The cross product in ${\mathbb{R}}^3$

Depends on

• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$

Used by

• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• Vector forms: the boundary integrals of fn and of n× F 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
• Divergence and curl of a C¹ vector field Definition
• Regular parametrized surface patches on compact Jordan parameter regions 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
• Angular momentum as the moment map for rotations of a cotangent bundle 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
• The two-sphere as a coadjoint orbit of SO(3) Example
• The volume of a closed ball recovered from the outward flux of the position field Example
• 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
• The general reduced dimension is dim M minus two dim G False statement
• 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
• The cross product is bilinear, alternating, and orthogonal to both factors Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
• The curl measures the antisymmetric part of the total derivative Lemma
• The divergence and curl of a cross product Lemma
• The second homotopy group of SO(3) vanishes Lemma
• A divergence-free C¹ field on a star-shaped open subset of ℝ³ has a vector potential Theorem
• The squared cross-product norm is the Gram determinant of two vectors Theorem

Dependency tree · two levels

28 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)