3. The Laplace-Beltrami operator on Poincare disk (of unit radius) is given by (1– 72)² 2 (1 –- r²)² [1 a where V = (r) + is the Laplace operator on the plane, in polar coordinates r and 0. The Poisson kernel is defined by 1- r2 1- r2 P(r,6 - 0) = A 1- 2rCos (o – 0) + r²" where A = 1- 2rCos (6 – 6) + r? and o is a fixed/constant angle. a) Find ApP(r,o- 0) =? b) Let H = PÀ, where A is a fixed number. Find A,H =? (Hint: Try to express VH interms of (@P/dr)? + [(1/r)(@P/d0)}² and use what you have found above.) [Homework: Study/Learn Poincare disk and calculate its Gauss curvature. (The Poincare disk is a surface of constant negative curvature.) (Though this will not help you either solving the problem above or gaining any points, this is part of Mathematics and Physics culture.)]

3. The Laplace-Beltrami operator on Poincare disk (of unit radius) is given by (1– 72)² 2 (1 –- r²)² [1 a where V = (r) + is the Laplace operator on the plane, in polar coordinates r and 0. The Poisson kernel is defined by 1- r2 1- r2 P(r,6 - 0) = A 1- 2rCos (o – 0) + r²" where A = 1- 2rCos (6 – 6) + r? and o is a fixed/constant angle. a) Find ApP(r,o- 0) =? b) Let H = PÀ, where A is a fixed number. Find A,H =? (Hint: Try to express VH interms of (@P/dr)? + [(1/r)(@P/d0)}² and use what you have found above.) [Homework: Study/Learn Poincare disk and calculate its Gauss curvature. (The Poincare disk is a surface of constant negative curvature.) (Though this will not help you either solving the problem above or gaining any points, this is part of Mathematics and Physics culture.)]

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

3. The Laplace-Beltrami operator on Poincare disk (of unit radius) is given by
(1 – r²)² [1 a
where V = (r) + is the Laplace operator on the plane, in polar coordinates r
and 0. The Poisson kernel is defined by
1- r2
1-r
P(r, – 0) =
A
1- 2rCos (o – 0) + r² '
where A = 1- 2rCos (6 – 0) +r? and o is a fixed/constant angle.
a) Find ApP(r, o – 0) =?
b) Let H = PA, where A is a fixed number. Find AH =? (Hint: Try to express VH
interms of (@P/dr)? + [(1/r)(@P/d0)]² and use what you have found above.)
[Homework: Study/Learn Poincare disk and calculate its Gauss curvature. (The Poincare
disk is a surface of constant negative curvature.) (Though this will not help you either solving
the problem above or gaining any points, this is part of Mathematics and Physics culture.)]

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