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...ssed in Cartesian coordinates. If one expresses regular solid harmonics in spherical polar coordinates, the cosine-type functions contain the factor cos |''m''| φ and the sin

In [[spherical polar coordinates]] ...bi determinant) of this transformation from [[Cartesian coordinates]] to [[spherical polar coordinates]] is

...nction ''V''(''x'',''y'',''z'') depends on ''r'' only, when expressed in [[spherical polar coordinates]] (''r'', θ, φ). ...of the torque more explicitly by expressing the gradient '''∇''' in spherical polar coordinates and writing the cross product as a determinant, using that the derivatives

Indeed, in terms of [[spherical polar coordinates]] the operator is,

the operator ∇² expressed in [[spherical polar coordinates]], can be written as

where ''r'', θ, and φ are [[spherical polar coordinates]] of a point with respect to a system of axes centered on the central ion.

{{rpr|Spherical polar coordinates}} 1698

where θ and φ are the [[spherical polar coordinates|spherical polar angles]] of '''r''' with respect to a coordinate system in ...le-dipole interaction this is most easily proved by integrating over the [[spherical polar coordinates|spherical polar angles]] of the dipole vector, while using the volume eleme

* [[Spherical polar coordinates]]

...ession of the problem; however, someone less stubborn might see the [[Talk:Spherical polar coordinates|discussion]] and decide that greater caution was called for. [[User:Anthony