(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 harmonicfunction, then so is V (r, 0) = U ( ½, –0).(c) Suppose that U is a solution to the Laplace equation in the disk = {r ≤ 1} andthat U(1,0) = 5 - sin² 0.(i) Without finding the solution to the equation, compute the value of U at theorigin i.e. at r = = 0.(ii) Without finding the solution to the equation, determine the location of themaxima and minima of U in 2.(Hint: sin² 0 =1-cos 20².)

Given that, what form does the Laplace equation take in polar coordinates:

Answered: (a) What form does the Laplace equation…

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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Question

(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 ( ½, –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
origin i.e. at r = = 0.
(ii) Without finding the solution to the equation, determine the location of the
maxima and minima of U in 2.
(Hint: sin² 0 =
1-cos 20
².)

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Follow-up Question

Suppose that U is a solution to the Laplace equation in the disk Ω = {r ≤ 1} and
that U(1, θ) = 5 − sin2
θ.
(i) Without finding the solution to the equation, compute the value of U at the
origin – i.e. at r = 0.
(ii) Without finding the solution to the equation, determine the location of the
maxima and minima of U in Ω.
(Hint: sin2
θ =
1−cos 2θ
2
.)

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