Regular parametrized surface patches on compact Jordan parameter regions

Depends on

• The cross product in $\mathbb R^3$
• $C^k$ Euclidean maps and diffeomorphisms
• The Riemann integral of a bounded function over a bounded Jordan measurable set
• Interior, closure, boundary, limit point, isolated point and dense subset of a metric space
• A bounded set in $\mathbb{R}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero

Used by

• Every patch of an elementary solid region's presentation is a graph face in some direction, and at interior base points its normal is outward Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• A degenerate two-parameter map can collapse its image to a curve Counterexample
• Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
• Boundary presentations adapted to a simple solid region in a coordinate direction Definition
• Finite gluings of elementary solid regions and their outward boundary presentation Definition
• Surface area and scalar surface integrals on a regular patch 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 tangent plane of a regular surface patch Definition
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• Distributional laplacian of the newtonian kernel Example
• Opposite parametrizations preserve area and negate flux Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The closed unit box, with its six faces, is an elementary solid region Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
• The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components Example
• The outward flux of the inverse-square field through a sphere centred at the origin is 4π Example
• The surface area of a torus is 4π²ab Example
• FALSE: continuity alone makes the regular-patch surface-area formula applicable False statement
• FALSE: flux is independent of the parametrization without an orientation condition False statement
• FALSE: Stokes' theorem requires the surface to be a graph over a coordinate plane False statement
• FALSE: the patches of a finite presentation can always be reoriented to make their normals agree on overlaps False statement
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• Content-zero parameter-boundary exceptions do not affect surface integrals Lemma
• The flux of a single-component field through a graph face is a base integral of its trace Lemma
• The single-direction flux identity on a simple solid region Lemma
• At interior base points, the graph faces of an adapted presentation induce the outward unit normal Proposition
• Regular level surfaces have local regular parametrizations with the same tangent plane Theorem
• Scalar surface integrals on a surface of revolution Theorem
• Surface area, scalar integrals, and flux over a C¹ graph Theorem
• The classical Stokes theorem for a C² patch over a finite elementary Green region Theorem

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

• M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)
• University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)