雅可比行列式【2】Jacobian行列式的意义

2.1 线性变换将面积伸缩

对于一个\(\R^2\to\R^2\)的线性变换：
\[
T(x,y)=
\left[
\begin{array}{c}
4x-2y\\
2x+3y
\end{array}
\right]
\]
设区域\(S_1=\{(x,y)|0\leq x,y\leq1\}\)，若想要求\(\iint_{S_1}T(x,y)\ d\sigma\).可以通过基底表示单位正方形：\(e_1=(1,0)',e_2=(0,1)'\)，则：
\[
S_1=\{xe_1+ye_2|0\leq x,y\leq1\}
\]
设\(A\)为线性变换\(T\)参考标准基地的表示矩阵，即有：
\[
T(xe_1+ye_2)=A(xe_1+ye_2)=xAe_1+yAe_2=xa_1+ya_2
\]
于是：
\[
T(S_1)=\{xa_1+ya_2|0\leq x,y\leq1\}
\]
这表明\(T(S_1)\)是以\(A=(a_1,a_2)\)表示的平行四边形，二阶行列式的绝对值为平行四边形的面积，因此\(v(T(S_1))=|detA|\)。这个结果表明平行四边形\(S_1\)经过线性变换\(T\)，面积伸缩了\(|detA|\)倍。

2.2 Jacobian行列式的意义

if \(F:\R^n\to\R^n\) is derivable, then the Jacobian matrix is in \(n\times n\) form in which we could express a number of it. We set the n is equal to 2, and vector function is: \(F:u\to x\)
\[
det\ J(u,v)=
\left|
\begin{matrix}
\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\
\end{matrix}
\right|=\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial v}
\]
若令\(R=\{r_1,r_2\}\)，其中\(r_1=(du,0)',r_2=(0,dv)'\)表示长方形，则\(F(R)=\{F(u)|u\in R\}\)近似如下面向量所表示的平行四边形：
\[
J(u,v)(du,0)'=
\left|
\begin{matrix}
\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\
\end{matrix}
\right|(du,0)'=
\left|
\begin{matrix}
\frac{\partial x}{\partial u}du\\
\frac{\partial y}{\partial u}du
\end{matrix}
\right|\\
J(u,v)(0,dv)'=\left|\begin{matrix}\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\\end{matrix}\right|(0,dv)'=\left|\begin{matrix}\frac{\partial x}{\partial v}dv\\\frac{\partial y}{\partial v}dv\end{matrix}\right|
\]
若令\(dA\)表示平行四边形\(F(R)\)的面积， 因为二阶行列式的行向量所形成的平行四边形面积等于行列式的绝对值，则：
\[
dA=\left|
det
\left[
\begin{matrix}
\frac{\partial x}{\partial u}du&\frac{\partial x}{\partial v}dv\\
\frac{\partial y}{\partial u}du&\frac{\partial y}{\partial v}dv\\
\end{matrix}
\right]
\right|=
\left|
det
\left[
\begin{matrix}
\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\
\end{matrix}
\right]
\right|dudv=|det\ J(u,v)|dudv
\]
所以微笑区域\(R\)经过向量函数\(F:R\to F(R)\)，其面积伸缩了\(|det\ J(u,v)|\)倍。对于\(f:\R^2\to\R\)我们可以得出变换积分公式：
\[
\int_{F(R)} f(x,y)dxdy=\int_{R} f(x(u,v),y(u,v))\left|J(u,v)\right|dudv\\
|J(u,v)|=\left|\frac{\partial(x,y)}{\partial(u,v)}
\right|
\]