Differential Form

Forms are multilinear mappings from a vector spce to scalars.

• bilinear form: a bilinear map from two vectors to $\mathbb{R}$
• multilinear form: generalization of the bilinear map to take any number of vectors
• quadratic form(symmetricl bilinear form): a homogeneous polynomial of degree two; the polarization identity gives a 1-1 correspondence with it
• algebraic form(completely symmetric multilinear form): generalization of quadratic form to arbitrary degree
• 2-form: an anti-symmetric bilinear form
• exterior form(completely anti-symmetric multilinear form)

Examples of k-forms in differential forms:

• 0-form: $\varphi=x^2y+e^z$
• 1-form: $\varphi=x^2dx+(yz+1)dz$
• 2-form: $\varphi=xyzdydz+xe^ydzdx+2dxdy$
• 3-form: $\varphi=(x^2+xyz+z^3)dxdydz$

Hodge star

• orientation (vector space): choosing a direction for bases in a real $n$-dimensional $V$. Specifically, selecting one of two equivalence classes formed by grouping ordered bases under transformations with $\det>0$. (= choosing the $GL^+(n)$ side among the two connected components of $GL(n,\mathbb{R})$)

• Equivalence: selecting up to sign a non-zero top-degree $n$-form ($=\Lambda^n V^* \backslash{0}$). That is, choosing one of the two components ${\Omega, -\Omega}$. With an inner product, we can set $

  \Omega

  =1$ and call it a volume form (unit pseudoscalar).

• Hodge star $\star$ is defined by “inner product + orientation” For the unit $n$-vector $\Omega=e_1\wedge\cdots\wedge e_n$ of the chosen inner product and orientation, define $\star:\Lambda^kV\rightarrow \Lambda^{n-k}V$ by $A\wedge \star B=\langle A,B\rangle\Omega$. \(A\wedge(\star B)=\langle A,B\rangle\Omega, \quad A,B\in\Lambda^k V\) where $\Omega$ is the unit $n$-form of the chosen orientation. Changing orientation gives $\Omega\mapsto -\Omega$, so there is a global sign reversal for all $k$-forms: $\star\mapsto -\star$.
  • Properties (signature $(r,s)$): $\star\star A=(-1)^{k(n-k)+s}A$; $\langle A,B\rangle=(-1)^s\langle\star A,\star B\rangle$, etc. For example, in $\mathbb{R}^3$, adopting the standard basis $(e_1,e_2,e_3)$ with the ‘right-hand’ convention is precisely choosing an orientation. Then $\star e^1=e^2\wedge e^3$, $\star e^2=e^3\wedge e^1$, $\star e^3=e^1\wedge e^2$. Flipping the orientation adds a $(-)$ sign to all equations. Therefore, in definitions like $v\times w=\star(v^\flat\wedge w^\flat)^\sharp$, we can see that changing orientation flips the sign of the cross product, showing that it is a pseudo-vector.

The Hodge star operator(Hodge isomorphism) is an element of linear maps based on the exterior product. The result produces a Hodge dual.

In real $\mathbb{R}^n$ space, let’s transform a k-form function $\varphi$ into an $(n-k)$-form function $\varphi^*$:

1. Given a set ${X_1,X_2,\dots,X_n}$, if there is a subset ${X_{i_1},X_{i_2},\dots,X_{i_k}}$, then its complement set is denoted as ${X_{j_1},X_{j_2},\dots,X_{j_{n-k}}}$.

   The differential form for these two sets gives:
   \(dX_{i_1}\cdots dX_{i_k}dX_{j_1}\cdots dX_{j_{n-k}}=dX_1dX_2\cdots dX_n\).
   Applying the Hodge star operator to this differential form:
   \((dX_{i_1}\cdots dX_{i_k})^*=dX_{j_1}\cdots dX_{j_{n-k}}\).
   These two differential forms have a Hodge dual relationship.

2. If $\varphi=A_1dX_{I_1}+\cdots+A_mdX_{I_m}$ is a k-form, then \(\varphi^*=A_1dX_{I_1}^*+\cdots+A_mdX_{I_m}^*\).
   For example:

   • In real \(\mathbb{R}^1\): \(1^*=dx\), \(dx^*=1\)
   • In real $\mathbb{R}^2$: \(1^*=dx\cdot dy\), \(dx^*=dy\), \(dy^*=-dx\), \((dx\cdot dy)^*=1\)

TBA