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What is 2-Form

  1. Jul 23, 2014 #1
    Definition/Summary

    The 1-forms (or covectors or psuedovectors) of a vector space with local basis [itex](dx_1,dx_2,\dots,dx_n)[/itex] are elements of a vector space with local basis [itex](dx^1,dx^2,\dots,dx^n)[/itex]

    The 2-forms are elements of the exterior product space with local basis [itex](dx^1\wedge dx^2,\ \dots)[/itex]

    In flat space with a global basis, the "d"s in the bases may be omitted.

    In ordinary 3-dimensional space, a 2-form is a directed area, whose normal covector (1-form) is the dual (Hodge dual) of the 2-form.

    Equations

    In ordinary 3-dimensional space with basis [itex](i,j,k)[/itex]:

    the 2-forms have the basis:
    [tex](j\wedge k,\ k\wedge i,\ i\wedge j)[/tex]

    and the 3-forms are all multiples of:
    [tex]i\wedge j \wedge k[/tex]

    and there are no higher forms.

    The curl [itex]\mathbf{\nabla}\times\mathbf{a}[/itex] of a vector and the cross product [itex]\mathbf{a}\times\mathbf{b}[/itex] of two vectors are covectors, or 1-forms, whose duals (Hodge duals) are 2-forms which are, respectively, the exterior derivative and exterior product of their covectors:

    [tex]\ast(\mathbf{\nabla}\times\mathbf{a}_i)\ =\ d \mathbf{a}^i[/tex]

    [tex]\ast(\mathbf{a}_i\times\mathbf{b}_i)\ =\ \mathbf{a}^i\wedge \mathbf{b}^i[/tex]

    Extended explanation

    p-forms (differential forms):

    Generally, for any number p, the p-forms are elements of the exterior product space with basis [itex](dx^1\wedge dx^2\wedge\cdots \wedge dx^p,\cdots )[/itex]

    p-forms in 4-dimensional space-time:

    The 2-forms in 4-dimensional space-time (Newtonian or Einsteinian) with basis [itex](t,i,j,k)[/itex] have the basis:
    [tex](t\wedge i,\ t\wedge j,\ t\wedge k,\ j\wedge k,\ k\wedge i,\ i\wedge j)[/tex]

    and the 3-forms have the basis:
    [tex](i\wedge j \wedge k,\ t\wedge i \wedge j,\ t\wedge j\wedge k,\ t\wedge k\wedge i,)[/tex]

    and the 4-forms are all multiples of:
    [tex]t\wedge i\wedge j \wedge k[/tex]

    and there are no higher forms.

    Electromagnetic 2-forms

    The best-known 2-forms are the Faraday 2-form for electromagnetic field strength [itex]\mathbf{F}\,=\,\frac{1}{2} F_{ij}dx^i\wedge dx^j[/itex], with coordinates [itex](E_x,E_y,E_z,B_x,B_y,B_z)[/itex], and its dual (Hodge dual), the Maxwell 2-form [itex]\ast\mathbf{F}[/itex], with coordinates [itex](-E_x,-E_y,-E_z,B_x,B_y,B_z)[/itex]

    Maxwell's equations may be written:

    [tex]d \mathbf{F}\ =\ 0[/tex]

    [tex]d(\ast\mathbf{F})\ =\ \mathbf{J}[/tex]

    where [itex]\mathbf{J}[/itex] is the current 3-form:

    [tex]\mathbf{J}\ =\ \ast(\rho,\ J_x,\ J_y,\ J_z) = \rho i\wedge j\wedge k\ +\ J_x t\wedge j\wedge k\ +\ J_y t\wedge k\wedge i\ +\ J_z t\wedge i\wedge j[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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