(a) What form does the Laplace equation take in polar coordinates (r, 0)?
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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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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdddfff36-bd46-4e8f-b66f-7a4677280cea%2F93f249af-250a-4af0-bcbc-bc522e03d083%2Fvs0jbbn_processed.jpeg&w=3840&q=75)
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 ( 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 Questions
Read through expert solutions to related follow-up questions below.
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
(Hint: sin2
θ =
1−cos 2θ
2
.)
Solution
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