Curve $\displaystyle \mathcal{K}$ is defined as an intersection of surfaces $\displaystyle z^2 = 2x^2 + y^2$ and $\displaystyle z = x + 1$. The curve's projection on plane $\displaystyle z = 0$ is positive oriented.

- Parametrise the curve $\displaystyle \mathcal{K}$.
- Find the Frenet-Serret basis of the curve $\displaystyle \mathcal{K}$ at point $\displaystyle A(2, 1, 3)$.
- Find the $\displaystyle \int_{\mathcal{K}}\vec{F}\mathrm{d}\vec{r}$ of the vector field $\displaystyle F(x, y, z) = (z, x - 1, -2y)$ along the curve $\displaystyle \mathcal{K}$. ∫