# Spherical coordinates

The numbers $\rho , \theta , \phi$ which are related to the Cartesian coordinates $x, y, z$ by the formulas

$$x = \rho \cos \phi \sin \theta ,\ \ y = \rho \sin \phi \sin \theta ,\ \ z = \rho \cos \theta ,$$

where $0 \leq \rho < \infty$, $0 \leq \phi < 2 \pi$, $0 \leq \theta \leq \pi$.

Figure: s086660a

The coordinate surfaces are (see Fig.): concentric spheres with centre $O$ $( \rho = OP = \textrm{ const } )$; half-planes that pass through the axis $Oz$ $( \phi = \textrm{ angle } xOP ^ \prime = \textrm{ const } )$; circular cones with vertex $O$ and axis $Oz$ $( \theta = \textrm{ angle } zOP = \textrm{ const } )$. The system of spherical coordinates is orthogonal.

The Lamé coefficients are

$$L _ \rho = 1,\ \ L _ \phi = \rho \sin \theta ,\ \ L _ \theta = \rho .$$

The element of surface area is

$$d \sigma = \ \sqrt {\rho ^ {2} \sin ^ {2} \theta \ ( d \rho d \phi ) ^ {2} + \rho ^ {2} ( d \rho d \theta ) ^ {2} + \rho ^ {4} \sin ^ {2} \theta ( d \phi d \theta ) ^ {2} } .$$

The volume element is

$$dV = \rho ^ {2} \sin \theta d \rho d \phi d \theta .$$

The basic operations of vector calculus are

$$\mathop{\rm grad} _ \rho f = \ \frac{\partial f }{\partial \rho } ,\ \ \mathop{\rm grad} _ \phi f = \frac{1}{\rho \sin \theta } \frac{\partial f }{ \partial \phi } ,\ \ \mathop{\rm grad} _ \theta f = \frac{1} \rho \frac{\partial f }{\partial \theta } ;$$

$$\mathop{\rm div} \mathbf a = \frac{2} \rho a _ \rho + \frac{\partial a _ \rho }{\partial \rho } + \frac{1}{\rho \sin \theta } \frac{\partial a _ \phi }{\partial \phi } + \frac{1}{\rho \mathop{\rm tan} \theta } a _ \theta + \frac{1} \rho \frac{\partial a _ \theta }{\partial \theta } ;$$

$$\mathop{\rm rot} _ \rho \mathbf a = \frac{1}{\rho \sin \theta } \frac{\partial a _ \theta }{\partial \phi } - \frac{1} \rho \frac{\partial a _ \phi }{ \partial \theta } - \frac{1}{\rho \mathop{\rm tan} \theta } a _ \phi ;$$

$$\mathop{\rm rot} _ \phi \mathbf a = \frac{1} \rho \frac{\partial a _ \rho }{\partial \theta } - \frac{\partial a _ \theta }{\partial \rho } - \frac{a _ \theta } \rho ;$$

$$\mathop{\rm rot} _ \theta \mathbf a = \frac{\partial a _ \phi }{\partial \rho } + \frac{a _ \phi } \rho - \frac{1}{\rho \ \sin \theta } \frac{\partial a _ \rho }{\partial \phi } ;$$

$$\Delta f = \frac{\partial ^ {2} f }{\partial \rho ^ {2} } + \frac{2} \rho \frac{\partial f }{\partial \rho } + \frac{1}{\rho ^ {2} \sin ^ {2} \theta } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } + \frac{1}{\rho ^ {2} } \frac{\partial ^ {2} f }{\partial \theta ^ {2} } + \frac{ \mathop{\rm cot} \theta }{\rho ^ {2} } \frac{\partial f }{\partial \theta } .$$

The numbers $u , v, w$, called generalized spherical coordinates, are related to the Cartesian coordinates $x, y, z$ by the formulas

$$x = au \cos v \sin w,\ \ y = bu \sin v \sin w,\ \ z = cu \cos w,$$

where $0 \leq u < \infty$, $0 \leq v < 2 \pi$, $0 \leq w \leq \pi$, $a > b$, $b > 0$. The coordinate surface are: ellipsoids $( u = \textrm{ const } )$, half-planes $( v= \textrm{ const } )$ and elliptical cones $( w = \textrm{ const } )$.

If the surface has been given by $R = R( \phi , \theta )$, then the element of surface area can be written as:
$$dS = R \sqrt {\left \{ R ^ {2} + \left ( \frac{\partial R }{\partial \theta } \right ) ^ {2} \right \} \sin ^ {2} \theta + \left ( \frac{\partial R }{\partial \theta } \right ) ^ {2} } \ d \theta d \phi .$$