Mathematical Methods for Physics and Engineering - Matematica.NET

10.10 GENERAL CURVILINEAR COORDINATESIn the last step we have used the chain rule for partial differentiation. Therefore e i · ɛ j =1if i = j, ande i · ɛ j = 0 otherwise. Hence {e i } and {ɛ j } are reciprocal systems of vectors. ◭We now derive expressions for the standard vector operators in orthogonalcurvilinear coordinates. Despite the useful properties of the non-unit bases discussedabove, the remainder of our discussion in this section will be in terms ofthe unit basis vectors {ê i }. The expressions for the vector operators in cylindricaland spherical polar coordinates given in tables 10.2 and 10.3 respectively can befound from those derived below by inserting the appropriate scale factors.GradientThe change dΦ in a scalar field Φ resulting from changes du 1 ,du 2 ,du 3 in thecoordinates u 1 ,u 2 ,u 3 is given by, from (5.5),dΦ = ∂Φ du 1 + ∂Φ du 2 + ∂Φ du 3 .∂u 1 ∂u 2 ∂u 3For orthogonal curvilinear coordinates u 1 ,u 2 ,u 3 we find from (10.57), and comparisonwith (10.27), that we can write this aswhere ∇Φ is given bydΦ =∇Φ · dr, (10.59)∇Φ = 1 ∂Φê 1 + 1 ∂Φê 2 + 1 ∂Φê 3 . (10.60)h 1 ∂u 1 h 2 ∂u 2 h 3 ∂u 3This implies that the del operator can be written∇ = ê1h 1∂∂u 1+ ê2h 2∂∂u 2+ ê3h 3∂∂u 3.◮Show that for orthogonal curvilinear coordinates ∇u i = ê i /h i . Hence show that the twosets of vectors {ê i } and {ˆɛ i } are identical in this case.Letting Φ = u i in (10.60) we find immediately that ∇u i = ê i /h i . Therefore |∇u i | =1/h i ,andso ˆɛ i = ∇u i /|∇u i | = h i ∇u i = ê i . ◭DivergenceIn order to derive the expression for the divergence of a vector field in orthogonalcurvilinear coordinates, we must first write the vector field in terms of the basisvectors of the coordinate system:The divergence is then given by∇ · a = 1h 1 h 2 h 3[ ∂∂u 1(h 2 h 3 a 1 )+a = a 1 ê 1 + a 2 ê 2 + a 3 ê 3 .∂∂u 2(h 3 h 1 a 2 )+367∂ ](h 1 h 2 a 3 ) .∂u 3(10.61)