Find a conformal mapping from the following set onto the upper half plane S' = {(u, v) | v>
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- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).3. Find points at which the mapping defined by f(z) = nz+z^n (n E N) is not conformal.Let F = <x-y+3z, x-2z, 3y-4x>, where C is the positively oriented, closed, triangular curve with vertices (3,0,0), (0,2,0), and (0,0,6), and S is part of the first octant bounded by C. Find the flux of F up through S.
- Use Green’s theorem to evaluate ∮C(ye2xy−5y)dx+ (xe2xy−2x)dy, where Cis the counterclockwise oriented boundary curve of the square with vertices at(0,0), (0,1), (1,0), and (1.1).compute ∫C F · dr for the oriented curve specified. F(x, y) = (ey^-x, e2x), piecewise linear path from (1, 1) to (2, 2) to (0, 2).Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our conformal mapping theory?
- Justify the following statement: The flux of F = (x3, y3, z3) through every closed surface is positive.Find the circulation of F=3xi+8zj+5yk around the closed path consisting of the following three curves traversed in the direction of increasing t.Evaluate∮C (x + 3y)dx + ydy where C is the Jordan curve given by thegraphs of y = e^x, y = e^−x and the horizontal line y = e^−1a) By Green’s theoremb) By direct computation