前言

1 傅里叶级数

$$
f \sim \sum a_n e^{inx}
$$
$$
a_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)e^{-inx}dx
$$
由此我们可以得到Parseval恒等式：
$$
\sum_{n=-\infty}^{+\infty} |a_n|^2 = \frac{1}{2\pi}\int_{-\pi}^\pi |f(x)|^2 dx
$$
在常庚哲,史济怀的《数学分析》中也有相关论述：
但是这样得到的度量，并非完备

---

前置知识储备

集合与度量

点 point

A point $x \in \mathbb R^n$ consists of a $d$-tuple of real numbers
$$
x=(x_1,x_2,\dots,x_d), x_i \in \mathbb R,\text{ for }i=1,\dots,d
$$

范数 norm

The norm of x is denoted by $|x|$ and is defined to be the standard Euclidean norm given by
$$
|x| = (x_1^2+\cdots+x_d^2)^{1/2}
$$

补集 complement

The complement of a set $E$ in $\mathbb R^d$ is denoted by $E^c$ and defined by
$$
E^c = {x \in \mathbb R^d : x \not \in E}
$$

距离 distance

The distance between two points $x$ and $y$ is then simply $|x-y|$.
The distance beteween two sets $E$ and $F$ is defined by
$$
d(E,F) = \inf_{x\in E,y \in F} |x-y|
$$

点与点之间的位置关系——开、闭、紧致集

The open ball in $\mathbb R ^ d$ centered at $x$ and of radius $r$ is defined by
$$
B_r(x) = {y \in \mathbb R^d : |y-x| < r}
$$
A subset $E \subset \mathbb R^d$ is **open** if for every $x \in E$ there exists $r > 0$ with $B_r(x) \subset E$.
A subset $E \subset \mathbb R^d$ is closed if its complement is open.
A subset $E \subset \mathbb R^d$ is bounded if it is contained in some ball of finite radius.

A subset $E \subset \mathbb R^d$ is compact if it is closed and bounded.warning: 这只在欧氏空间中成立
A subset $E \subset \mathbb R^d$ is compact if there is a finite subcovering for any open covering.

A point $x \in\mathbb R^d$ is a limit point of the set $E$ if for every $r> 0$, the ball $B_r(x)$ contains points of $E$.
An isolated point of E is a point $x \in E$ such that there exists an $r > 0$ where $B_r(x) \cap E$ is equal to ${x}$.
A point $x \in E$ is an interior point of $E$ if there exists $r > 0$ such that $B_r(x) \subset E$.

The set of all interior points of $E$ is called the interior of E.
the closure $E$ of the $E$ consists of the union of $E$ and all its limit points.
The boundary of a set $E$, denoted by $\partial E$, is the set of points which are in the closure of $E$ but not in the interior of $E$.
内部是被$E$包含的最大开集，闭包是包含$E$的最小闭集。

a closed set $E$ is perfect if $E$ does not have any isolated points.

最直接的体积计算——长方形

A (closed) rectangle $R$ in $\mathbb R^d$ is given by the product of $d$ one-dimensional closed and bounded intervals
$$
R = [a_1, b_1] \times [a_2, b_2] \times \cdots \times [a_d, b_d]
$$
![[Pasted image 20230622215919.png]]
The volume of the rectangle $R$ is defined as
$$
|R| = (b_1 - a_1)\cdots(b_d-a_d)
$$
a cube is a rectangle for which $b_1 - a_1 = \cdots = b_d-a_d$, whose volume is $(b_1-a_1)^d$

A union of rectangles is said to be almost disjoint if the interiors of the rectangles are disjoint.

