please i need the question 15 for the detailed proof and explaination ! thanks ! 233 42 Isometries, Conformal Maps 14,...
233 42 Isometries, Conformal Maps 14, we say that a differentiable map ф: S,--S2 preserves angles when for every p e Si and every pair vi, v2 E T (S,) we have cos(u, 2) cos(dp, (vi). do,()). Prove that pis locally conformal if and only if it preserves angles. 15. Letp: R2 R2 be given by ф(x, y)-(u (x, y), u(x, y), where u and v are differentiable functions that satisfy the Cauchy-Riemann equations Show that φ is a local conformal map from R2-Q into R, where R3, where 16. Let x: U CR sin ф, cos ), cosp, sin (sin X(9, ф) be a parametrization of the unit sphere S2. Let
233 42 Isometries, Conformal Maps 14, we say that a differentiable map ф: S,--S2 preserves angles when for every p e Si and every pair vi, v2 E T (S,) we have cos(u, 2) cos(dp, (vi). do,()). Prove that pis locally conformal if and only if it preserves angles. 15. Letp: R2 R2 be given by ф(x, y)-(u (x, y), u(x, y), where u and v are differentiable functions that satisfy the Cauchy-Riemann equations Show that φ is a local conformal map from R2-Q into R, where R3, where 16. Let x: U CR sin ф, cos ), cosp, sin (sin X(9, ф) be a parametrization of the unit sphere S2. Let