外微分形式 | {Differential Forms} [3]

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 4 Applications: Moving Frames, Laplacian, Coordinates and Surfaces

Chapter 4 Applications: Moving Frames, Laplacian, Coordinates and Surfaces

1. In \(\mathsf E^3\), \(\vec x\) is an a point described by \(\mathrm d \vec x=\sigma_1\hat e_1+\sigma_2\hat e_2+\sigma_3\hat e_3\). It should be made clear that \(\sigma_i\) are 1-forms and \(\hat e_i\) are bases. In fact, \(\mathrm d\vec x\) is a 1-form here.

\(\mathrm d \hat e_i=\omega_{i1}\hat e_1 +\omega_{i2}\hat e_2+\omega_{i3}\hat e_3\) is the exterior differential of the basis.

For convenience of denoting, we collect the terms as matrices.
\begin{eqnarray}
\hat e = \left (\begin{matrix}
\hat e_1\\
\hat e_2\\
\hat e_3\\
\end{matrix} \right ), \sigma=(\sigma_1,\sigma_2,\sigma_3), \Omega=||\omega_{ij}||
\end{eqnarray}

Structure equations
\begin{eqnarray}
\mathrm d \vec x &=& \sigma \vec x \\
\mathrm d \hat e &=& \Omega \hat e \\
\Omega &=& ^t\Omega
\end{eqnarray}

The last equation says \(\Omega\) is a skrew-symmetric or anti-symmetric matrix. This is deeply rooted in the linear property of the exterior diff and the orthogonal property of the basis.

Another set of equations that counts is the integrability equations.
\begin{eqnarray}
\mathrm d\sigma &=& \sigma \Omega \\
\mathrm \Omega &=& \Omega^2
\end{eqnarray}

These equations are results of Poncare Lemma.

Examples of the structure eqns and the integrablilty eqns are given in the book.

2. Flanders gives an example of 6-dim frame space by combining \(\mathsf E^1,\mathsf E^2, \mathsf E^3\).

3. \(\vec x (u, v ,w)\) is a point in \(\mathsf U\), which is a domain in \(\mathsf E^3\). We can always use basis that is orthonormal,
\begin{eqnarray}
\hat e_1=\frac 1\lambda \frac{\partial \vec x}{\partial u}, \hat e_2=\frac 1 \mu \frac{\partial \vec x}{\partial v}, \hat e_3=\frac 1\nu \frac{\partial \vec x}{\partial w}
\end{eqnarray}

Under such basis,
\begin{eqnarray}
\mathrm d\vec x&=&\mathrm du\frac{\partial \vec x}{\partial u}+\mathrm dv\frac{\partial \vec x}{\partial v}+\mathrm dw\frac{\partial \vec x}{\partial w} \\
&=&(\lambda \mathrm d u)\hat e_1+(\mu\mathrm dv)\hat e_2+(\nu\mathrm dw)\hat e_3 \\
&=&\sigma_1\hat e_1+\sigma_2\hat e_2+\sigma_3\hat e_3
\end{eqnarray}

Here \(\lambda, \mu,\nu\) are called scale factors sometimes.

Laplacian is
\begin{equation}
\delta f =\mathrm d *\mathrm d f=(\delta f)\sigma_1\sigma_2\sigma_3
\end{equation}

Just apply the constants, one would get the expression for Laplacian.

4. Gaussian curvature etc. This part is familiar to most of us.

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