Lemma 1.1

If a rectangle is the almost disjoint union of finitely many other rectangles, say $R = \bigcup_{k=1}^N R_k$, then

$$
|R| = \sum_{k=1}^N |R_k|
$$

proof.
如下图，进行有限次划分，进而变成我们更加容易处理的形式。
$$
|R| = \sum_{j=1}^M |\overline{R_j}| = \sum_{k=1}^N\sum_{j \in J_k}|\overline{R_j}| = \sum_{k=1}^{N}|R_k|
$$

Lemma 1.2

If $R, R_1, \dots , R_N$ are rectangles, and $R \subset \bigcup^N_{k=1} R_k$, then $|R| \le \sum^N_{k=1} |R_k|$.

proof. 和Lemma 1.1相似的方法，将矩形分割，将面积累加。而重叠的部分则是不等的缘由

Lemma 1.3

Every open subset $\mathcal{O}$ of $\mathbb R$ can be writen uniquely as a countable union of disjoint open intervals.

proof.
Consider
$$
a_x = \inf{a<x:(a,x) \subset \mathcal{O}}, b_x = \inf{b>x:(x,b) \subset \mathcal{O}}
$$
Since $I_x = (a_x,b_x)$ is a subset of $\mathcal O$, then we have
$$
\mathcal O = \bigcup_{x\in\mathcal O} I_x
$$
If $I_x \cap I_y \not = \emptyset$, then $I_x \cup I_y$ is also a interval. Then we have $I_x \cup I_y \subset I_x$ and $I_x \cup I_y \subset I_x$ by definition. This only happen when $I_x = I_y$.
Since every open interval contain a rational number, we can choose disjoint rational number for every element in ${I_x}{x \in \mathcal O}$, since they are disjoint.
Therefore, ${I_x}{x \in \mathcal O}$ is countable.

Lemma 1.4

Every open subset $\mathcal O$ of $\mathbb R^d, d \ge 1$, can be written as a countable union of almost disjoint closed cubes.

proof.
先用长度为1的正方形划分$\mathbb R^d$，标记被$\mathcal O$包含的正方形为$Q_{1,1},Q_{1,2},Q_{1,3},\dots$
再使用更细的划分去填补剩下的空间，每一次都将边长变成原来的1/2.
因为每一层都最多使用可数个正方形，而总共有可数层，于是总计使用的正方形数量不会超过可数个。
而对于开集$\mathcal O$中的每一个点$x$, 都存在一个以$x$为中心半径为$\delta$的球被包含在开集$\mathcal O$内, 因此必然存在边长大于$\delta / 2$的划分使得该点被正方形覆盖。
因此，the open set $\mathcal O$ can be written as a countable union of almost disjoint closed cubes

康托尔集 Cantor set

The total length of Cantor set is 0. （但这个说法对于现在的我们来说很符合直觉但不严谨，你如何去测量一个有无穷断点的物体的长度呢？

---

外测度 exterior measure

由上面的定理，我们知道，一个相对规整的集合（比如开集）可以表示为可数个正方形的无交并，这预示着我们去利用已知的正方形的体积去测量任意集合的体积
当我们用多个正方形从外侧去逼近一个图形的时候，也许就可以得到一些与面积相关的信息

定义外测度

The precise definition is as follows: if $E$ is any subset of $\mathbb R^d$, the exterior measure of $E$ is
$$
m_*(E) = inf{\sum_{j=1}^\infty |Q_j|}
$$

值得注意

1 这里使用有限和是有可能得不到我们想要的面积的

比如说$[0,1]$中所有有理数构成的集合$E$，在有限和定义的测度下测度为1，但是数学分析告诉我们这应该是一个零测集。

2 事实上可以将正方形替换成长方形，这个定义依然生效

可以试试将长方形用可数个正方形表达出来

试试手

Example 1

The exterior measure of a point is zero.

Example 2

The exterior measure of a closed cube is equal to its volume.

Example 3

If $Q$ is an open cube, the result $m_∗(Q) = |Q|$ still holds.

Example 4

The exterior measure of $\mathbb R^d$ is infinite.

Example 5

The Cantor set $\mathcal C$ has exterior measure $0$.

