Unit normal fields, orientations, and flux through a regular surface patch

Depends on

• Surface area and scalar surface integrals on a regular patch
• The tangent plane of a regular surface patch
• The surface area density is the norm of the cross product of the parameter tangents
• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

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
• Boundary presentations adapted to a simple solid region in a coordinate direction Definition
• Finitely patched regular surfaces, their area, scalar integrals, and flux Definition
• The induced boundary chain and circulation of a C² patch over a finite elementary Green region Definition
• A closed cylinder as a finitely patched oriented surface Example
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• A U-shaped prism is a finite gluing of three boxes and is not simple in every coordinate direction Example
• Both sides of the divergence theorem for F(x,y,z)=(x²,y²,z²) on the closed unit box 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 volume of a closed ball recovered from the outward flux of the position field Example
• FALSE: a field with vanishing divergence has zero outward flux through the boundary of every solid it surrounds 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
• Internal faces cancel and volume integrals add when elementary solid regions are glued 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
• Agreement with classical Gauss flux in Euclidean space Proposition
• At interior base points, the graph faces of an adapted presentation induce the outward unit normal Proposition
• Flux is invariant under orientation-preserving reparametrization and changes sign under reversal 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
• The divergence theorem on an elementary solid region Theorem

Dependency tree · two levels

22 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

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