Mathematical Methods for Physics and Engineering - Matematica.NET

LINE, SURFACE AND VOLUME INTEGRALSzh 1 ∆u 1 ê 1RS TQh 2 ∆u 2 ê 2P h 3 ∆u 3 ê 3yxFigure 11.10 A general volume ∆V in orthogonal curvilinear coordinatesu 1 ,u 2 ,u 3 . PT gives the vector h 1 ∆u 1 ê 1 , PS gives h 2 ∆u 2 ê 2 and PQ givesh 3 ∆u 3 ê 3 .◮By considering the infinitesimal planar surface element PQRS in figure 11.10, show that(11.17) leads to the usual expression for ∇×a in orthogonal curvilinear coordinates.The planar surface PQRS is defined by the orthogonal vectors h 2 ∆u 2 ê 2 and h 3 ∆u 3 ê 3at the point P . If we traverse the loop in the direction PSRQ then, by the right-handconvention, the unit normal to the plane is ê 1 .Writinga = a 1 ê 1 + a 2 ê 2 + a 3 ê 3 , the lineintegral around the loop in this direction is given by∮PSRQa · dr = a 2 h 2 ∆u 2 +[− a 2 h 2 +[ ∂= (a 3 h 3 ) −∂u 2[a 3 h 3 +∂ ](a 3 h 3 )∆u 2 ∆u 3∂u 2∂ ](a 2 h 2 )∆u 3 ∆u 2 − a 3 h 3 ∆u 3∂u 3]∂∂u 3(a 2 h 2 )∆u 2 ∆u 3 .Therefore from (11.17) the component of ∇×a in the direction ê 1 at P is given by[]1(∇×a) 1 = lima · dr∆u 2 ,∆u 3 →0 h 2 h 3 ∆u 2 ∆u 3∮PSRQ= 1 [ ∂(h 3 a 3 ) −∂ ](h 2 a 2 ) .h 2 h 3 ∂u 2 ∂u 3The other two components are found by cyclically permuting the subscripts 1, 2, 3. ◭Finally, we note that we can also write the ∇ 2 operator as a surface integral bysetting a = ∇φ in (11.15), to obtain( ∮ )1∇ 2 φ = ∇ · ∇φ = lim ∇φ · dS .V →0 V400S