外测度的性质

观察所得

观察所见，加上少许证明，即为基本的性质。

Obs1

$A \subset B \Rightarrow m_*(A) \le m_*(B)$

因为任何一个$B$的覆盖也是$A$的一个覆盖。

Obs2

若$J$可数，则$m_*(\bigcup_{j \in J} A_j) \le \sum_{j \in J} m_*(A_j)$

By definition, we get a covering $E_j \subset \bigcup_{k=1}^\infty Q_{k,j}$ such that
$$
\sum_{k=1}^{\infty} |Q_{k,j}| < m_*(E_j) + \frac{\epsilon}{2^j}
$$
Then, we have
$$
m_*(E) \le \sum_{j=1}^\infty \sum_{k=1}^\infty|Q_{k,j}| \le \sum_{j=1}^\infty \left(m_*(E) + \frac{\epsilon}{2^j}\right) \le \sum_{j=1}^\infty m_*(E) + \epsilon
$$

Obs3

If $E \subset \mathbb R^d$, then $m_∗(E) = \inf m_∗(\mathcal O)$, where the infimum is taken over all open sets $\mathcal O$ containing $E$.

Obs4

If $E = E_1 \cup E_2$, and $d(E_1, E_2) > 0$, then $m_∗(E) = m_∗(E_1) + m_∗(E_2)$.

By Obs2, we have $m_*(E) \le m_*(E_1) + m_*(E_2)$.

Obs5

If a set $E$ is the countable union of almost disjoint cubes $E = \bigcup_ {j=1}^\infty Q_j$, then

$$
m_∗(E) = \sum_ {j=1}^\infty |Q_j |
$$

测度公理化

1. （Borel性质）$\mathbb R^n$中每一个开集都是可测集，每一个闭集都是可测集。
2. （补性质）如果$\Omega$是可测的，那么$\mathbb R^n / \Omega$也是可测的
3. （Boole代数性质）如果$(\Omega_j){j \in J}$是可测集的有限族，那么$\bigcup{j \in J} \Omega_j$和$\bigcap_{j \in J} \Omega_j$都是可测集
4. （$\sigma$代数性质）如果$(\Omega_j){j \in J}$是可测集的可数族，那么$\bigcup{j \in J} \Omega_j$和$\bigcap_{j \in J} \Omega_j$都是可测集
5. $m(\emptyset) = 0$
6. $0\le m(\Omega) \le \infty$
7. $A \subset B \Rightarrow m(A) \le m(B)$
8. 若$J$可数，则$m(\bigcup_{j \in J} A_j) \le \sum_{j \in J} m(A_j)$
9. 若$(A_j){j\in J}$ 是互不相交的可数族，则$m(\bigcup{j \in J} A_j) =\le= \sum_{j \in J} m(A_j)$
10. （正规化性质）$m([0,1]^n)=1$
11. 若$\Omega$是可测集而$x \in \mathbb R^n$，那么$x+\Omega := {x+y:y\in\Omega}$是可测的，而且$m(x+\Omega) = m(\Omega)$.

We denote a prime $p$, $q = p^n$ and $F$ is a field with $q$ element.

Lemma 1

$x = y^2 \text{ for some } y \in F \Leftrightarrow x^{\frac{q-1}{2}} = 1$

Lemma 2

For any generator $g$ of multiplication group of $F^\times$, there exist a element $x \in F$ such that $(g - x^2) = y^2$ for some $y \in F$.

Consider
$$
\begin{array}{l}
\sum_{x \in F^\times} \left(g - x^2\right)^{\frac{q-1}{2}}
\end{array}
$$
$$
= \sum_{k=1}^{q-1} \left(g - (g^{k})^2\right)^{\frac{q-1}{2}}
$$
$$
=g^{\frac{q-1}{2}} \sum_{k=1}^{q-1} \left(\sum_{i=0}^{\frac{q-1}{2}}\binom{\frac{q-1}{2}}{i} (-g)^{(2k-1)i} \right)
$$
$$
= - \sum_{i=1}^{\frac{q-1}{2}-1}\binom{\frac{q-1}{2}}{i}\sum_{k=1}^{q-1} (-g)^{(2k-1)i} -\binom{\frac{q-1}{2}}{0}\sum_{k=1}^{q-1} (-g)^{0}-\binom{\frac{q-1}{2}}{\frac{q-1}{2}}\sum_{k=1}^{q-1} (-g)^{(2k-1)\frac{q-1}{2}}
$$
$$
= -\sum_{i=1}^{\frac{q-1}{2}-1}\binom{\frac{q-1}{2}}{i}\frac{1-(-g)^{(2q-1)i}}{1-(-g)^{2i}} (-g)^i - (q-1) - (q-1)
$$
$$
= -\sum_{i=1}^{\frac{q-1}{2}-1}\binom{\frac{q-1}{2}}{i} (-g)^i +2
$$
$$
= -(1-g)^{\frac{q-1}{2}} + 1 + (-g)^{\frac{q-1}{2}} + 2
$$
$$
= 3 - (1-g)^{\frac{q-1}{2}} - (-1)^{\frac{q-1}{2}} \in {1,3,5}
$$

