Mathematical Methods for Physics and Engineering - Matematica.NET

10.9 CYLINDRICAL AND SPHERICAL POLAR COORDINATESzê rê φPê θkθriOjyφxFigure 10.9 Spherical polar coordinates r, θ, φ.Substituting these relations and (10.44) into the expression for a we finda = zρ sin φ (cos φ ê ρ − sin φ ê φ ) − ρ sin φ (sin φ ê ρ +cosφ ê φ )+z 2 ρ cos φ ê z=(zρsin φ cos φ − ρ sin 2 φ) ê ρ − (zρsin 2 φ + ρ sin φ cos φ) ê φ + z 2 ρ cos φ ê z .Substituting into the expression for ∇ · a given in table 10.2,∇ · a =2z sin φ cos φ − 2sin 2 φ − 2z sin φ cos φ − cos 2 φ +sin 2 φ +2zρcos φ=2zρcos φ − 1.Alternatively, and much more quickly in this case, we can calculate the divergencedirectly in Cartesian coordinates. We obtain∇ · a = ∂a x∂x + ∂a y∂y + ∂a z=2zx − 1,∂zwhich on substituting x = ρ cos φ yields the same result as the calculation in cylindricalpolars. ◭Finally, we note that similar results can be obtained for (two-dimensional)polar coordinates in a plane by omitting the z-dependence. For example, (ds) 2 =(dρ) 2 + ρ 2 (dφ) 2 , while the element of volume is replaced by the element of areadA = ρdρdφ.10.9.2 Spherical polar coordinatesAs shown in figure 10.9, the position of a point in space P , with Cartesiancoordinates x, y, z, may be expressed in terms of spherical polar coordinatesr, θ, φ, wherex = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, (10.53)361