工科数学分析-多重积分换元法
前置
雅可比行列式
\[\frac{\partial \left(F_1, F_2, \dots, F_n\right)}{\partial\left(x_1, x_2, \dots, x_i\right)} = \begin{vmatrix} \dfrac{\partial F_1}{\partial x_1} & \dfrac{\partial F_1}{\partial x_2} & \ldots & \dfrac{\partial F_1}{x_i} \\ \\ \dfrac{\partial F_2}{\partial x_1} & \dfrac{\partial F_2}{\partial x_2} & \ldots & \dfrac{\partial F_2}{\partial x_i} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \dfrac{\partial F_n}{\partial x_1} & \dfrac{\partial F_n}{\partial x_2} & \dots & \dfrac{\partial F_n}{\partial x_i} \end{vmatrix}\]多元积分换元法
以二元积分为例
\[\iint_R f(x, y)\mathrm{d}x\mathrm{d}y = \iint_G f(g(u, v), h(u, v)) \left| \frac{\partial \left(x, y\right)}{\partial \left(u, v\right)}\right| \mathrm{d}u\mathrm{d}v\]其中的雅可比行列式称为伸缩系数，以下为常见的伸缩系数
二重积分
极坐标系
从直角坐标系转换到极坐标系下
\[\mathrm{d}x\mathrm{d}y = r\mathrm{d}r\mathrm{d}\theta\]