$ in the $(x, u)$-plane is given by

$$\kappa=\frac{u_{x x}}{\left(1+u_{x}^{2}\right)^{3 / 2}}$$

Rotations in the $(x, u)$-plane are generated by the vector field

$$W=x \frac{\partial}{\partial u}-u \frac{\partial}{\partial x}$$

Show that the curvature $\kappa$ at a point along a plane curve is invariant under such rotations. Find two further transformations that leave $\kappa$ invariant.