# 数学分析 Mathematical Analysis  MA140-10/MA152-15

Proof As is shown in linear algebra, the matrix $A$ that represents $T$ is a product of elementary matrices
$$A=E_{1} \cdots E_{k} .$$
Each elementary $2 \times 2$ matrix is one of the following types:
$$\left[\begin{array}{ll} \lambda & 0 \ 0 & 1 \end{array}\right] \quad\left[\begin{array}{ll} 1 & 0 \ 0 & \lambda \end{array}\right] \quad\left[\begin{array}{ll} 0 & 1 \ 1 & 0 \end{array}\right] \quad\left[\begin{array}{ll} 1 & \sigma \ 0 & 1 \end{array}\right]$$
where $\lambda>0$. The first three matrices represent isomorphisms whose effect on $I^{2}$ is obvious: $I^{2}$ is converted to the rectangles $\lambda I \times I, I \times \lambda I, I^{2}$. In each case, the area agrees with the magnitude of the determinant. The fourth isomorphism converts $I^{2}$ to the parallelogram$\Pi$ is Riemann measurable since its boundary is a zero set. By Fubini’s Theorem, we get
$$|\Pi|=\int \chi_{\Pi}=\int_{0}^{1}\left[\int_{x=\sigma x}^{x=1+\sigma y} 1 d x\right] d y=1=\operatorname{det} E .$$

## MA140-10/MA152-15 COURSE NOTES ：

$$d x_{I}: \varphi \mapsto \int_{l^{k}} \frac{\partial \varphi_{I}}{\partial u} d u$$
where this integral notation is shorthand for
$$\int_{0}^{1} \ldots \int_{0}^{1} \frac{\partial\left(\varphi_{i_{1}}, \ldots, \varphi_{i_{k}}\right)}{\partial\left(u_{1}, \ldots, u_{k}\right)} d u_{1 \ldots} . d v_{k}$$
If $f$ is a smooth function on $\mathbb{R}^{n}$ then $f d x_{l}$ is the functional
$$f d x_{I}: \varphi \mapsto \int_{I^{k}} f(\varphi(u)) \frac{\partial \varphi_{I}}{\partial u} d u$$