(3) The Partial Differential Equationf, f= 0is called the Laplace equation. Any function f = (x, y) of class C² that satisfies theLaplace equation is called a harmonic function. Let the functions u = u(x, y)and v = v(r, y) be of class C2 and satisfy the Cauchy-Riemann equationsdudvdvduanddyShow that u and v are both harmonic.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(9) If f(z) is analytic function expressed in polar form, show that both u and v satisfy Laplace equation in two dimensions given by: 1 ди 1 №²u + r²00² d²u + Or² rər = 0, and = d²v 1 dv + dr² r dr + 1 d²v r.² 80² = 0
The Cauchy-Riemann equations for f(z) is ди ne dx ду dv ди dx he dv dv dx ду ди ду
Parametrize the intersection of the surfaces y² - z² = x - 6, y² + z² = 25 using trigonometric functions. (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the parametrization of the y variable in the form a cos (t).) x(t) = y(t) = z(t) =
5. Sketch the curve r(t) = (you can plot it in Geogebra first and then transfer on to paper). Can you calculate the vectors T, B and N at t = 0 and at t 2πT? -
Express the given parametric equations of a line using bracket notation and also using i, j, k notation. (a) x = 6t, y = - 4 + 3t O (x, y) = (0, 4) +1 (6,3) ai + bj = -4j + t (6i + 3j) O (x, y) = (0, 4) + t (6, 3) ai + bj = 4j+ t (6i + 3j) O (x, y) = (0, 6) + (-4, 3) ai + bj = − 4j + t (6i + 3j) O(x, y) =(4,0) + t (3,6) ai + bj = −4i + t (3i + 6j) O (x, y) = (6,3)+1(0, - 4) ai + bj (6i + 3j) +t(-4j) (b)x= 4 + 6t,y=4+9t, z = 87t O (x, y, z) = (-4, 4, − 8) +1(-6, ai + bj + ck = 9,7) ( − 4i + 4j - 8 k) + t ( − 6i - 9j + 7k) O (x, y, z) = (4, ai + bj + ck = 4, 8) + t (6, 9, - 7) (4i − 4j + 8 k) + t (6i + 9j – 7k) O (x, y, z) = (6,9, −7) +1(4, - 4,8) ai + bj + ck = (6i + 9j − 7k) + t(4i - 4j + 8k) O (x, y, z) = (-6, 9, 7) + 1 - 4, 4, 8) ai+bj + ck = ( – 6i – 9j + 7k) + t( − 4i + 4j − 8k) O (x, y, z) = (4, 4, 8) + t (6,9,7) ai + bj + ck = (4i +4j + 8 k) + t (6i + 9j + 7k)
1
1. State and prove the Lagrange's identity. 2. Prove that any straight line in space can represented by linear vector function whose components are linear scalar functions. 3. Find the points of intersection of the two cylinders: F₁ = y - x² = 0, F₂ ==-x³ = 0.
Let f(2) = +. Use the polar form of the Cauchy-Riemann equations to determine where f is differentiable. %3D

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Laplace equation is called a harmonic function. Let the functions u = u(x, y) and v = v(x, y) be of class C² and satisfy the Cauchy–Riemann equations ди dv and dv du ду Show that u and v are both harmonic.