Differential Forms

The definition of Differential Forms

Differential 1-forms

A differential 1 form on a manifold M is a sooth map

(1)   \begin{equation*} \omega: TM \to \mathbb{R} \end{equation*}

of the tangent bundle of M to the line, linear on each tangent space TM_{\mathbf{x}}

One could say that a differential 1-form on M is an algebraic 1-form on TM_{\mathbf{x}} which is “differentiable with respect to \mathbf{x}.”

Also we can define the differential forms dx_1, \dots, dx_n, which act on a tangent vector \boldsymbol{\xi} \in T\mathbb{R}_{\mathbf{x}}^n will produce its components \xi_1,\dots,\xi_n, that is

(2)   \begin{equation*} dx_1(\boldsymbol{\xi})=\xi_1,\dots,dx_n(\boldsymbol{\xi})=\xi_n \end{equation*}

Every differential 1-form on the space \mathbb{R}^n with a given coordinate system x_1,\dots,x_n can be written uniquely in the form

(3)   \begin{equation*} \omega=a_1(x)dx_1+\cdots+a_n(x)dx_n \end{equation*}

where the coefficients a_i(x) are smooth functions.

The usual differential df|_{\mathbf{x}} of a function f at \mathbf{x} is in fact a differential 1-form.

(4)   \begin{equation*} df|_{\mathbf{x}}:TM_{\mathbf{x}}\to\mathbb{R} \end{equation*}

Notice that df itself will not give a value on \mathbb{R}, it should act on a vector in the tangent space and then give the value.

For example, let \boldsymbol{\xi} \in TM_{\mathbf{x}} be the velocity vector of the curve \mathbf{x}(t). Then, the value of differential 1-form df|_{\mathbf{x}} on \dot{\mathbf{x}}(t) at point \mathbf{x} is

(5)   \begin{align*} df|_{\mathbf{x}}(\boldsymbol{\xi}) =df|_{\mathbf{x}}(\boldsymbol{\dot{\mathbf{x}}(t)}) =f'(\mathbf{x})\dot{\mathbf{x}}(t) \end{align*}

Differential k-forms

A differential k-form \omega^k|_\mathbf{x} at a point \mathbf{x} of a manifold M is an exterior k-form on the tangent space TM_{\mathbf{x}} to M at \mathbf{x}, i.e., a k-linear skew-symmetric function of k vectors \boldsymbol{\xi}_1, \dots, \boldsymbol{\xi}_k tangent to M at \mathbf{x}.

Every differential k-form on the space \mathbf{R}^n with a given coordinate system x_1,\dots,x_n can be written uniquely in the form

(6)   \begin{equation*} \omega^k=\sum_{i_1<\cdots<i_k}a_{i_1,\dots,i_k}(\mathbf{x})dx_{i_1}\wedge\cdots\wedge dx_{i_k} \end{equation*}

where the a_{i_1,\dots,i_k}(\mathbf{x}) are smooth functions on \mathbb{R}^n.

Integration of Differential Forms

Let

(7)   \begin{equation*} \omega^k=\sum_{i_1<\cdots<i_k}a_{i_1,\dots,i_k}(\mathbf{x})dx_{i_1}\wedge\cdots\wedge dx_{i_k} \end{equation*}

be a differential form and S a differentiable k-manifold over which we wish to integrate, where S has the parameterization

(8)   \begin{equation*} S({\mathbf {u} })=(x_{1}({\mathbf {u} }),\dots ,x_{k}(\mathbf{u})) \end{equation*}

for \mathbf{u} in the parameter domain D. Then (Rudin 1976) defines the integral of the differential form over S as

(9)   \begin{equation*} \int _{S}\omega^k =\int _{D}\sum _{i_{1}<\cdots <i_{k}}a_{i_{1},\dots ,i_{k}}(S({\mathbf {u} })){\frac {\partial (x_{i_{1}},\dots ,x_{i_{k}})}{\partial (u_{1},\dots ,u_{k})}}\,du_{1}\cdots du_{k} \end{equation*}

where the integral on the right-hand side is the standard Riemann (or Lebesgue) integral over D, and

(10)   \begin{equation*} J={\frac {\partial (x_{i_{1}},\dots ,x_{i_{k}})}{\partial (u_{1},\dots ,u_{k})}} \end{equation*}

is the determinant of the Jacobian. The Jacobian exists because S is differentiable.

Immediately, we can see that the integral of a k-form \omega^k=\phi(\mathbf{x})dx_1\wedge\cdots\wedge dx_k on euclidean space \mathbb{R}^k over a domain D is

(11)   \begin{equation*} \int_D \omega^k= \int_D \phi(\mathbf{x})dx_1\cdots dx_k \end{equation*}

Exterior Derivative

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

If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k+1)-parallelotope.

The exterior derivative of a differential form of degree k is a differential form of degree k+1.

In a local coordinate system x_1, \dots, x_n on M, the form \omega^k is written as

(12)   \begin{equation*} \omega^k=\sum_{i_1<\cdots<i_k}a_{i_1,\dots,i_k}(\mathbf{x})dx_{i_1}\wedge\cdots\wedge dx_{i_k} \end{equation*}

then the exterior derivative is

(13)   \begin{equation*} d\omega^k=\sum_{i,i_1<\cdots<i_k}\frac{\partial}{\partial x_i} a_{i_1,\dots,i_k}dx_i\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k} \end{equation*}

References

[1] Arnol’d, Vladimir Igorevich. Mathematical methods of classical mechanics. Vol. 60. Springer Science \& Business Media, 2013.
[2] https://en.wikipedia.org/wiki/Differential\_form

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