工程力学部分知识点

极坐标

$$\left[\begin{array}{c}Px\\ Py\end{array}\right]=\left[\begin{array}{cc}c\theta & -s\theta \\ s\theta & c\theta \end{array}\right]\left[\begin{array}{c}Pr\\ R\theta \end{array}\right]$$其中$$\beta =\left[\begin{array}{cc}c\theta & s\theta \\ -s\theta & c\theta \end{array}\right]$$$${\beta }^{-1}=\left[\begin{array}{cc}c\theta & -s\theta \\ s\theta & c\theta \end{array}\right]$$

所以$$P^{2}x=(Prc\theta -R\theta s\theta {)}^2=P^{2}r{c}^2\theta -PrR\theta (2s\theta c\theta )+Rr{s}^2\theta +\mathbb{R}\theta (2s\theta c\theta )+R^2\theta s^2\theta$$

重调和方程就是$$(P^{2}r+Rr+R^2\theta {)}^2\varphi =0$$

拉普拉阿斯算子$${\nabla }^2=P^{2}r+Rr+R^2\theta$$