# 外微分形式 | {Differential Forms} [3]

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

\delta f =\mathrm d *\mathrm d f=(\delta f)\sigma_1\sigma_2\sigma_3

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

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

Reference:

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