If $p \not = 2$, then $\sum_{x \in F^\times} \left(g - x^2\right)^{\frac{q-1}{2}} \not = q-1$ (calculate in $F$). Thus, there at least one element $x$ of $F$ such that $(g-x^2) = y^2$ for some $y \in F$.
If $p = 2$, then every element of $F$ are square.
we finish our proof.

Thm

Every element of $F$ is a sum of two square.

Consider $g$ as a generator of $F^\times$,
By Lemma 2, there exist $a,b \in F$ such that $g = a^2 + b^2$.
for any $x = g^r \in F$, if $r$ is even ,then $x = (g^{\frac{r}{2}})^2 + 0^2$.
If $r$ is odd, $x = (g^{\frac{r-1}{2}})^2 g= ((g^{\frac{r}{2}})^2 + 0^2)(a^2+b^2) = (g^{\frac{r}{2}}a)^2+(g^{\frac{r}{2}}b)^2$.

Hence, every element of $F$ can write as a sum of two square.

In commucative algebra, we usually want to embbed a ring into a field to analysis its property. Remark how we construct $\mathbb Q$ on $\mathbb Z$ (从整数构造有理数域：公理化 - 知乎 (zhihu.com)). We try to find out a method to construct the quotient field of a ring.
In this way, mathmatician came up the localization.

Def of Localization

Let $A$ be a ring, S be a multiplicative subset of A.
Now, we construct from an ordered pair $(a,s)$ with $a \in A, s \in S$. And define a relation on then,
$$
(a,s) \equiv (b,t) \Leftrightarrow (at - bs) u = 0 \text{ for some } u\in S
$$
This is a equivalence relation. It implies a equivalence class set $S^{-1}A = {a/s|\text{the equivalence class of }(a,s) a \in A, s \in S}$.
Furthermore, we can define addiction and multication,
$$
(a/s) + (b/t) = (at + bs)/st
$$
$$
(a/s)(b/t)=ab/st
$$
Then, $S^{-1}A$ satisfies the axioms of a commutative ring with identity.

Similarly, we can define localization of a A-module M as above, and denote as $S^{-1}M$.

局部化究竟保留了什么

Prop 3.11 iv)
The prime ideals of $S^{-1}A$ are in one-to-one correspondence $(\mathfrak{p} \leftrightarrow S^{-1}\mathfrak{p} )$ with the prime ideals of A ideals of $A$ ehich don’t meet $S$.

That is said, quotient ring $A/\mathfrak{a}$ kill the ideals not containing the ideal $\mathfrak{a}$ and keep the ideals containing $\mathfrak{a}$.
And, localization kill the prime ideals intersect $S$, and keep the prime ideals not intersect $S$.

局部化的代数特性

Denote A as a ring, S as a multiplicative closed subset of A.
M,N is A-modole.

Prop 3.3 The operation $S^{-1}$ is exact.

1. $S^{-1}(N+P) = S^{-1}(N) + S^{-1}(P)$
2. $S^{-1}(N \cap P) = S^{-1}(N) \cap S^{-1}(P)$
3. $S^{-1}(M/N) \simeq (S^{-1}M)/(S^{-1}N)$
4. $S^{-1}M \simeq S^{-1}A \otimes_A M$

循环图 Circulant Graphs

之前提到过 cycle，我们这里再把 cycle 拿出来做一个更加详细的定义

n点循环 $C_n$

The cycle on n vertices is the graph Cn with vertex set {0, … , n- 1} and with i adjacent to j if and only if j - i = ±1 mod n.

