向量

$\rm vector$ 表示

$\vec{a}/\mathbf{a}$

$||\vec{a}||$为向量的长度

在直角坐标系中

$||\vec{a}||=\sqrt{x^2+y^2}\\ 单位向量\hat{a}=\vec{a}/||\vec{a}||$

两向量之间的夹角为$0\le\left \langle \alpha,\beta \right \rangle \le \pi$

点乘

$\rm Dot\;Product$

$\vec{a} \cdot \vec{b}=\|\vec{a}\|\|\vec{b}\| \cos \theta =x_1x_2+y_1y_2 \\ \cos \theta=\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|}\\ \theta=\arccos (\hat{a}\cdot\hat{b})$

矩阵

$\vec{a} \cdot \vec{b}=a^Tb$

运算性质

$交换律：\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a} \\ 分配律：\vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c} \\ (k \vec{a}) \cdot \vec{b}=\vec{a} \cdot(k \vec{b})=k(\vec{a} \cdot \vec{b})$

叉乘

$\rm Cross \;Product$

$||\vec{a}\times\vec{b}||=||\vec{a}||||\vec{b}||\sin\theta$

运算性质

$\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}\\ \vec{k}=\vec{i}\times \vec{j}$ $\begin{array}{l}\vec{a} \times \vec{b}&=\left(\begin{array}{c} y_{a} z_{b}-y_{b} z_{a} \\ z_{a} x_{b}-x_{a} z_{b} \\ x_{a} y_{b}-y_{a} x_{b} \end{array}\right)=\begin{vmatrix} i& j & k\\ x_a &y_a &z_a \\ x_b & y_b &z_b \end{vmatrix}\\ &=\begin{bmatrix} 0& -z_a & y_a\\ z_a &0 &-x_a \\ y_a & x_a &0 \end{bmatrix}\begin{bmatrix} x_b\\ y_b \\ z_b \end{bmatrix}=[a]_\times b\end{array}\\ [a]_\times+[a]_\times^\text T=0$

混合积

$[\vec{a}\;\vec{b}\; \vec c]=\vec{a}\times\vec{b}\cdot \vec c\\ =\begin{vmatrix} x_c& y_c & z_c\\ x_a &y_a &z_a \\ x_b & y_b &z_b \end{vmatrix}$

根据Levi-Civita

$\vec{a}\times\vec{b}\cdot \vec c=\vec{b}\times\vec{c}\cdot \vec a=\vec{c}\times\vec{a}\cdot \vec b$

范数

$||\vec{x}||_p=\sqrt[p]{\sum_{i=1}^n|x_i|}$ $\|x\|_{1}=\left|x_{1}\right|+\cdots+\left|x_{n}\right| \\ \|x\|_{2}=\sqrt{\left|x_{1}\right|^{2}+\cdots+\left|x_{n}\right|^{2}}=\sqrt{x^{H} x} \\ \|x\|_{\infty}= \underset{1 \leq i \leq n}\max|x_{i}|$

表示

根据三角形法则，对于任何一个向量

都可以分解为两个基向量的线性组合

$\vec{OP}=x\boldsymbol{i}+y\boldsymbol{j}$

所有线性组合构成集合$\text{span}(\boldsymbol{i},\boldsymbol{j})$，称为张成空间

这就是向量的坐标表示法

$\vec{OP}=\left(\begin{array}l x\\y\end{array}\right)\\ \boldsymbol{i}=\left(\begin{array}l 1\\0\end{array}\right),\boldsymbol{j}=\left(\begin{array}l 0\\1\end{array}\right)$

直线的向量方程

对于直线，有如下关系

$\vec{OR}=\vec{OA}+t\vec{AR}\quad(t\in \mathbf{R})$

其中非零向量$\vec{AR}$为该直线的方向向量

写成坐标形式，可得

$\left(\begin{array}l x\\y\end{array}\right)=\left(\begin{array}l x_0\\y_0\end{array}\right)+t\left(\begin{array}l v_1\\v_2\end{array}\right)$

导数

求导的本质只是把标量求导的结果排列起来，至于是按行排列还是按列排列都是可以的

最基本的求导布局有两个：

分子布局(numerator layout)和分母布局(denominator layout )

