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# endofbook - APPENDIX A VECTOR ANALYSIS A.1 GENERAL...

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APPENDIX A VECTOR ANALYSIS A.1 GENERAL CURVILINEAR COORDINATES Let us consider a general orthogonal coordinate system in which a point is located by the intersection of three mutually perpendicular surfaces (of unspeci- fied form or shape), u constant v constant w constant where u , v , and w are the variables of the coordinate system. If each variable is increased by a differential amount and three more mutually perpendicular sur- faces are drawn corresponding to these new values, a differential volume is formed which is closely a rectangular parallelepiped. Since u , v , and w need not be measures of length, such as, for example, the angle variables of the cylindrical and spherical coordinate systems, each must be multiplied by a gen- eral function of u , v , and w in order to obtain the differential sides of the parallelepiped. Thus we define the scale factors h 1 , h 2 , and h 3 each as a function of the three variables u , v , and w and write the lengths of the sides of the differential volume as 529

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dL 1 h 1 du dL 2 h 2 dv dL 3 h 3 dw In the three coordinate systems discussed in Chap. 1, it is apparent that the variables and scale factors are Cartesian: u x v y w z h 1 1 h 2 1 h 3 1 Cylindrical: u ± v ² w z h 1 1 h 2 ± h 3 1 (A.1) Spherical: u r v ³ w ² h 1 1 h 2 r h 3 r sin ³ The choice of u , v , and w above has been made so that a u ± a v a w in all cases. More involved expressions for h 1 , h 2 , and h 3 are to be expected in other less familiar coordinate systems. 1 A.2 DIVERGENCE, GRADIENT, AND CURL IN GENERAL CURVILINEAR COORDINATES If the method used to develop divergence in Secs. 3.4 and 3.5 is applied to the general curvilinear coordinate system, the flux of the vector D passing through the surface of the parallelepiped whose unit normal is a u is D u 0 dL 2 dL 3 1 2 @ @ u D u dL 2 dL 3 du or D u 0 h 2 h 3 dv dw 1 2 @ @ u D u h 2 h 3 dv dw du and for the opposite face it is ² D u 0 h 2 h 3 dv dw 1 2 @ @ u D u h 2 h 3 dv dw du giving a total for these two faces of @ @ u D u h 2 h 3 dv dw du Since u , v , and w are independent variables, this last expression may be written as 530 ENGINEERING ELECTROMAGNETICS 1 The variables and scale factors are given for nine orthogonal coordinate systems on pp. 50±59 in J. A. Stratton, ``Electromagnetic Theory,'' McGraw-Hill Book Company, New York, 1941. Each system is also described briefly.
@ @ u h 2 h 3 D u du dv dw and the other two corresponding expressions obtained by a simple permutation of the subscripts and of u , v , and w . Thus the total flux leaving the differential volume is @ @ u h 2 h 3 D u @ @ v h 3 h 1 D v @ @ w h 1 h 2 D w ± ² du dv dw and the divergence of D is found by dividing by the differential volume r ³ D 1 h 1 h 2 h 3 @ @ u h 2 h 3 D u @ @ v h 3 h 1 D v @ @ w h 1 h 2 D w ± ² A : 2 The components of the gradient of a scalar V may be obtained (following the methods of Sec. 4.6) by expressing the total differential of V , dV @ V @ u du @ V @ v dv @ V @ w dw in terms of the component differential lengths, h 1 du , h 2 dv , and h 3 dw , dV 1 h 1 @ V @ u h 1 du 1 h 2 @ V @ v h 2 dv 1 h 3 @ V @ w h 3 dw Then, since d L h 1 du a u h 2 dv a v h

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