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such that ) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, (i) Au= 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uVu. Can you make the same conclusion if we replace replace condition (ii) with “Vu · ñ = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen-

such that ) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, (i) Au= 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uVu. Can you make the same conclusion if we replace replace condition (ii) with “Vu · ñ = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen-

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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4 such that ) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, (i) Au= 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uVu. Can you make the same conclusion if we replace replace condition (ii) with “Vu · ñ = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen- tial equations.
Transcribed Image Text:4 such that ) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, (i) Au= 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uVu. Can you make the same conclusion if we replace replace condition (ii) with “Vu · ñ = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen- tial equations.
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