机器学习主要为分子布局

分子布局：分子为列向量，分母为行向量

$\mathbf{x}=\begin{bmatrix}x_1\\x_2\\ \vdots \\x_n\end{bmatrix} \\\frac{\partial y}{\partial \mathbf{x}}=\begin{bmatrix}\frac{\partial y}{\partial x_1} \frac{\partial y}{\partial x_2} \cdots \frac{\partial y}{\partial x_n} \end{bmatrix}$

方向导数，定义的$\rm shape$，例

$\frac{\partial}{\partial \mathbf{x}}(x_1^2+2x_2^2)=[2x_1,4x_2]$

求导运算是线性算子

$\mathbf{y}=\left[\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{m} \end{array}\right] \quad \frac{\partial \mathbf{y}}{\partial x}=\left[\begin{array}{c} \frac{\partial y_{1}}{\partial x} \\ \frac{\partial y_{2}}{\partial x} \\ \vdots \\ \frac{\partial y_{m}}{\partial x} \end{array}\right]$

雅可比矩阵

$\text{jacobi matrix} \\\frac{\partial \mathbf{y}}{\partial \mathbf{x}}=\left[\begin{array}{c} \frac{\partial y_{1}}{\partial \mathbf{x}} \\ \frac{\partial y_{2}}{\partial \mathbf{x}} \\ \vdots \\ \frac{\partial y_{m}}{\partial \mathbf{x}} \end{array}\right]=\left[\begin{array}{c} \frac{\partial y_{1}}{\partial x_{1}}, \frac{\partial y_{1}}{\partial x_{2}}, \ldots, \frac{\partial y_{1}}{\partial x_{n}} \\ \frac{\partial y_{2}}{\partial x_{1}}, \frac{\partial y_{2}}{\partial x_{2}}, \ldots, \frac{\partial y_{2}}{\partial x_{n}} \\ \vdots \\ \frac{\partial y_{m}}{\partial x_{1}}, \frac{\partial y_{m}}{\partial x_{2}}, \ldots, \frac{\partial y_{m}}{\partial x_{n}} \end{array}\right]$

线性回归

$x , w \in \mathbf{R }^ { n } , y \in \mathbf{R } \\ z = (\left \langle \mathbf{x} ,\mathbf{w} \right \rangle - y ) ^ { 2 }$ $\begin{aligned}\frac{\partial z}{\partial \mathbf{w}}\end{aligned}=2(\left \langle \mathbf{x} ,\mathbf{w} \right \rangle - y )\mathbf{w}^{\rm T}$

求导

采用分子布局

$\frac{\partial A x}{\partial x}=A\\ \frac{\partial A x}{\partial x^{T}}=A^{T} \\ \frac{\partial x^{T} A}{\partial x}=A^{T}$

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场论

$\vec{F}(x,y,z)=f_x(x,y,z)\mathbf{i}+f_y(x,y,z)\mathbf{j}+f_z(x,y,z)\mathbf{k}$

梯度

$(x',y')=F(x,y)$

向量场里每个点赋予一个向量

$\nabla=\begin{bmatrix}\frac{\partial }{\partial x_1} \frac{\partial}{\partial x_2} \cdots \frac{\partial }{\partial x_n}\end{bmatrix}$

散度

$\text{div}F(x,y)=\nabla\cdot F$

描述净流量，若流入大于流出则为负

$(x,y)$点生成的流量

例如对于不可压缩的流体

$\text{div}V=0$

旋度

$\text{curl}F(x,y) =\nabla\times F$

$\begin{array}{l}\nabla\cdot F= {\left[\begin{array}{c} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{array}\right] \cdot\left[\begin{array}{c} \mathbf{F}_{x} \\ \mathbf{F}_{y} \end{array}\right]=\frac{\partial F_{x}}{\partial x}+\frac{\partial F_{y}}{\partial y}} \\ {\nabla\times F=\left[\begin{array}{c} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{array}\right] \times\left[\begin{array}{c} \mathbf{F}_x \\ \mathbf{F}_{y} \end{array}\right]=\frac{\partial F_{y}}{\partial x}-\frac{\partial F_{x}}{\partial y}} \end{array}$