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 } ) $.

Comments

If the surface has been given by $ R = R( \phi , \theta ) $, then the element of surface area can be written as:

$$ dS = R \sqrt {\left \{ R ^ {2} + \left ( \frac{\partial R }{\partial \theta } \right ) ^ {2} \right \} \sin ^ {2} \theta + \left ( \frac{\partial R }{\partial \theta } \right ) ^ {2} } \ d \theta d \phi . $$

A general method to transform vector functions when new coordinates are introduced is, e.g., given in [a1].

References

[a1]	D.E. Rutherford, "Vector methods" , Oliver & Boyd (1949)
[a2]	M.R. Spiegel, "Vector analysis and an introduction to tensor analysis" , McGraw-Hill (1959)
[a3]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) pp. 11; 258