2.1 線性變換將面積伸縮
對於一個\(\R^2\to\R^2\)的線性變換：
\[T(x,y)= \left[ \begin{array}{c} 4x-2y\\ 2x+3y \end{array} \right] \]
設區域\(S_1=\{(x,y)|0\leq x,y\leq1\}\)，若想要求\(\iint_{S_1}T(x,y)\ d\sigma\).可以通過基底表示單位正方形：\(e_1=(1,0)',e_2=(0,1)'\)，則：
\[S_1=\{xe_1+ye_2|0\leq x,y\leq1\} \]
設\(A\)為線性變換\(T\)參考標准基地的表示矩陣，即有：
\[T(xe_1+ye_2)=A(xe_1+ye_2)=xAe_1+yAe_2=xa_1+ya_2 \]
於是：
\[T(S_1)=\{xa_1+ya_2|0\leq x,y\leq1\} \]
這表明\(T(S_1)\)是以\(A=(a_1,a_2)\)表示的平行四邊形，二階行列式的絕對值為平行四邊形的面積，因此\(v(T(S_1))=|detA|\)。這個結果表明平行四邊形\(S_1\)經過線性變換\(T\)，面積伸縮了\(|detA|\)倍。
2.2 Jacobian行列式的意義
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\(r_1=(du,0)',r_2=(0,dv)'\)表示長方形，則\(F(R)=\{F(u)|u\in R\}\)近似如下面向量所表示的平行四邊形：
\[J(u,v)(du,0)'= \left| \begin{matrix} \frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\ \end{matrix} \right|(du,0)'= \left| \begin{matrix} \frac{\partial x}{\partial u}du\\ \frac{\partial y}{\partial u}du \end{matrix} \right|\\ J(u,v)(0,dv)'=\left|\begin{matrix}\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\\end{matrix}\right|(0,dv)'=\left|\begin{matrix}\frac{\partial x}{\partial v}dv\\\frac{\partial y}{\partial v}dv\end{matrix}\right| \]
若令\(dA\)表示平行四邊形\(F(R)\)的面積， 因為二階行列式的行向量所形成的平行四邊形面積等於行列式的絕對值，則：
\[dA=\left| det \left[ \begin{matrix} \frac{\partial x}{\partial u}du&\frac{\partial x}{\partial v}dv\\ \frac{\partial y}{\partial u}du&\frac{\partial y}{\partial v}dv\\ \end{matrix} \right] \right|= \left| det \left[ \begin{matrix} \frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\ \end{matrix} \right] \right|dudv=|det\ J(u,v)|dudv \]
所以微笑區域\(R\)經過向量函數\(F:R\to F(R)\)，其面積伸縮了\(|det\ J(u,v)|\)倍。對於\(f:\R^2\to\R\)我們可以得出變換積分公式：
\[\int_{F(R)} f(x,y)dxdy=\int_{R} f(x(u,v),y(u,v))\left|J(u,v)\right|dudv\\ |J(u,v)|=\left|\frac{\partial(x,y)}{\partial(u,v)} \right| \]