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Problem 2. Let C R² be a bounded smooth domain and p, 4 € C²(N). Consider a vector field v = (v₁, v₂): → R2 such that v E C¹ (N) and V· v=0. at √ (v(x) · Vp(x)) y(x) dæ - √ (v(x) · Vy(x)) p(x) dx . Ω (2) Assume now that p(t, ∙) E C'! (N) solves dtp + v · Vp = 0. Show that d 1/√₁\p(t, x) |²dx = 0. (1) Prove that ==

Problem 2. Let C R² be a bounded smooth domain and p, 4 € C²(N). Consider a vector field v = (v₁, v₂): → R2 such that v E C¹ (N) and V· v=0. at √ (v(x) · Vp(x)) y(x) dæ - √ (v(x) · Vy(x)) p(x) dx . Ω (2) Assume now that p(t, ∙) E C'! (N) solves dtp + v · Vp = 0. Show that d 1/√₁\p(t, x) |²dx = 0. (1) Prove that ==

Advanced Engineering Mathematics
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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Problem 2. Let N C R² be a bounded smooth domain and p,y E C(N)", Consider a vector field (v1, v2) : N → R² such that v E C'(N) and V · v = 0. V = (1) Prove that (v(x) · Vp(x)) (x) dæ = – -,(v(x) · Vy(x)) p(æ) dæ (2) Assume now that p(t, ·) E C-(N) solves dp+ v · Vp = 0. Show that d le(t, æ)l²dæ = 0. dt
Transcribed Image Text:Problem 2. Let N C R² be a bounded smooth domain and p,y E C(N)", Consider a vector field (v1, v2) : N → R² such that v E C'(N) and V · v = 0. V = (1) Prove that (v(x) · Vp(x)) (x) dæ = – -,(v(x) · Vy(x)) p(æ) dæ (2) Assume now that p(t, ·) E C-(N) solves dp+ v · Vp = 0. Show that d le(t, æ)l²dæ = 0. dt
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