外微分形式 | {Differential Forms} [1]

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

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Contents:
Chapter 1 Why forms?
Chapter 2 Exterior Algebra

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Chapter 1 Why forms?

Orientation of the integration, determined by the argument.

More simple for some calculation. I always encounter calculations hard to be solved only with tensor analysis. That is why I started to learn a little about exterior differential forms.

…

Chapter 2 Exterior Algebra

1. Set up the space we will work on.

$\mathcal{R}$: field of real numbers.

$\mathcal{L}$: n dimensional space over $\mathcal{R}$ with elements $\alpha$, $\beta$, $\dots$

$\wedge^p \mathcal{L}$: space of p-vectors on $\mathcal{L}$. This will be made clear later.

2. Basic rules for basis.

Linear: $(a \alpha +b \beta)\wedge \alpha_2\wedge\cdots\wedge\alpha_p = a(\alpha\wedge\alpha_2\wedge\cdots\wedge\alpha_p)+b(\beta\wedge\alpha_2\wedge\cdots\wedge\alpha_p)$

Any of the terms repeated in the expression $\alpha_1\wedge\alpha_2\wedge\cdots\wedge\alpha_p$ leads the whole to 0.

Interchanging between two terms in $\alpha_1\wedge\alpha_2\wedge\cdots\wedge\alpha_p$ changes the whole’s sign.

3. Properties of p-vector space

Function g can be defined on $\mathcal{L}$ and mapping f on $\wedge^p \mathcal{L}$, satisfying $f(\alpha_1\wedge\cdots\wedge\alpha_p)=g(\alpha_1, \cdots, \alpha_p)$.

4. Determinants

Function $g_A$: $\bigotimes^n \mathcal{L} \rightarrow \wedge^n \mathcal{L}$ and $f_A$: $\wedge^n\mathcal{L}\rightarrow\wedge^n\mathcal{L}$ can be related by

$$\begin{equation} f_A(\alpha_1\wedge\cdots\wedge\alpha_p)=g_A(\alpha_1,\cdots, \alpha_p)\equiv A\alpha_1\wedge\cdots\wedge A\alpha_p \end{equation}$$

Since only one kind of transformation the linear function $f_A$ has, the only possible expression for $f_A$ is $|A|\alpha_1\wedge\cdots\wedge\alpha_p$, where $|A|$ is a scalar. Define $\wedge^p(\alpha_1\wedge\cdots\wedge\alpha_p)\equiv A\alpha_1\wedge\cdots\wedge A\alpha_p$.

Another way of looking at $g_A$ is $A\alpha_1\wedge\cdots\wedge A\alpha_p = |a^i_j|\alpha_1\wedge\cdots\wedge\alpha_p$, if we have $A\sigma^i=\Sigma a^i_j \sigma^j$.

Comparing the two understandings, we find $|A|=|a^i_j|$. Here $|A|$ is the determinant.

5. Exterior products

Basic rules: ($\mu$ and $\lambda$ are p vector and q vector respectively)

a. $\mu\wedge\lambda$ distribuative

b. associative

c. $\mu\wedge\lambda=(-1)^{pq}\lambda\wedge\mu$

6. Linear transformations

Linear spaces: $\mathcal{M},\mathcal{N}$ with dimension m and n respectively. The basis of $\mathcal{M}$ is $\sigma^i$.

Define a linear transformation A : $\mathcal M \rightarrow \mathcal N$, which is

$$\begin{equation} (\alpha_1, \cdots, \alpha_p) \rightarrow A\alpha_1\wedge\cdots\wedge A\alpha_p . \end{equation}$$

This is a map from $\bigotimes^p\mathcal M$ to $\wedge^p \mathcal N$.

One would find a useful linear transformation can be denoted $\wedge^p A$ : $\wedge^p \mathcal M\rightarrow \wedge^p\mathcal N$.

Note that $\wedge^p A$ has matrix representation.

7. Inner product spaces

Basic rules

a. Linear in each variable

b. Symmetric: $(\alpha, \beta)=(\beta, \alpha)$

c. Nondegenerate: if for fiexed $\alpha$, $(\alpha,\beta)=0$ for all $\beta$, then $\alpha=0$.

Properties:

a. Each inner product space $\mathcal L$ has an orthonormal basis.

b. $f$: linear function on $\mathcal L$. Then there is a vector $\beta$ in $\mathcal L$ such that $f(\alpha)=(\alpha, \beta)$. (This property is very important.)

8. Inner product of p-vectors

Vector space $\mathcal L$ has inner product $(\alpha,\beta)$ on it. Define $(\lambda,\mu)=|(\alpha_i,\beta_j)|$ on $(\wedge^p \mathcal L)\bigotimes (\wedge^p\mathcal L)$, in which $\lambda=\alpha_1\wedge\cdots\wedge\alpha_p$, $\mu=\beta_1\wedge\cdots\wedge\beta_p$.

9. Hodge star

Linear transformation on $\wedge^p\mathcal L\rightarrow \wedge^{n-p}\mathcal L$. (Depends on inner product and orientation. Orientation of $\mathcal L$ which will remain fixed means we hake on basis for $\mathcal L$ and only consider other basis which are expressed in terms of this one by a matrix with positive determinant.)

$$\begin{equation} \lambda\wedge\mu=(*\lambda, \mu)\sigma (=f_\lambda (\mu)\sigma) \end{equation}$$

here * map is a linear one $\wedge^p\mathcal L\rightarrow \wedge^{n-p}\mathcal L$.

注： $f_\lambda(\mu)$把$\mu$从$\wedge^p\mathcal L$映射到$\wedge^n\mathcal L$。$\lambda$是$\wedge^p\mathcal L$上，$\mu$是$\wedge^{n-p}\mathcal L$上的，为了定义出这个线性的$f_\lambda(\mu)$，想到inner product,这样需要把$\lambda$，$\mu$放在同样的space中，故通过*来把$\lambda$从$\wedge^p\mathcal L$映射到$\wedge^{n-p}\mathcal L$上。

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Reference:

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