(i) Give definitions for the unit tangent vector $\hat{\mathbf{T}}$ and the curvature $\kappa$ of a parametrised curve $\mathbf{x}(t)$ in $\mathbb{R}^{3}$. Calculate $\hat{\mathbf{T}}$ and $\kappa$ for the circular helix

$$\mathbf{x}(t)=(a \cos t, a \sin t, b t),$$

where $a$ and $b$ are constants.

(ii) Find the normal vector and the equation of the tangent plane to the surface $S$ in $\mathbb{R}^{3}$ given by

$$z=x^{2} y^{3}-y+1$$

at the point $x=1, y=1, z=1$.