83(x) Curvilinear-coordiante 83(x) 分成温 X(r)ECP(D D,) 82(x) 胚 线 g1(xa 坐 标 故而知新 g2 gt 系 local Co var iant-Basis DY(x)=[81,82:3](x) 微 x Problem 1(Practices on Orthogonal Curvilinear Coordinates) To study the following curvilinear coordinate that sets up the diffeomorphism between a cube in the parametric space and a filled torus in the physical space (B+ srcos)cosφ 摘自2012 (x):另3x=0→叫=x2(,n)=(R+5rcos)sino 2013学年第 s·rsin 学期试卷 where2={5,0,v∈(0,1,0∈(0,2x,∈(.,2m)}, R and r are constants I. To Calculate the Fundamental Geometric Quantities 1. To calculate the jacobian matrin of the above vector valued mapping and to confirm that this is an orthogonal coordinates 2. To sketch the coordinate curves and the local vectors of variant basis1 x 2 x 3 x X1 X3 X2 b c a f g e h d a b c d e f g h 1   a g x 2   a g x 3   a g x 1 x 2 x 3 x o     ; p x y Curvilinear coordiante Xx C DD       123   var : ,, local Co iant Basis DX x g g g x   1 x 3 x 1   d g x 3   d g x 2   d g x 3 x 1 x 温故而知新 —— “曲线坐标系” ～ 微 分同胚 摘自 2012－ 2013学年第一 学期试卷