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MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 10

Tham khảo tài liệu 'mathematical method in science and engineering episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | MAPPINGS 313 Fig. 12.12 Conformal mapping 12.4.1 Conformal Mappings To see an interesting property of analytic functions we differentiate w f z 12.115 at Z where the modulus and the arguments of the derivative are given as and a respectively. We now use polar coordinates to write the modulus z I ZQ lim AzAo Aw Ă7 and the argument Fig. 12.12 as arg 72 dz . Aw a arg lim I aAzAo Az a Jim arg Aw Jim0 arg Az . 12.116 12.117 12.118 12.119 Since this function f z maps a curve c2 in the z-plane into another curve cw in the w-plane from the arguments Eq. 12.119 it is seen that if the slope of c2 at Zq is 00 then the slope of cw at Wo is a ffo- For a pair of curves intersecting at Zo the angle between their tangents in the w- and z-planes will be equal that is ự 2 - ỉ ĩ Ỡ2 a - Ổ1 a 2 1- 12.120 12.121 Hence analytic functions preserve angles between the curves they map Fig. 12.12 . For this reason they are also called conformal mappings or transformations. 314 COMPLEX VARIABLES AND FUNCTIONS 12.4.2 Electrostatics and Conformal Mappings Conformal mappings are very useful in electrostatic and laminar irrotational flow problems where the Laplace equation must be solved. Even though the method is restricted to cases with one translational symmetry it allows one to solve analytically some complex boundary value problems. Example 12.10. Conformal mappings and electrostatics Let US consider two conductors held at potentials Vj and V2 with hyperbolic cross sections X2 y2 Cj and x2 y2 C2- 12.122 We want to find the equipotentials and the electric field lines. In the complex z-plane the problem can be shown as in Figure 12.13. We use the conformal mapping w z2 12.123 x2 -y2 i 2xy 12.124 to map these hyperbolae to the straight lines u Cj and u C 2 12.125 in the w-plane Fig. 12.14 . The problem is now reduced to finding the equipotentials and the electric field lines between two infinitely long MAPPINGS 315 Fig. 12.14 Equipotentiais and electric field lines in the w-plane parallel plates