Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 7 / a / i

Set $H(x,y,z)=x^3+xy+yz-1$. The gradient $\nabla H=(3x^2+y,x+z,y)$ is a normal vector to its level set. Since $y\ge 1/2$ on the given patch, division by its last component is valid and gives the unique normalized fieldIts dot product with $\mathbf{k}$ is one, as required; it is not generally a unit vector.