Solution
= Solution
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 $y\geq1/2$ on the given patch, division by its last component is valid and gives the unique normalized field
$$
\boxed{\mathbf u=\frac{\nabla H}{y}=\left(\frac{3x^2+y}{y},\frac{x+z}{y},1\right).}
$$
Its dot product with $\mathbf k$ is one, as required; it is not generally a unit vector.