Let Re: R² R² be the map that takes a vector (x, y) and rotates it by the angle (in radians) counterclockwise about the origin, where 0 € [0,2). Write down a formula for R₂(x, y). Let Rø,x, Rø,y : R² → R be the component functions of Re so that R₁(x, y) = (Ro,z(x, y), Rø,y(x, y)) for all (x, y) = R². Show that: əxR0,x(x, y) = dyRo,y(x,y) and Oy Ro, z(x, y) = −ðÂRo,y(x, y) Suppose u: R2 → R is a C² function that satisfics: Ru(x, y) + Ru(x, y) = 0 Show that the same is true of the function u o Re for any 0 € R. for all (x, y) = R²
Let Re: R² R² be the map that takes a vector (x, y) and rotates it by the angle (in radians) counterclockwise about the origin, where 0 € [0,2). Write down a formula for R₂(x, y). Let Rø,x, Rø,y : R² → R be the component functions of Re so that R₁(x, y) = (Ro,z(x, y), Rø,y(x, y)) for all (x, y) = R². Show that: əxR0,x(x, y) = dyRo,y(x,y) and Oy Ro, z(x, y) = −ðÂRo,y(x, y) Suppose u: R2 → R is a C² function that satisfics: Ru(x, y) + Ru(x, y) = 0 Show that the same is true of the function u o Re for any 0 € R. for all (x, y) = R²
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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Transcribed Image Text:1. (a) Let Re: R2 R2 be the map that takes a vector (x, y) and rotates it by the angle (in radians)
counterclockwise about the origin, where € [0, 27). Write down a formula for R(x, y). Let
Ro,r, Roy R² → R be the component functions of Re so that Re(x, y) = (Ro,z(x, y), Ro,y(x, y)) for
all (x, y) E R2. Show that:
dr Ro,z(x, y) = dy Roy(x, y) and dyRo,x(x, y) = −d Ro,y (x, y)
(b) Suppose u: R²R is a C² function that satisfics:
u(x, y) + u(x, y) )=0
Show that the same is true of the function u o Re for any 0 € R.
(c) More generally, suppose that U and V are open subsets of R2 and f: U → V is a C² diffeomorphism
so that if f (f1, f2),
for all (x, y) = R²
d₁f1(u₁, U₂) = d2f2(u₁, u2) and ₂f1(u₁, u₂) = −0₁f2(u₁, U₂)
If : V→ R is a C² function that satisfies the equation:
(v₁, v₂)+(v₁, v₂) = 0 for all (v1, V₂) € V
= (po f): U → R. That is,
(₁, ₂) + 2/(U₁, U₂) = 0 for all (u₁, U₂) EU
Show that the same is true of
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