The level sets of a differentiable function is a manifold

I have seen stronger propositions that imply this, but both their statement and their proofs require more advanced tools than I’d like to use in my text, which is aimed at a general scientific audience.

I want to prove: With $fcolon mathbb R^k to mathbb R$ smooth (infinitely differentiable) and $x$ a non-critical point (i.e. $nabla f(x) neq 0$), then the level set $f^{-1}(x)$ is a manifold. Preferrably providing a procedure to construct charts.

It seems that there should be a proof using undergraduate calculus only. Any ideas?

MathOverflow Asked by user8948 on January 31, 2021

Proof of the implicit function theorem in several variables calculus requires the contraction mapping theorem, so is probably not suitable for your audience. You need to use an iterative method and take a limit. You can look for a complete proof in Spivak, Calculus on Manifolds.

If you just replace one of the coordinate functions by $$f$$, you get a chart, but the proof that it is a chart requires the implicit function theorem.

Answered by Ben McKay on January 31, 2021

Related Questions

Commutator estimates regarding pseudo-differential operators

0  Asked on December 15, 2020 by shaoyang-zhou

English translation of Borel-Serre Le theoreme de Riemann-Roch?:

1  Asked on December 15, 2020

Upper bound for an exponential sum involving characters of a finite field

1  Asked on December 14, 2020 by nahila

Euler function summation

0  Asked on December 13, 2020 by andrej-leko

A problem about an unramified prime in a Galois extension

1  Asked on December 9, 2020 by neothecomputer

Reference request: discretisation of probability measures on $mathbb R^d$

1  Asked on December 9, 2020 by mb2009

Smoothness of a variety implies homological smoothness of DbCoh

0  Asked on December 8, 2020 by dbcohsmoothness

Reference for matrices with all eigenvalues 1 or -1

1  Asked on December 7, 2020

Characterizations of groups whose general linear representations are all trivial

1  Asked on December 7, 2020 by qsh

Continuous time Markov chains and invariance principle

0  Asked on December 6, 2020 by sharpe

Continuity property for Čech cohomology

0  Asked on December 6, 2020 by xindaris

Reference request: superconformal algebras and representations

0  Asked on December 6, 2020 by winawer

Extending rational maps of nodal curves

1  Asked on December 5, 2020 by leo-herr

How large is the smallest ordinal larger than any “minimal ordinal parameter” for any pair of an Ordinal Turing Machine and a real?

1  Asked on December 4, 2020 by lyrically-wicked

Almost geodesic on non complete manifolds

1  Asked on December 4, 2020 by andrea-marino

With Khinchine’s inequality, prove Fourier basis is unconditional in $L^{p}[0,1]$ only for $p=2$

0  Asked on December 3, 2020 by eric-yan

Kähler manifolds deformation equivalent to projective manifolds

0  Asked on December 3, 2020 by user164740

Duality of eta product identities: a new idea?

2  Asked on December 1, 2020 by wolfgang

Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?

2  Asked on December 1, 2020 by bernhard-boehmler