外微分形式 | {Differential Forms} [2]

从今天开始，会不定期的贴出比较完整的外微分形式(exterior differential forms)的大纲。大部分只是简单的从书本中摘出来的，以便以后快速回想和查阅使用。初学者可能会犯很多错误，强烈要求指正。
所读书本为Harley Flanders的书，Google Books链接在文末给出。

第一部分：http://multiverse.lamost.org/blog/?p=2091
第二部分：http://multiverse.lamost.org/blog/?p=2167
第三部分：http://multiverse.lamost.org/blog/?p=2173

Contents:
Chapter 3 The Exterior Derivative

Chapter 3 The Exterior Derivative

1. Exterior derivatives are mappings from smooth manifold \(\mathcal{M}\), which is m dimensional, to itself, that satisfy the following conditions.

\begin{eqnarray}
&&\mathrm d(\omega+\eta)=\mathrm d\omega+\mathrm d \eta \\
&&\mathrm d(\lambda\wedge\mu)=\mathrm d\lambda\wedge\mu+(-1)^{\text{deg}\lambda}\lambda\wedge\mathrm d\mu \\
&&\text{For each } \omega, \mathrm d (\mathrm d\omega)=0 \text{. This is the Poncare Lemma.}\\
&&\text{For each smooth function }f, \mathrm d f=\Sigma \frac{\partial f}{\partial x^i}\mathrm d x^i \\
\end{eqnarray}

A reminder: do not try to prove the Poncare lemma using the ordinary partial derivatives. It is easy to prove it using the defination.

2. Mappings are very important here. Two of the most important mappings are pull back and push forwad. A figure is useful for understanding.

\begin{equation}
\mathcal{R} \xleftarrow{\phi^*g=g\circ \phi} \mathcal{U} \xrightarrow{\phi} \mathcal{V} \xrightarrow{g} \mathcal{R}
\end{equation}

In this schematic figure, \(\phi\) is pull back mapping and \(\phi^*\) is the push forward mapping. Since \(g\) is a function, \(\phi^*\) should be a map from \(F^0(V)\) to \(F^0(U)\).

We need these mappings because we want to move p-forms from one manifold to another. Thus extending this from scalar function to p-forms is useful. (\(\mathcal {W,U,V}\) are p-form spaces.)

\begin{equation}
\mathcal{W} \xleftarrow{\psi\circ \phi} \mathcal{U} \xrightarrow{\phi} \mathcal{V} \xrightarrow{\psi} \mathcal{W}
\end{equation}

Since \(\mathcal W\) is a p-form space now, the push forward map can not be as simple as the previous one.

Now we will define it,
\begin{equation}
F^p(\mathcal W) \xrightarrow{(\psi\circ\phi)^*=\phi^*\circ\psi^*} F^p(\mathcal{U}) \xleftarrow{\phi^*} F^p(\mathcal{V}) \xleftarrow{\psi^*} F^p\mathcal{(W)}
\end{equation}

All the mappings about p-forms are done under corresponding points if they are in different domains.

3. An important relation that shows the exterior derivative of a differential form is independent of the coordinate system in which it is calculated is shown here.
\begin{equation}
\mathrm d \phi^*\omega=\phi^*\mathrm d \omega
\end{equation}

Differential forms with applications to the physical sciences
By By Harley Flanders