也就是说，我们利用了同模的概念来表达一种周而复始的“循环递增”。

循环图 circulant graphs

Let $\mathbb{Z}_n$ denote the additive group of integers modulo n. If $C$ is a subset of $\mathbb{Z}_n \setminus 0$, then construct a directed graph $X = X (\mathbb{Z}_n , C)$ as follows. The vertices of X are the elements of $\mathbb{Z}_n$ and $( i, j)$ is an arc of $X$ if and only if $j - i \in C$. The graph $X(\mathbb{Z}_n, C)$ is called a circulant of order $n$, and $C$ is called its connection set.

大致可以这样来理解这套定义，$\mathbb{Z}_n$事实上表达了这张图的点的个数，是自由度，而$C$则是间隔多少个来连边，是连线方式。

对于一个比较大的循环群$R$我们有
$$
\left | \mathrm{Aut}(X) \right | \ge 2n
$$
因为一般的循环图至少可以进行“旋转”与“反转”的操作，这构成一个与$D_{2n}$同构的自同构子群。只是在$n\le2$时有可能出现谬误，因为点数太少了。

平面图 Planar graph

A graph is called planar if it cann be drawn without crossing edges.

对于平面图有一个很经典的结果

欧拉公式

Theorem 1.8.1 (Euler)
If a connected plane graph has $n$ vertices, $e$ edges and $f$ faces,
then $n - e + f = 2$.

证明也很直接。

1. 我们注意到对一个多边形进行三教剖分并不会改变它 $(n-e+f)$ 的结果，因此我们可以假设我们面对的都是已经被三角剖分的图形。
2. 我们又注意到，对于一个已经三角剖分过的平面图删去一个顶点以及与这个顶点相关的边，$(n-e+f)$ 也不会改变，因此我们可以用这个手段来减少顶点数。
3. 递归直到剩下一个三角形（即只剩下三个顶点），我们得知 $(n-e+f)=2$

平面图的性质

Lemma_1

A planar graph with $n$ vertices has no more than $3n-6$ edges.

Proof:
注意到$2e\ge3f$
$f\le\frac 2 3 e$
由$(n-e+f)=2$可知$e-(n-2)=f\le\frac 2 3 e$.
所以我们有$e\ge 3n-6$.

---

利用这些引理，我们可以验证一些图像是否是平面图

$K_5$ 不是一个平面图

Proof：
由Lemma 1即可验证

$K_{3,3}$不是平面图

Proof：
考虑$K_{3,3}$里的顶点数和边数，那么有$n=6,e=9$.
并不满足Lemma 1中给出的不等关系，因此不可能是平面图

对偶图

Given a plane graph $X*$, we can form another plane graph called the dual graph $X*$ . The vertices of $X*$ correspond to the faces of $X$, with each vertex being placed in the corresponding face . Every edge e of X gives rise to an edge of $X*$ joining the two faces of X that contain $e$.

大致就是点变成面，面变成点。只是在变化的时候，可能要把平面图的标准放松一点点。因为两个面之间可能不止有一条共同边。

射影平面

The real projective plane is a nonorientable surface, which can be rep­ resented on paper by a circle with diametrically opposed points identified.

在这个平面上，画出$K_5$便成为可能。

感觉像装了传送门一样

学完拓扑就学代数拓扑这很合理（确信

动机 motivations

代数拓扑是对拓扑空间的定性分析
$$
\begin{array}{c}
\text{Solution sets } \left { x^2+y^2+z^2 = 1 \right } \
\downarrow \
\text{manifolds}\
\downarrow \
\text{topological spaces}
\end{array}
$$

我们将要做什么

• Have machinery that gives us general information about spaces that is insentive to collapse and various other things.
• We also need to decide on a family of topological space that we are interested in Cell complex and CW complexes.

胞腔复形 cell complex

Definitions/notations

• Open $n$-cell: topolical space homemeomorphic to the open unit ball $\mathbb B ^n$
• Close $n$-cell: topolical space $\overline{\mathbb B ^n} = \mathbb B ^n \cup \mathbb S ^{n-1}$.

