(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(1, -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
(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(1, -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
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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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(1, -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
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