using Milne-Thompson Method
[3](4 pts) Let f(x) = u(x, y) + iv(x,y) be differentiable for all z = x...
1. Write f(z) in the form f(x) = u(x, y) +iv(x, y). (a) f(x) = 23+2+1 (b) f(3) = 2,270. Suppose f(z) = x2 - y2 - 2y +i (2x - 2xy), where z = x + iy, and express () in terms of .
Problem 8. Let f(z) = u(x, y) iv(x, y) be an entire function with real and imaginary parts u(x, y) and v(x, y). Assume that the imaginary part is bounded v(x, y) < M for every z = x+ iy. Prove that f is a constant 1
Consider the function Let where f(t) is differentiable for all t ∈ R. Show that z satisfies the partial differential equation (x2 − y2 ) ∂z/∂x + xy ∂z/∂y = xyz for all (x, y) ∈ R2 \ { (t, 0)|t ∈ R }.
7. Let f:D + C be a complex variable function, write f(x) = u(x, y) +iv(x,y) where z = x +iy. (a) (9 points) (1) Present an equivalent characterization(with u and v involved) for f being analytic on D. (Just write down the theorem, you don't need to prove it.) (2) Let f(z) = (4.x2 + 5x – 4y2 + 3) +i(8xy + 5y – 1). Show that f is an entrie function. (3) For the same f as above,...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top hemisphere of the unit sphere oriented upward, and C the unit circle in the xy-plane with positive orientation. (a) Compute div(F) and curl(F). (b) Is F conservative? Briefly explain. (c) Use Stokes’ Theorem to compute ? F · dr by converting it to a surface integral. (The integral is easy if C you set it up correctly) 4. (23 pts)...
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that: 1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a 3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...
Given f(x, y, z) = 4yz - x2, and that x, y, z are each differentiable functions of u and v. Suppose (x, y, z) = (-1,0,1) at (u, v) = (1, 2), and that af 09 = 70. au |(1,2) = -50 and av (1,2) What is the value of x,(1, 2)? a) 230/3 b)-230/3 c) 40 d) 110 e) -120