$C^k$ Euclidean maps and diffeomorphisms

Depends on

• $C^k$ maps and multi-index derivative notation in Euclidean space
• Continuously differentiable maps, local inverses, and local diffeomorphisms
• Injection, surjection, bijection
• If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative

Used by

• An injective regular Cᵏ map is a Cᵏ diffeomorphism onto its image Corollary
• Green's first identity on a glued elementary solid region Corollary
• Green's theorem is the curl statement for a planar field lifted to ℝ³ Corollary
• The divergence at a point is the limit of outward flux per unit volume Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• The planar divergence theorem: the flux form of Green's theorem Corollary
• An isolated boundary point obstructs pointwise-zero Green data Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The fundamental Hessian is not absolutely locally integrable Counterexample
• x↦ x³ is a C¹ bijection whose inverse is not differentiable at zero Counterexample
• Divergence and curl of a C¹ vector field Definition
• Regular parametrized surface patches on compact Jordan parameter regions Definition
• Surface reparametrizations and their orientation sign Definition
• The induced boundary chain and circulation of a C² patch over a finite elementary Green region Definition
• The Laplacian of a C² function and of a C² vector field Definition
• The rank of a derivative and constant-rank Euclidean maps Definition
• Vector potentials of a continuous field on an open subset of ℝ³ Definition
• Both sides of the divergence theorem for F(x,y,z)=(x²,y²,z²) on the closed unit box Example
• Dilation determines the Riesz-potential target exponent Example
• Flux normalization on every centered sphere Example
• Newton shell theorem from harmonic mean values Example
• Newtonian potential of radial compact data Example
• The graph of a Cᵏ Euclidean map is a regular level set Example
• The orthogonal group is a regular level set of dimension n(n-1)/2 Example
• The two-dimensional logarithmic kernel has unit normalized flux Example
• A C¹ map sends a compact set of content zero to a set of content zero Lemma
• A nonzero rank minor supplies the source coordinates for the constant-rank theorem Lemma
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• Choice-free smooth inverse function theorem in Euclidean space Lemma
• Divergence and curl are linear and satisfy the scalar product rules Lemma
• Each coordinate of the oriented area vector is the Jacobian determinant of the matching cyclic projection Lemma
• Matrix inversion preserves Cᵏ regularity where the determinant is nonzero Lemma
• Near and far bounds for a Riesz potential Lemma
• Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
• The divergence and curl of a cross product Lemma
• The Laplace fundamental solution is harmonic off its pole Lemma
• The pointwise norm on a tangent space is smooth off the zero vector Lemma
• The single-direction flux identity on a simple solid region Lemma

…and 9 more results.

Dependency tree · two levels

17 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. Lebl, Basic Analysis II, §§8.5–8.6 (standard reference, not scraped)
• University of Toronto MAT237, §3.3 (standard reference, not scraped)