Prop:
If $D \subset \mathbb R ^n$ is a compact convex subset weith nonempty interior, then $D$ is a closesd $n$-cell and its interior is an open $n$-cell.
In fact, given any point $p \in \mathrm{Int} D$, there exist a homeomorphism $F:\overline{\mathbb B ^n} \to D$ that sends $0$ to $p$, $\overline{\mathbb B ^n}$ to $\mathrm{Int} D$ and $\mathbb S ^{n-1}$ to $\partial D$.

Example

• Every closed interval in $\mathbb R$ is a closed 1-cell
• Every polygons in $\mathbb R ^2$ are closed 2-cells

Cell decompositions

attaching cells to a given topological space $X$:
Given ${D_{\alpha}}_{\alpha \in A}$ and indexed collection of close $n$-cell foe some fixed $n \ge 1$. we can glue them to X via continous maps $\varphi _\alpha: \partial D \to X$ as follows: Let $\varphi :\bigsqcup_\alpha \partial D_\alpha \to X$ be the map restriction to each $\partial D_\alpha$ is $\varphi_\alpha$ We can then form the cadjunction space $X \cup_\alpha (\sqcup _\alpha D_\alpha)$.

Cell decomposition

Def:
If $X$ is a nonempty topological space, a cell decomposition of $X$ is a partition $\xi$ of $X$ into subspaces that are open cells of various dimensions, such that the following condition is satisfied: for each cell $e \in \xi$ of dimension $n \ge 1$, there exists a continuous map $\varPhi$ from some closed $n$-cell $D$ into $X$ (called a characteristic map for $e$) that restricts to a homeomomorphism from $\mathrm{Int}D$ onto $e$ and maps $\partial D$ into the union of all cells of $\xi$ of dimensions strictly less than $n$.

这个书的定义方式是，先给出一个拓扑空间$X$，然后我们用一些手段去展示这些胞腔之间是怎么拼到一起的。

顺便一提，这里事实上只是限制了，分解出来的空间是个开胞腔，我们对于这个区域在$X$中的边界依旧是一无所知。

这个要求保证，如果存在一个胞腔分解，那么我们就可以用之前的办法来构造这个空间. 事实上，$\varPhi$本身给出了一个将这个子部分与其他地方连接的“连接方式”。在有限的情况下，我们可以认为，这其实给出了用一堆胞腔去粘连出这个cell complex(定义如下)的方法，因此，我们可以用胞腔去分析它 了。

Cell complex

Def:
A cell complex is a Hausdorff space $X$ together with a specific cell decomposition of $X$.

Finite cell complex

Def:
A finite cell complex is one whose cell decomposition has only finitely many cells.

local finite

Def:
A cell complex is called locally finite if the connection of open cell is “locally finite”.

CW complex

CW complex

Def:
A CW complex is a complex $(X,\xi)$ satisfying:
(C, closure finiteness): The closure of each cell is contained in a union of finitely many cells.
(W, weak topology): The topology of X is coherent with the family of closed subspaces ${\overline e | e \in \xi}$

向量空间的定向 Orientation

Def:
$e = \left { e_i \right } _{i = 1} ^n$, $f = \left { f_i \right } _{i = 1} ^n$ is order base.
We say $e$ and $f$ have the same orientation if the matrix of change of basis has position determinant. Denoted as $e \sim f$.

$e \sim f$ 事实上是一个等价关系，因为它满足等价关系的三个条件。

Remark:
等价关系的三个条件：

1. $e \sim e$
2. $e \sim f \Rightarrow f \sim e$
3. $e \sim f, f \sim g \Rightarrow e \sim g$

由此，它把向量空间分成了两个等价类。

外积

Def: Vector product (Cross product) in $\mathbb R ^3$
For $v,u \in \mathbb R ^3$, the vector product of u and v is the unique vector $u\wedge v \in \mathbb R^3$

