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The cone $x^{2} + y^{2} = z^{2}$ has a rank drop at its apex

Statement refuted

A level set of a smooth map need not have constant derivative rank. For $F (x, y, z) = x^{2} + y^{2} - z^{2}$ , the zero level is regular away from its apex and has derivative rank $0$ at the apex.

Facts & Assumptions

Given: The smooth map $F : R^{3} \to R$ , $F (x, y, z) = x^{2} + y^{2} - z^{2}$ .

[L1]

The power rule and derivative algebra give the continuous Jacobian row $(2 x, 2 y, - 2 z)$ , which is the total derivative by the continuous-partials theorem, and polynomial coordinate expressions are smooth (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, $C^{k}$ Euclidean maps are closed under componentwise algebra and composition).

[L2]

A point is regular exactly when its derivative is surjective, and the regular-level graph theorem requires that hypothesis (Regular and critical points, regular and critical values, and level sets, A regular level set is locally a $C^{k}$ graph of dimension $m - n$ ).

Counterexample

technique · direct

1.1

The equation $F = 0$ is $x^{2} + y^{2} = z^{2}$ , the double cone, and [L1] gives $D F (0, 0, 0) = 0$ . Thus the derivative rank at the apex is $0$ .

givenL1

1.2

If $(x, y, z) \neq 0$ lies on the cone, then the row $(2 x, 2 y, - 2 z)$ is nonzero, so the derivative has rank $1$ and is surjective.

givenL1

2.1

The rank therefore drops at the apex. Moreover the cone contains the rays with directions $(1, 0, 1)$ , $(1, 0, - 1)$ , and $(0, 1, 1)$ , which span $R^{3}$ ; no single two-dimensional tangent plane at the apex contains all their velocities, so [L2] cannot supply a regular graph there.

step 1.1step 1.2L2algebra

3.1

This explicit smooth map refutes constant rank on its level and isolates the failure at the critical apex.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. M. Lee, Introduction to Smooth Manifolds, regular-level examples (standard reference, not scraped)

The cone x2+y2=z2x2+y2=z2 has a rank drop at its apex