球面座標

球座標の定義です。 \[ \begin{align} x&=r\sin\theta\cos\phi\\ y&=r\sin\theta\sin\phi\\ z&=r\cos\theta \end{align} \] つまり \[ \begin{align} r&=\sqrt{x^2+y^2+z^2}\\ \theta&=\arctan\frac{\sqrt{x^2+y^2}}{z}\\ \phi&=\arctan\frac{y}{x}\\ \end{align} \] ヤコビアンは \[ \begin{align} J&= \left|\begin{array}{rr} \frac{\partial{x}}{\partial r} & \frac{\partial{x}}{\partial\theta} & \frac{\partial{x}}{\partial\phi}\\ \frac{\partial{y}}{\partial r} & \frac{\partial{y}}{\partial\theta} & \frac{\partial{y}}{\partial\phi}\\ \frac{\partial{z}}{\partial r} & \frac{\partial{z}}{\partial\theta}& \frac{\partial{z}}{\partial\phi} \end{array}\right|\\ &= \left|\begin{array}{rr} \sin\theta\cos\phi&r\cos\theta\cos\phi&-r\sin\theta\sin\phi\\ \sin\theta\sin\phi&r\cos\theta\sin\phi&r\sin\theta\cos\phi\\ \cos\theta&-r\sin\theta&0 \end{array}\right|\\ &=\cos\theta \left|\begin{array}{rr} r\cos\theta\cos\phi & -r\sin\theta\sin\phi\\ r\cos\theta\sin\phi&r\sin\theta\cos\phi\\ \end{array}\right| \\ &+r\sin\theta \left|\begin{array}{rr} \sin\theta\cos\phi&-r\sin\theta\sin\phi\\ \sin\theta\sin\phi&r\sin\theta\cos\phi\\ \end{array}\right| \\ &=\cos\theta(r^2\sin\theta\cos\theta\cos^2\phi+r^2\sin\theta\cos\theta\sin^2\phi)\\ &\ \ \ +r\sin\theta(r\sin^2\theta\cos^2\phi+r\sin^2\theta\sin^2\phi)\\ &=\cos\theta\cdot r^2\sin\theta\cos\theta\\ &\ \ \ +r\sin\theta\cdot r\sin^2\theta\\ &=r^2\sin\theta\cos^2\theta+r^2\sin^3\theta\\ &=r^2\sin\theta \end{align} \] なので \[ dxdydz=r^2\sin\theta\ dr d\theta d\phi \] 単位ベクトルは \[ \begin{align} \hat r&=\sin\theta\cos\phi\ \hat x +\sin\theta\sin\phi\ \hat y+\cos\theta\ \hat z \\ \hat\theta&=\cos\theta\cos\phi\ \hat x +\cos\theta\cos\phi\ \hat y-\sin\theta\ \hat z\\ \hat\phi&=-\sin\theta\sin\phi\ \hat x +\sin\theta\cos\phi\ \hat y \end{align} \] 逆行列をとって \[ \begin{align} \hat x&=\sin\theta\cos\phi\ \hat r +\cos\theta\sin\phi\ \hat\theta -\sin\phi\ \hat\phi \\ \hat y&=\sin\theta\sin\phi\ \hat r +\cos\theta\sin\phi\ \hat\theta+\cos\phi\ \hat\phi\\ \hat z&=\cos\theta\ \hat r - \sin\theta\ \hat\theta \end{align} \]