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(a) What form does the Laplace equation take in polar coordinates (r, 0)? (b) Let (r, 0) denote the usual polar coordinates. Show that if U(r, 0) is a harmonic function, then so is V (r, 0) = U (⁄, −0). (c) Suppose that U is a solution to the Laplace equation in the disk N = {r ≤ 1} and that U(1,0) = 5-sin² 0. (i) Without finding the solution to the equation, compute the value of U at the origini.e. at r = 0. (ii) Without finding the solution to the equation, determine the location of the maxima and minima of U in N. (Hint: sin² 0 = 1-os 20) 2 [1

(a) What form does the Laplace equation take in polar coordinates (r, 0)? (b) Let (r, 0) denote the usual polar coordinates. Show that if U(r, 0) is a harmonic function, then so is V (r, 0) = U (⁄, −0). (c) Suppose that U is a solution to the Laplace equation in the disk N = {r ≤ 1} and that U(1,0) = 5-sin² 0. (i) Without finding the solution to the equation, compute the value of U at the origini.e. at r = 0. (ii) Without finding the solution to the equation, determine the location of the maxima and minima of U in N. (Hint: sin² 0 = 1-os 20) 2 [1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 98E
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(a) What form does the Laplace equation take in polar coordinates (r, 0)? (b) Let (r, 0) denote the usual polar coordinates. Show that if U(r, 0) is a harmonic function, then so is V(r, 0) = U(, -0). (c) Suppose that U is a solution to the Laplace equation in the disk = {r < 1} and that U (1,0) = 5 – sin² 0. (i) Without finding the solution to the equation, compute the value of U at the origini.e. at r 0. - (ii) Without finding the solution to the equation, determine the location of the maxima and minima of U in N. (Hint: sin2 0 - 1-cos 20 = 2 [1
Transcribed Image Text:(a) What form does the Laplace equation take in polar coordinates (r, 0)? (b) Let (r, 0) denote the usual polar coordinates. Show that if U(r, 0) is a harmonic function, then so is V(r, 0) = U(, -0). (c) Suppose that U is a solution to the Laplace equation in the disk = {r < 1} and that U (1,0) = 5 – sin² 0. (i) Without finding the solution to the equation, compute the value of U at the origini.e. at r 0. - (ii) Without finding the solution to the equation, determine the location of the maxima and minima of U in N. (Hint: sin2 0 - 1-cos 20 = 2 [1
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