Differential Forms

Exterior differential forms arise when concepts such as the work of a field along a path and the flux of a fluid through a surface are generalized to higher dimensions. Each of differential forms assigns an exterior form to each tangent space of a manifold.

Differential Forms

Differential Forms

A differential -form on a smooth manifold of dimension is a smooth map that is an exterior form restricted to each tangent space at each point . The exterior and interior product of differential forms inherits from the exterior and interior product of exterior forms respectively.

e.g.

• The simplest example of a differential form is the differential of a function. Specifically, if is a smooth map, then is a vector field with the property that for any vector field ,
• Every differential -form on with a given coordinate system can be expressed uniquely as where are smooth functions on and are the coordinate differentials. In fact, every differential -form on the space with a coordinate system can be expressed uniquely as where are smooth functions on and are the coordinate differentials.

Volume Form

A volume form is a top degree differential form on a smooth manifold of dimension , that is non-vanishing everywhere.

Proposition

Given a differential -form , and vector fields , there is a canonical way to evaluate on , denoted by , which is a smooth function on . This is skew-symmetric, and respects the multiplication by smooth functions.

Proof The canonical evaluation is given by where is the -form at point . The skew-symmetry follows from the skew symmetry of . The evaluation respects the multiplication by smooth functions because is -linear:

Proposition

Differential forms on a smooth manifold forms a super-commutative graded algebra over the ring of smooth functions on with the exterior product as multiplication.

Pullback Invariance

Pullback of Differential Forms

Given a smooth map between smooth manifolds, the pullback of a differential -form on is a differential -form on , defined as where is the pushforward of the vector field on to .

Pullback Preserves the Exterior Product and Addition

For any two differential forms on , we have

Proof This directly follows from the pullback invariance of exterior forms.

A Smooth Function is a Differential -Form

We can treat any smooth function as a differential -form, so that

Pullback Preserves The Differential

For a smooth function and a smooth map , there holds

Proof By definition of the pullback, we have

e.g. We can use the above properties to determine the pullback of a differential form. For example, if , then

Lemma

Suppose is a smooth family of linear maps, is a -form on . We define Then is a -form on , and that is a linear map from -forms on to -forms on , and

Lie Derivative of Differential Forms

Lemma

Given vector spaces , , and , a bilinear map , and smooth families of vectors and , there holds

Proposition

Suppose is a diffeomorphism, and is a vector field on , then

Proof For all , we have LHSand RHS

Lie Derivative of Differential Forms

Suppose is a smooth manifold, and is a global flow on . Let be a vector field generated by , then we define the Lie derivative of a differential form on as

We can get the following property of the Lie derivative immediately from the definition:

Proposition

Suppose is a global flow on . Let be a vector field generated by , then

Proof $$\begin{aligned}\ddt{}\bigg|{t=s}f{t}^{}\omega &= \ddtz f_{s+t}^{}\omega\ &= \ddtz (f_{s}^{}\circ f_{t}^{}) \omega \ &= f_{s}^{}\left( \ddtz f_{t}^{}\omega \right)\&= f_{s}^{*}\L_{v}\omega.\end{aligned}$$$\square$

Lemma

For a smooth function , there holds , and .

Proof This is in fact a special case of the Cartan’s formula.

e.g. We can use these two properties to evaluate a Lie derivative. Suppose , then Note that and similarly, , . So we have

Proposition

for any differential -form .

Proof By the [[Differential Forms#^bfdff9]|proposition]], we know that , substituting into the LHS, we obtain Note that is bilinear as a map , so by question (9), we can write where the last equality holds because is the identity map.

Proposition

For all , and vector fields ,

Proof Fix some arbitrary , consider the vector field at : where the fourth equation follows from linearity of , and the sixth equation is by the product rule. Since the above holds for all , we have $$L_{v}xw=(L_{v}x)w + x L_{v}w.$$$\square$

Identifying Lie Derivative with Lie Bracket

For any smooth function , and vector fields , If we identify vector fields with their action on functions, this is the formula

Proof By the above proposition, for the 1-form , we have: Using the lemma, we identify the terms as follows: Substituting into the original equation yields: Rewriting, we find: which implies as expected.

e.g. can be seen easily as because commutes with .

The Exterior Derivative

On a smooth manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree:

Exterior Derivative of Differential Forms

The exterior derivative of a differential -form is a differential -form. It satisfies the following properties:

• The exterior derivative of a -form (i.e. smooth functions on ) is its differential.
• It is a degree derivation of the graded commutative algebra ,
• is local in the sense that if vanishes on a neighborhood of , vanishes on a neighborhood of .

e.g. We can use the above properties to compute:

Exterior Derivative Commutes with Pullback

For any smooth map , and a differential form on , we have

Exact Form

We call a differential form exact, if there exists a differential form , such that .

Cartan's Formula

Suppose is a smooth manifold, and is a vector field on . Then for any differential form ,