雅克比行列式与二重积分换元法

雅可比行列式

现有函数

f_1(x_1,x_2,x_3,\cdots,x_n)\\f_2(x_1,x_2,x_3,\cdots,x_n)\\f_3(x_1,x_2,x_3,\cdots,x_n)\\\cdots\\f_m(x_1,x_2,x_3,\cdots,x_n)

则雅可比行列式为:

J=\left|\begin{array}{cc} \dfrac{\partial f_1}{\partial x_1}&\dfrac{\partial f_1}{\partial x_2}&\dfrac{\partial f_1}{\partial x_3}&\cdots&\dfrac{\partial f_1}{\partial x_n}\\ \dfrac{\partial f_2}{\partial x_1}&\dfrac{\partial f_2}{\partial x_2}&\dfrac{\partial f_2}{\partial x_3}&\cdots&\dfrac{\partial f_2}{\partial x_n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial f_m}{\partial x_1}&\dfrac{\partial f_m}{\partial x_2}&\dfrac{\partial f_m}{\partial x_3}&\cdots&\dfrac{\partial f_m}{\partial x_n}\\ \end{array}\right|

与定积分换元法对比

定积分换元法 二重积分换元法
\int_{a}^{b} f(x) \mathrm{d} x \xlongequal{x=\varphi(t)} \int_{\alpha}^{\beta} f[\varphi(t)] \varphi^{\prime}(t) \mathrm{d} t \iint_{D x y} f(x, y) \mathrm{d} x \mathrm{~d} y \xlongequal[y=y(u, v)]{x=x(u, v)}\iint_{D_{u v}} f[x(u, v), y(u, v)]\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v

a. f(x) \rightarrow f[\varphi(t)]
b. \int_{a}^{b} \rightarrow \int_{\alpha}^{\beta}
c. \mathrm{d} x \rightarrow \varphi^{\prime}(t) \mathrm{d} t

a. f(x, y) \rightarrow f[x(u, v), y(u, v)]
b. \iint_{D x y} \rightarrow \iint_{D u v}
c. \mathrm{d} x \mathrm{~d} y \rightarrow\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v

\mathrm{d} x \rightarrow \varphi^{\prime}(t) \mathrm{d} t 的意思是 \mathrm{d} x 每改变一个单位, \mathrm{d} t 改变 \varphi^{\prime}(t) 个单位。

\mathrm{d} x \mathrm{~d} y \rightarrow\left\vert\frac{\partial(x, y)}{\partial(u, v)}\right\vert \mathrm{d} u \mathrm{~d} v 的意思是 \mathrm{d} x \mathrm{~d} y 每改变一个单位, \mathrm{d} x \mathrm{~d} y 改变 \left\vert\frac{\partial(x, y)}{\partial(u,v)}\right\vert 个单位。

换元后面积改变量

作换元

x=x(u, v)\\y=y(u, v)

可以看到x与u相关也与v相关,这相当于两个一元换元的和,dx与du的比例关系是 \dfrac{\partial x}{\partial u} ,dx与dv的比例关系是 \dfrac{\partial x}{\partial v} ,y也是同样的道理,所以有:

\left[\begin{array}{l}d x \\ d y\end{array}\right]=\left[\begin{array}{l}\dfrac{\partial x}{\partial u} d u+\dfrac{\partial x}{\partial v} d v \\ \dfrac{\partial y}{\partial u} d u+\dfrac{\partial y}{\partial v} d v\end{array}\right]=\left[\begin{array}{ll}\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{array}\right]\left[\begin{array}{l}d u \\ d v\end{array}\right]

整理成矩阵相乘的形式,我们发现其本质就是一个线性变换,因为当x和y都趋近无限小的时候,面积元会趋近于平行四边形,所以从微分层面来看换元就是线性变换。

行列式的几何意义是线性变换后面积的变换,你会发现这个矩阵的行列式就是雅可比行列式:

J=\left|\begin{array}{ll}\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}\end{array}\right|

二重积分换元法还要在雅可比行列式外面加个绝对值,这是为了保证积分下限小于积分上限。

卷积公式

其实卷积公式就是二重积分换元法,例如:

Z=X+Y

可以如下换元:

\iint_{D x y} f(x, y) \mathrm{d} x \mathrm{~d} y \xlongequal[y=z-x]{x=x}\iint_{D_{x z}} f(x, z-x)\left\vert\frac{\partial(x, y)}{\partial(x, z)}\right\vert \mathrm{d} x \mathrm{~d} z

这相当于,换元后的联合概率密度 g(x,z) 为:

g(x,z) = f(x, z-x)\left\vert\frac{\partial(x, y)}{\partial(x, z)}\right\vert

于是我们可以用 g(x,z) ,来求边缘概率密度 f_{Z}(z)

f_{Z}(z) =\int_{-\infty}^{+\infty} g(x,z) \mathrm{d} x

(2007年真题)设二维随机变量 (X, Y) 的概率密度为

f(x, y)= \begin{cases}2-x-y, & 0<x<1,0<y<1, \\[1ex] 0, & \text { 其他. }\end{cases}

(1) 求 P\{X>2 Y\} ;
(2) 求 Z=X+Y 的概率密度 f_{Z}(z) .

posted @ 2021-09-19 17:05:52
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