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The graph of a $C^{k}$ Euclidean map is a regular level set

Example

Let $ψ : U \subseteq R^{m} \to R^{n}$ be $C^{k}$ , $k \geq 1$ , and define $G : U \times R^{n} \to R^{n}$ by $G (x, y) = y - ψ (x)$ . Then $0$ is a regular value, $G^{- 1} (0)$ is the graph of $ψ$ , and $T_{(x, ψ (x))} G^{- 1} (0) = {(v, D ψ (x) v) : v \in R^{m}} .$

Facts & Assumptions

Given: The map $ψ$ and the associated map $G$ .

[L1]

Finite sums and scalar multiples of $C^{k}$ Euclidean maps are $C^{k}$ , coordinate maps are $C^{k}$ componentwise, and total-derivative algebra gives $D G (x, y) (v, w) = w - D ψ (x) v$ ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition, $C^{k}$ Euclidean maps and diffeomorphisms, Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).

[L2]

A regular level is locally a graph and has tangent space equal to the derivative kernel (A regular level set is locally a $C^{k}$ graph of dimension $m - n$ , The tangent space to a regular level set).

Verification

technique · direct

1.1

The equation $G (x, y) = 0$ is equivalent to $y = ψ (x)$ , so $G^{- 1} (0)$ is precisely the graph.

givenalgebra

1.2

By [L1], $D G (x, y) (0, w) = w$ for every $w \in R^{n}$ , so $D G (x, y)$ is surjective at every point and $0$ is a regular value.

givenL1

2.1

Solving $D G (x, ψ (x)) (v, w) = 0$ gives $w = D ψ (x) v$ , and [L2] identifies this kernel with the displayed tangent space.

step 1.2L1L2

3.1

The graph conclusion holds on the whole open set $U$ , including when $U$ is empty, in which case both sides are empty.

step 1.1step 2.1∎

Depends on

Used by

A critical value can have a smooth level set Counterexample
FALSE: a critical value must have a singular level set False statement

Dependency tree · two levels

26 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, graph and regular-level examples (standard reference, not scraped)
L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)

The graph of a CkCk Euclidean map is a regular level set