请给出Laplace算符在柱坐标下的表示，即$x=\rho\cos\varphi,y=\rho\sin \varphi,z=z$时的$\nabla^2=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}$的表示
解答
\[\rho^2=x^2+y^2,\dfrac{y}{x}=\tan \varphi,z=z\\ \dfrac{\partial}{\partial x}=\dfrac{\partial}{\partial \rho}\dfrac{\partial \rho}{\partial x}+\dfrac{\partial}{\partial \varphi}\dfrac{\partial \varphi}{\partial x}+\dfrac{\partial}{\partial z}\dfrac{\partial z}{\partial x}=\dfrac{x}{\rho}\dfrac{\partial}{\partial \rho}+(-\dfrac{y}{\rho^2})\dfrac{\partial}{\partial \varphi}\\ [Explain:对\dfrac{y}{x}=\tan \varphi两边对x求偏导可得\dfrac{\partial \varphi}{\partial x}=-\dfrac{1}{x^2}\cos^2\varphi=-\dfrac{y}{\rho^2}]\\ \dfrac{\partial}{\partial y}=\dfrac{\partial}{\partial \rho}\dfrac{\partial \rho}{\partial y}+\dfrac{\partial}{\partial \varphi}\dfrac{\partial \varphi}{\partial y}+\dfrac{\partial}{\partial z}\dfrac{\partial z}{\partial y}=\dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}+\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\\ \therefore \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}\\=\left(\dfrac{x}{\rho}\dfrac{\partial}{\partial \rho}-\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\left(\dfrac{x}{\rho}\dfrac{\partial}{\partial \rho}-\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\\+\left( \dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}+\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\left( \dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}+\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\\ =\dfrac{x}{\rho }\dfrac{\partial}{\partial \rho}\left(\dfrac{x}{\rho }\dfrac{\partial}{\partial \rho} \right)-\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\left(\dfrac{x}{\rho }\dfrac{\partial}{\partial \rho} \right)-\dfrac{x}{\rho }\dfrac{\partial}{\partial \rho}\left(\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)+\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\left(\dfrac{y}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\\ +\dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}\left(\dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}\right)+\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi} \left(\dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}\right)+\dfrac{y}{\rho}\dfrac{\partial}{\partial \rho}\left(\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)+\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\left(\dfrac{x}{\rho^2}\dfrac{\partial}{\partial \varphi}\right)\\ =+\cos \varphi\dfrac{\partial}{\partial \rho}\left(\cos \varphi\dfrac{\partial}{\partial \rho}\right)\\ -\dfrac{\sin \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\left(\cos \varphi\dfrac{\partial}{\partial \rho}\right)\\ -\cos \varphi\dfrac{\partial}{\partial \rho}\left(\dfrac{\sin \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\right)\\ +\dfrac{\sin \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\left(\dfrac{\sin \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\right)\\ +\sin \varphi\dfrac{\partial}{\partial \rho}\left(\sin \varphi\dfrac{\partial}{\partial \rho}\right)\\ +\dfrac{\cos \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\left(\sin \varphi\dfrac{\partial}{\partial \rho}\right)\\ +\sin \varphi\dfrac{\partial}{\partial \rho}\left(\dfrac{\cos \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\right)\\ +\dfrac{\cos \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\left(\dfrac{\cos \varphi}{\rho}\dfrac{\partial}{\partial \varphi}\right)\] \[一项一项按照上式形式求偏导，化简后可得:\\ \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}\\=+\left(\cos^2\varphi\dfrac{\partial^2}{\partial \rho^2}\right)\\ +\left(\dfrac{\sin^2 \varphi}{\rho}\dfrac{\partial}{\partial\rho}-\dfrac{\sin \varphi\cos\varphi}{\rho}\dfrac{\partial^2}{\partial \varphi \partial\rho}\right)\\ +\left(\dfrac{\cos\varphi\sin\varphi}{\rho^2}\dfrac{\partial}{\partial\varphi}-\dfrac{\cos\varphi\sin\varphi}{\rho}\dfrac{\partial^2}{\partial\rho\partial\varphi}\right)\\ +\dfrac{1}{\rho^2}\left(\cos\varphi\sin\varphi\dfrac{\partial}{\partial \rho}+\sin^2\varphi\dfrac{\partial^2}{\partial^2 \varphi^2}\right)\\ +\left(\sin^2\varphi\dfrac{\partial^2}{\partial\rho^2}\right)\\ +\left(\dfrac{\cos^2 \varphi}{\rho}\dfrac{\partial}{\partial\rho}+\dfrac{\sin \varphi\cos\varphi}{\rho}\dfrac{\partial^2}{\partial \varphi \partial\rho}\right)\\ +\left(-\dfrac{\cos\varphi\sin\varphi}{\rho^2}\dfrac{\partial}{\partial\varphi}+\dfrac{\cos\varphi\sin\varphi}{\rho}\dfrac{\partial^2}{\partial\rho\partial\varphi} \right)\\ +\dfrac{1}{\rho^2}\left(-\cos\varphi\sin\varphi\dfrac{\partial}{\partial \rho}+\cos^2\varphi\dfrac{\partial^2}{\partial^2 \varphi^2}\right)\\ =\left(\cos^2\varphi\dfrac{\partial^2}{\partial \rho^2}\right)+\left(\sin^2\varphi\dfrac{\partial^2}{\partial\rho^2}\right)+\dfrac{\sin^2 \varphi}{\rho}\dfrac{\partial}{\partial\rho}+\dfrac{\cos^2 \varphi}{\rho}\dfrac{\partial}{\partial\rho}+\dfrac{1}{\rho^2}\left(\sin^2\varphi\dfrac{\partial^2}{\partial^2 \varphi^2}+\cos^2\varphi\dfrac{\partial^2}{\partial^2 \varphi^2} \right)\\ =\dfrac{\partial^2}{\partial \rho^2}+\dfrac{1}{\rho}\dfrac{\partial}{\partial\rho}+\dfrac{1}{\rho^2}\dfrac{\partial^2}{\partial^2 \varphi^2}\\\] \[\therefore \nabla^2=\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}=\dfrac{\partial^2}{\partial \rho^2}+\dfrac{1}{\rho}\dfrac{\partial}{\partial\rho}+\dfrac{1}{\rho^2}\dfrac{\partial^2}{\partial^2 \varphi^2}+\dfrac{\partial^2}{\partial z^2}\]编者注：大家可以查一下有一个东西叫拉梅系数……