2. Let U C R³ be the upper hemisphere, i.e.,U = {(x, y, z) € R³: x² + y² + z² = 1, z ≥ 0}.ELet D C R² be the closed disk, i.e., D = {(x, y): x² + y² ≤ 1}.Give an explicit formula for a homeomorphismf:U → D. and its inverse f¯¹:D → U. Draw a simple picture.

Answered: Image /qna-images/answer/06a9e049-4271-4bd3-8812-546f8b1d3d74.jpg

Answered: 2. Let U C R³ be the upper hemisphere,…

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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(c) Let z(x, y) = x be a function defined on a disk D in the positive quadrant containing the point (1,2). Prove whether z(x, y) satisfies the Clairaut Theorem at (1, 2). [3,4,4] Question Five
1? b. Find a conformal map f (z) which takes the disk {z : z < 3} onto the left half plane {w: Re(w) <0} and satisfies f (3) 0.
If G: R3 --> R3 is a function, define the Jacobian associated to G viewed as a change of variables. Compute the Jacobian of the function G, sending polar coordinates (ρ, θ, Φ) to their respective Cartesian coordinates (x, y, z).
3. Consider the following graphs of rational functions. Write down the horizontal and vertical asymptotes, zeros, and y-intercept, as well as any holes (if applicable). Use this information to find a function R(r) for each graph. Note some graphs have the vertical and horizontal asymptotes marked for you, but the others are not marked. a. b. -9 -8 -7 -6-5 -6 -5 4 -3 -1 3 -2 1 2 3 4 56 -1 10 6 -4 2 0 2 4 -6 -8 -10- -12- 1 2 3 4 5 6
Let V be the vector space of polynomials in over R of degree ≤ 3 and define an inner product f . g = f¹ f(x)g(x)dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa: V → V by a(ƒ)(x) = f(x) + xƒ'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product? Select one: O O O a is self-adjoint a is orthogonal a is 'anti-self adjoint' meaning a* = -α a = I-a, where I denotes the identity map None of the others apply
Answer this with some brief explanation please. Thank you!
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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2. Let U C R³ be the upper hemisphere, i.e., с U = {(x, y, z) € R³: x² + y² + z² = 1, z ≥ 0}. Let D C R² be the closed disk, i.e., D = = {(x, y): x² + y² ≤ 1}. Give an explicit formula for a homeomorphism f:U → D. and its inverse f¯¹:D → U. Draw a simple picture.