Exploring 1-forms & 2-forms in Vector Spaces

[Greg Bernhardt]

Definition/Summary

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

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

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 $(i,j,k)$:

the 2-forms have the basis:
$$(j\wedge k,\ k\wedge i,\ i\wedge j)$$

and the 3-forms are all multiples of:
$$i\wedge j \wedge k$$

and there are no higher forms.

The curl $\mathbf{\nabla}\times\mathbf{a}$ of a vector and the cross product $\mathbf{a}\times\mathbf{b}$ 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:

$$\ast(\mathbf{\nabla}\times\mathbf{a}_i)\ =\ d \mathbf{a}^i$$

$$\ast(\mathbf{a}_i\times\mathbf{b}_i)\ =\ \mathbf{a}^i\wedge \mathbf{b}^i$$

Extended explanation

p-forms (differential forms):

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

p-forms in 4-dimensional space-time:

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

and the 3-forms have the basis:
$$(i\wedge j \wedge k,\ t\wedge i \wedge j,\ t\wedge j\wedge k,\ t\wedge k\wedge i,)$$

and the 4-forms are all multiples of:
$$t\wedge i\wedge j \wedge k$$

and there are no higher forms.

Electromagnetic 2-forms

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

Maxwell's equations may be written:

$$d \mathbf{F}\ =\ 0$$

$$d(\ast\mathbf{F})\ =\ \mathbf{J}$$

where $\mathbf{J}$ is the current 3-form:

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

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[fresh_42]

I always find a calculation as an example helpful. Here is one:
https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443
Also a good read is the geometry behind it:
https://arxiv.org/pdf/1205.5935.pdf