> [!NOTE] Definition (Spherical Coordinates) > Let $(x,y,z)\in\mathbb{R}^{3}$ be an element of the [[Real n-Space|real 3-space]]. Then its spherical coordinate is the triple $(r,\theta,\phi)$ where $x = r\sin \phi \cos \theta , \quad y = r \sin \phi \sin \theta , \quad z = r \cos \phi $and $r,\theta, \phi$ are called the radial, azimuthal and polar coordinates respectively. # Properties By [[Radial Coordinate of Element of Real 3-Space equals Length]], $r=||(x,y,z)||$ since $ x^{2}+y^{2}+z^{2} =r^{2}(\cos^{2}\phi + \sin^{2}\phi (\cos^{2} \theta+\sin^{2}\theta) )=r^{2}.$ By [[Azimuthal Coordinate of Element of Real 3-Space]], $\theta$ is the angle measured from the positive $x$-axis to $(x,y).$ By [[Polar Coordinate of Element of Real 3-Space]], $\phi$ is the angle measured from the positive $z$-axis to $(x,y,z).$ **Visualisation**: ![[Spherical Coordinates.png|300]]