Continuously differentiable maps, local inverses, and local diffeomorphisms

Depends on

• The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
• The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
• Injection, surjection, bijection

Used by

• Change of variables on bounded open Jordan sets when both integrands are bounded and Riemann integrable Corollary
• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• The real complex-squaring map is locally but not globally invertible off the origin Counterexample
• x↦ x³ is a C¹ bijection whose inverse is not differentiable at zero Counterexample
• Cᵏ Euclidean maps and diffeomorphisms Definition
• The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix Definition
• The regular locus of a square-dimensional C¹ map Definition
• The unit circle is locally a C¹ graph at every point Example
• FALSE: an everywhere-invertible derivative gives a global inverse False statement
• FALSE: an invertible derivative at one point gives a local inverse False statement
• A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets Lemma
• A C¹ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes Lemma
• Choice-free smooth inverse function theorem in Euclidean space Lemma
• Newton maps are uniform contractions near a point with invertible derivative Lemma
• An injective C¹ map with invertible derivative sends compact Jordan sets to compact Jordan sets Theorem
• An injective regular C¹ map is a diffeomorphism onto its image Theorem
• Change of variables for an injective C¹ map on a compact Jordan set Theorem
• The Euclidean implicit function theorem with derivative formula Theorem
• The Euclidean inverse function theorem Theorem
• The Jacobian sign of a regular C¹ map is constant on a connected domain Theorem

Dependency tree · two levels

12 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

• Tillmann, Notes of Lectures on Multivariable Calculus, Inverse and Implicit Function Theorems (standard reference, not scraped)