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- Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our conformal mapping theory?Let f(x, y, z) = ln(x2 + z) + y2exz − cos(2yz). Find fzyx.Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate ∮C F • dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).
- But the definition of subharmonic is -laplace(v) less than or equal 0 in U. How could we obtain that?Compute the line integral∫C [2x3y2 dx + x4y dy]where C is the path that travels first from (1, 0) to (0, 1) along the partof the circle x2 + y2 = 1 that lies in the first quadrant and then from(0, 1) to (−1, 0) along the line segment that connects the two points.Use the limit definition to show that the partial derivatives of F(x,y) = with respect to x and y are/aren’t the same any point(a,b). (i.e show that Fx(a,b) =/≠ Fy(a,b))
- Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the integral (Image 1) , fulfills that: (image 2)Show that if φ is continuously differentiable in a given region V and on itsboundary S, then∫S φ dS =∫V ∇φ dVSuppose that the second order partial derivatives, fx,y and fy,x, are both continuous on an open set V in R2. Use Fubini’s theorem to prove that fx,y = fy,x in V . Hint: if fx,y(a) − fy,x(a) > 0, there is a rectangle R containing a on which fx,y − fy,x > 0.