Advanced Calculus fi..

Advanced Calculus, Fifth Edition: sThe divergence of p is the sum of the divergences of the terms on the right-handside. By (3.22) and (3.50) the divergence of the first term isBy (3.60) this can be written asa 1 ar a 1 a 1 a- (grad Byp,). -- = - [grad Byp,l, = --(Byp,) = --J a a~ J J a~ ~ B Yau (BYP~).This gives the first term on the right-hand side of (3.61); the others are found in thesame way.The proof of (3.62) is left to Problem 5 following.From (3.60) and (3.61) we obtain an expression for the Laplacian in orthogonalcurvilinear coordinates:Remark. If the curvilinear coordinates are not orthogonal, the very notion of vectorcomponents must be generalized. This leads to tensor analysis (see Section 3.9).The formulas (3.60), (3.61), (3.62) can be applied to the special case in whichthe new coordinates are obtained simply by choosing new rectangular coordinatesu, v, w in space. This can be accomplished by choosing a positive triple il, jl, kl ofunit vectors and then choosing axes u, v, w through an origin 0, : (XI, yl , zl) havingthe directions il , jl , kl , respectively (Problems 13 and 14 following Section 1.15).The new coordinates (u, v, w) of a point P : (x, y , z) are defined by the equation:One then findssimilar expressions are found for v and w. Also,1and similar expressions are found for y and z. These two sets of equations correspondto (3.43) and (3.42). We note that in all cases the functions involved are linear.The quantities a, B, y can be evaluated for this case without computation. For,since the new coordinates are rectangular and no change of scale is made, one musthavefor element of arc on a general curve. Hencerl