# 外微分形式 | {Differential Forms} [2]

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.

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

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.)

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

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,

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)}

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.

\mathrm d \phi^*\omega=\phi^*\mathrm d \omega

Reference:

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