用行列式表达的话，
$$
u\wedge v = \begin{vmatrix}
u_1 & u_2 &u_3 \
v_1 & v_2 &v_3 \
\hat{e_1} & \hat{e_2} &\hat{e_3}
\end{vmatrix}
$$
由此，由行列式的性质，我们可以得到以下运算规律

1. $u\wedge v = -v\wedge u$
2. $u\wedge v = 0 \Leftrightarrow u,v\text{线性相关}$
3. $(u\wedge v)\cdot u = 0,(u\wedge v)\cdot v = 0$
4. $(u \wedge v) \wedge w = (u \cdot w)v - (v \cdot w)u$

几何意义

$(u\wedge v)$事实上是$u$和$v$生成平面的一个法向量。

Lagrange’s identity

$$
(u\wedge v) \cdot (x \wedge y) = \begin{vmatrix}
u\cdot x & v\cdot x\
u\cdot y & v\cdot y
\end{vmatrix}
$$

proof

因为外积与内积都是线性运算，因此我们只需要对一组积进行验证即可
$$
(e_i\wedge e_j) \cdot (e_k \wedge e_l) = \begin{vmatrix}
e_i\cdot e_k & e_j\cdot e_k\
e_i\cdot e_l & e_j\cdot e_l
\end{vmatrix}
$$

外积的模长

用拉格朗日的公式，我们即可获得
$$
\left | u \wedge v \right | ^2 = \begin{vmatrix}
u\cdot u & v\cdot u\
u\cdot v & v\cdot v
\end{vmatrix} = |u|\cdot|v|(1-\cos ^2(\theta))
$$

---

对于单个点附近的函数变化的研究，往往会求助于线性代数。
但当这些局部性质沿着曲线动起来的时候，就会出现微积分了。

---

微分下的结构变换

$$
\frac{\mathrm{d} }{\mathrm{d} t} \left ( u(t) \wedge v(t) \right ) = \left ( \frac{\mathrm{d} }{\mathrm{d} t} u(t) \right ) \wedge v(t) + u(t) \wedge \left ( \frac{\mathrm{d} }{\mathrm{d} t} v(t) \right )
$$

自同构 isomorphism

定义

An isomorphism from a graph $X$ to itself is called an automorphism of $X$. An automorphism is therefore a permutation of the vertices of $X$ that maps edges to edges and nonedges to nonedges.

所谓自同构，就是自己到自己的一个同构。

至于为什么要考虑自同构，是因为我们需要利用这里天然形成的一个群结构。也就是所谓“代数”。

自同构群 isomorphism group

The set of all automorphisms of $X$ forms a group, which is called the automorphism group of $X$ and denoted by $\mathrm{Aut}( X)$.

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什么是图 Graph

A graph $X$ consits of a vertex set $V(X)$ and an edgeset $E(X)$, where an edge is an unordered pair of distinct vertices of $X$.

图$X$由两部分组成，一部分是顶点集$V(X)$，另一部分是边集$E(X)$。每条边本质上都是包含两个顶点的无序集。

依照边的关系所自然导出的一个拓扑，事实上正是这里的离散拓扑。因此，有限图在自然导出的拓扑下是一个零维空间。

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曲线 Curve

事实上，对于三维空间里的任意曲线，很有可能会变得任意复杂，或者有无限多个拐角。这会让事情变得很复杂。也许可以研究，但是，我们现在还是从比较简单的情况开始入手。

参数化曲线 Parametrized differential Curves

Def A parametrized differntiable curve is a differential map $\alpha:I \to \mathbb{R}^3$ of an open interval $I = (a,b)$ of the real line $R$ into $R$.

简单来说就是，整个曲线可以用一个从R上的开区间$(a,b)$到$\mathbb{R}^3$的可微函数$\alpha$来表示。这里的开区间并没有排除掉整个实数轴的情况。

参数化之后的一个显而易见的好处就是可以应用我们的分析工具了。

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高代上的一些不等式

假设

如无特殊说明，一下符号即代表如下含义

矩阵

$$
A \in \mathrm {M} _{m \times n} (K),B \in \mathrm {M} {n \times l} (K),C \in \mathrm{M}{l \times t},P \in \mathrm {M} _{s \times m} (K)
$$

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