円柱座標

円柱座標系は\(x,y\)を円座標で、\(z\)を直交座標系のまま表した座標系です。変換式を示します。 \[ \begin{align} x&=\rho\cos\phi\\ y&=\rho\sin\phi\\ z&=z \end{align} \] つまり \[ \begin{align} \rho&=\sqrt{x^2+y^2}\\ \phi&=\arctan\frac{y}{x}\\ \end{align} \] ヤコビアンは \[ \left|\begin{array}{rr} \frac{\partial{x}}{\partial\rho} & \frac{\partial{x}}{\partial\phi}\\ \frac{\partial{y}}{\partial\rho} & \frac{\partial{y}}{\partial\phi} \end{array}\right|\\ = \left|\begin{array}{rr} \cos\phi&-\rho\sin\phi\\ \sin\phi&\rho\cos\phi \end{array}\right|\\ =\rho \] なので \[ dxdydz=\rho \ d\rho\phi dz \] ここで \[ \begin{align} \frac{\partial\rho}{\partial x} &= \frac{x}{r} = \cos\phi\\ \frac{\partial\phi}{\partial x} &= \frac{1}{(\frac{y}{x})^2+1}\cdot -\frac{y}{x^2}\\ &=-\frac{y}{\rho^2}=-\frac{\sin\phi}{\rho}\\ \\ \frac{\partial\rho}{\partial y} &= \frac{x}{r} = \sin\phi\\ \frac{\partial\phi}{\partial y} &= \frac{1}{(\frac{y}{x})^2+1}\cdot \frac{1}{x}\\ &=\frac{x}{\rho^2}=\frac{\cos\phi}{\rho}\\ \\ df&=\frac{\partial f}{\partial\rho}d\rho+\frac{\partial f}{\partial\phi}d\phi+\frac{\partial f}{\partial z}dz\\ \frac{\partial}{\partial x}&=\frac{\partial}{\partial\rho}\frac{\partial\rho}{\partial x}+\frac{\partial}{\partial\phi}\frac{\partial\phi}{\partial x}+\frac{\partial}{\partial z}\frac{\partial z}{\partial x}\\ &=\cos\phi\frac{\partial}{\partial\rho}-\frac{\sin\phi}{\rho}\frac{\partial}{\partial\phi}\\ \frac{\partial}{\partial y}&=\sin\phi\frac{\partial}{\partial\rho}+\frac{\cos\phi}{\rho}\frac{\partial}{\partial\phi}\\ \end{align} \] 単位ベクトルは \[ \begin{align} \hat\rho&=\cos\phi\ \hat x +\sin\phi\ \hat y \\ \hat\phi&=-sin\phi\ \hat x +\cos\phi\ \hat y \end{align} \] 逆行列をとって \[ \begin{align} \hat x &= \cos\phi\ \hat\rho -\sin\phi\ \hat\phi\\ \hat y &= \sin\phi\ \hat\rho + \cos\phi\ \hat\phi \end{align} \] 勾配をとって \[ \begin{align} \nabla &= \frac{\partial}{\partial x} \hat x + \frac{\partial}{\partial y} \hat y + \frac{\partial}{\partial z} \hat z \\ \nabla f &= (\cos\phi\frac{\partial}{\partial\rho}-\frac{\sin\phi}{\rho}\frac{\partial}{\partial\phi})(\cos\phi\ \hat\rho -\sin\phi\ \hat\phi)\\ &+(\sin\phi\frac{\partial}{\partial\rho}+\frac{\cos\phi}{\rho}\frac{\partial}{\partial\phi})(\sin\phi\ \hat\rho + \cos\phi\ \hat\phi)\\ &+ \frac{\partial}{\partial z} \hat z\\ &=\frac{\partial}{\partial \rho} \hat\rho +\frac{1}{\rho}\frac{\partial}{\partial\phi} \hat\phi+\frac{\partial}{\partial z} \hat z \end{align} \]