Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 3 / 9A / a

Away from the vertex, use the defining function $H=z^2-x^2-y^2$. Its gradient is normal to the smooth part of the double circular cone, andon the surface. Hence the field is tangent at every regular point of the cone. At the origin the cone has no unique tangent plane, but $\mathbf{F}(0)=0$, which belongs to its tangent cone. An equivalent argument, valid also at the vertex, uses the integral curves of a vector field: $\dot{\mathbf{x}}=\mathbf{x}$ has solution $\mathbf{x}(t)=e^t\mathbf{x}(0)$, and these dilations preserve $H=0$. This is an example of radial dilation tangency to homogeneous zero sets.