7. Show that the following functions u(x, y) monic functions v(x, y) and determine f(z) = u(x,y) + iv(x, y) are harmoni...
Solve the Dirichlet problem in an infinite strip uxx + uyy=0 for x ϵ R and 0 <y <b , u(x,0)=f(x) , u(x,b)=g(x). (Hint: first do the case f=0. The case g=0 reduces to this one by the substitution y→ b-y , and the case general is obtained by superposition) 4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
5. Find the derivative matrices of the following composition of functions. (а) fog where f (x, у) — 2х — 3у, g(u, v) - (usin u, U sin u) (Ъ) f.g where f(х, у, 2) %3D (x? + у? +2?,х— у+2:), g() %3 (2, 13, 2/4) (с) fog wherе f (x, у, z) 3D (хуz, ху + xz — yz) where g(u, v, w) %3D (uu, uw, vw) 5. Find the derivative matrices of the following composition of functions. (а)...
Show that the following PDE for u(x,y) is linear in u and homogeneous. ди ду ди = 3- дх Ә2 и + sin(у) дх2
Hw2 Q1 Show that the function f(z) = z2 + z is analytic. Also find its derivative. (Hint: check CR Equations for Analyticity, and then proceed finding the derivative as shown in video 8 by any of the two rules shown in video 7] Q2 Verify that the following functions are harmonic i. u = x2 - y2 + 2x - y. ii. v=e* cos y. Q3 Verify that the given function is harmonic, and find the harmonic conjugate function...
solve number 3 2. Show that u(x, y) = e = cos 12 + y2 is harmonic on the punctured complex plane D=C\{0}, and find a harmonic conjugate v of it. 1 2 3. Reveal, what is f(x) = u(x,y) + iv(I,y) in Problem 2?
Give a parametric description of the form r(u, v) = (x(u,v),y(u,v),z(u,v)) for the following surface. The cap of the sphere x² + y2 + z = 36, for 3/3 sz56 Select the correct choice below and fill in the answer boxes to complete your choice. (Type any angle measures in radians. Use angle measures greater than or equal to 0 and less than 21. Type exact answers in terms of ..) A. Fu.v) = (6 sin u cos V,6 sin...
(%) = u(x, y) + f 0(4,7) For each of the following functions, write as f(z) = u(x, y) + í v(x, y) and use the Cauchy-Riemann conditions to determine whether they are analytic (and if so, in what domain) a. f(z) = 2 + 1/(2+2) b. f(z) = Re z C. f(x) = e-iz d. f(z) = ez? 16 marks]
Calculus III 1) Identify each of the following surfaces: а) z' %3x? - 5у" b) z 4x2-4 y c) z2+3x2-5y = 4 d) z2x23-5y e) у3х* 2) Find and classify all of the critical points for f(x, y)=xy -x2 + y'. 3) Find the maximum and minimum values of f(x,y)=xy over the ellipse х* + 2y %3D1. 4) Let fx, y) x3 -cosy a) Find the first order Taylor polynomial for f(x,y) based at (1,7). b) Find the sccond order...
The Laplacian and harmonic functions The quantity V-Vu-V2u, called the Laplacian of the function u, is particularly useful in applications. (a) For a function u(x, y, z), compute V Vu (c) A scalar valued function u is harmonic on a region D if V a all points of D. Compare this to Laplace's equation eu +Pn=0 and ψ" + ψ”=0. The Laplacian and harmonic functions The quantity V-Vu-V2u, called the Laplacian of the function u, is particularly useful in applications....
a) Find the real part u(x,y) and imaginary part v(x,y) of f(2)= (1+2i )z? + (i – 1)2 +3 b) Verify if the above function is analytic c) Using Laplace's equation verify if the real part u(x,y) is harmonic.