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Handbook of mathematics for engineers and scienteists part 177. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1200 Orthogonal Curvilinear Systems of Coordinate T4.2.3. Coordinates of a Prolate Ellipsoid of Revolution T4.2.3-1. Transformations of coordinates. The metric tensor components. 1 . Transformations of coordinates x2 a2 a2 - 1 1 - r2 cos2 y2 a2 a2 - 1 1 - t2 sin2 z aar a 1 t -1 0 2n . Coordinate surfaces the z-axis is the axis revolution x2 y2 a2 a2 - 1 x2 y2 a2 r2 - 1 y x tan p Metric tensor components 2 a2 - t 2 9aa a 2 - g a -7 1 a2 ơ2 1 a2T 2 prolate ellipsoids of revolution hyperboloid of revolution of two sheets half-planes through the -axis . 2 ơ2 - T2 2 1 gr- a T-T 2 . Special coordinate system u v p gvv a2 ơ2 - 1 1 - T2 g a3 ơ2 - T2 . ơ cosh u T cos v p p 0 u TO 0 v n 0 p 2n x a sinh u sin v cos p y a sinh u sin v sin p z a cosh u cos v. Metric tensor components guu gvv a2 sinh2 u sin2 v g a2 sinh2 u sin2 v. T4.2.3-2. Basic differential relations. Gradient of a scalar f n . 1 Ơ2 - 1 df 1 1 - T2 df -f V f 7 7 . a ơ2 - T2 dơ a ơ2 - T2 dr 1 f. aự 1 - T2 ơ2 - 1 dg v Divergence of a vector v V v 1 - d vự ơ2 - T w - d Vt ơ2 - T2 1 - T2 1 Vp ơ - T 2 . 9t L J dp p ự ơ2 - 1 1 - T2 J J Gradient of a scalar f along a vector v v. V f Vl J ơ -1 df Vl J 1 - T2 df _ - df ay Ơ2 - T2 dơ ay Ơ2 - T2 dT a ơ2 - 1 1 - T2 dp Gradient of a vector w along a vector v v V w v V wơĩơ v V wTiT v V wpip. Laplacian of a scalar f Af 1 Ị d a2 ơ2 - T2 dơ ơ -1 f dơ d dĩ 2 df 1 ơ2 - T2 d2f i T dT ơ2 - 1 1 - T2 d 2 . T4.2. Special Curvilinear Coordinate Systems 1201 T4.2.4. Coordinates of an Oblate Ellipsoid of Revolution T4.2.4-1. Transformations of coordinates. The metric tensor components. 1 . Transformations of coordinates x2 a2 1 a2 1 - r 2 cos2 p y2 a2 1 a2 1 - r 2 sin2 p z aar a 0 -1 r 1 0 p 2n . Coordinate surfaces the z-axis is the axis of revolution x2 y2 a2 1 - a2 x2 y2 a2 1 - t 2 L-_ a2 a2 - 1 a2T 2 y x tan p oblate ellipsoids of revolution hyperboloid of revolution of one sheet half-planes through the z-axis . Components of the metric tensor 2 a2 r2 2 a2 r2 9aa a -----T grr a .
