行列式的定义及简单计算

行列式是一个数值，虽然形式上看起来比较复杂，比较不象数值；
- 将行列式可以转化为加减乘除的四则运算，同样也可以将四则运算转换为行列式的形式，这样转化的好处在什么呢，利用行列式的性质对其进行快速计算；

1. 三阶行列式的计算

2. 雅克比行列式到极坐标

从Jacobian矩阵、Hessian矩阵到Theano实现
雅可比行列式在积分坐标变换中的应用

Jn=∂(y1,y2,…,yn)∂(x1,x2,…,xn)=∣∣∣∣∣∣∣∣∣∣∂f1∂x1∂f2∂x1⋯∂fn∂x1∂f1∂x2∂f2∂x2⋯∂fn∂x2⋯⋯⋯⋯∂f1∂xn∂f2∂xn⋯∂fn∂xn∣∣∣∣∣∣∣∣∣∣

J_n=\frac{\partial (y_1,y_2,\ldots,y_n)}{\partial(x_1,x_2,\ldots,x_n)}=\begin{vmatrix} \frac{\partial f_1}{\partial x_1}&\frac{\partial f_1}{\partial x_2}&\cdots&\frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}&\frac{\partial f_2}{\partial x_2}&\cdots&\frac{\partial f_2}{\partial x_n} \\ \cdots&\cdots&\cdots&\cdots\\ \frac{\partial f_n}{\partial x_1}&\frac{\partial f_n}{\partial x_2}&\cdots&\frac{\partial f_n}{\partial x_n} \end{vmatrix}

雅克比行列式的每一行表示多元函数关于各个成员的偏导数；

对于二重积分而言，设极坐标变换 x=ρcosθ,y=ρsinθx=\rho \cos\theta,y=\rho \sin\theta，则其二阶行列式为：

∣∣∣cosθsinθ−ρsinθρcosθ∣∣∣=ρ

\begin{vmatrix} \cos\theta & -\rho\sin\theta\\ \sin\theta&\rho\cos\theta \end{vmatrix}=\rho

所以在积分运算中，便存在如下的变换形式：

∫∫Dxyf(x,y)dxdy=∫∫Dρ,θf[ρcosθ,ρsinθ]ρdρdθ

\int\int_{D_{xy}}f(x,y)dxdy = \int\int_{D_{\rho,\theta}}f\left[\rho\cos\theta,\rho\sin\theta\right]\rho d\rho d\theta

4. 计算

化简如下的表达式：

y=sin2xsin2ysin2z+sin(x+y)sin(y+z)sin(x+z)+sin(x+z)sin(y+z)sin(y+x)−sin(y+x)sin2zsin(x+y)−sin(y+z)sin(z+y)sin2x−sin(z+x)sin(x+z)sin2y

\begin{split} y=&\sin 2x\sin 2y\sin 2z+\sin(x+y)\sin(y+z)\sin(x+z)+\\ &\sin(x+z)\sin(y+z)\sin(y+x)-\sin(y+x)\sin2z\sin(x+y)-\\ &\sin(y+z)\sin(z+y)\sin 2x-\sin(z+x)\sin(x+z)\sin2y \end{split}

将其转化为行列式形式：

y==∣∣∣∣∣sin2xsin(y+x)sin(z+x)sin(x+y)sin2ysin(z+y)sin(x+z)sin(y+z)sin2z∣∣∣∣∣∣∣∣∣∣∣sinxcosx+sinxcosxsinycosx+sinxcosysinzcosx+sinxcoszsinxcosy+sinycosxsinycosy+sinycosysinzcosy+sinycoszsinxcosz+sinzcosxsinycosz+sinzcosysinzcosz+sinzcosz∣∣∣∣∣∣

\begin{split} y=&\begin{vmatrix} \sin2x&\sin(x+y)&\sin(x+z)\\ \sin(y+x)&\sin2y&\sin(y+z)\\ \sin(z+x)&\sin(z+y)&\sin2z \end{vmatrix}\\ =&\begin{vmatrix}\\ \sin x\cos x+\sin x\cos x&\sin x\cos y+\sin y\cos x&\sin x\cos z+\sin z\cos x\\ \sin y\cos x+\sin x\cos y&\sin y\cos y+\sin y\cos y&\sin y\cos z+\sin z\cos y\\ \sin z\cos x+\sin x\cos z&\sin z\cos y+\sin y\cos z&\sin z\cos z+\sin z\cos z \end{vmatrix} \end{split}

然后将行列式（两个方阵和的行列式）拆分为 6 个小行列式，其结果是十分清晰的，那就是 0.