Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / F (Multivariable Calculus)

A $k$-dimensional embedded submanifold of ${\mathbb{R}}^n$ is a subset $M$ such that near each $p\in M$ there is a smooth diffeomorphism of ambient open neighbourhoods that sends $M$ to the coordinate plane ${\mathbb{R}}^k\times \{0{\}}^{n-k}$. More precisely, for some open $U\ni p$ and diffeomorphism $\Phi :U\to V\subseteq {\mathbb{R}}^n$,Thus locally the subset has $k$ freely varying smooth coordinates.

A tangent vector to $M$ at $p$ is a velocity ${\gamma }^{\prime }(0)$ of a smooth curve $\gamma :(-\epsilon ,\epsilon )\to M$ with $\gamma (0)=p$. The tangent space is the set of all such velocities:It is a $k$-dimensional linear subspace of ${\mathbb{R}}^n$; for a local parametrization $\varphi$, it is the image of the injective derivative $D\varphi$ at the corresponding parameter point.

Let $g(x,y,z)=x^2+y^2-z^2$. The upper cone away from the apex is the part of $g^{-1}(0)$ with $z>0$. ThereThe regular level set theorem says that a level set of a smooth scalar function with nonzero gradient is an embedded submanifold of codimension one. Intersecting it with the open half-space $z>0$ preserves this local property. Thus $M$ is a two-dimensional submanifold of ${\mathbb{R}}^3$.

The tangent space to a regular level set is the kernel of its derivative. Writing $z_0=\sqrt{x_0^2+y_0^2}$ givesIf an affine tangent plane through $v_0$ is intended, its equation is $x_0(x-x_0)+y_0(y-y_0)-z_0(z-z_0)=0$. Since $x_0^2+y_0^2=z_0^2$, this simplifies to the same plane $x_{0}x+y_{0}y-z_{0}z=0$, which passes through the apex.

Every point of the indicated ray has the form $w_0=(r,2r,\sqrt{5}r)$ with $r>0$; the apex is excluded. Substitution into the preceding tangent space formula givesindependent of $r$. The affine tangent plane has the same equation, since the ray lies in this plane. Thus both interpretations of the requested tangent plane are constant along this generator of the cone.