11. Letn 22 and f: D" → R" be smooth. Suppose ||f(x)-x|| < 1 for all x € S-1. Prove that therefis some x € D" so that f(x) = 0. (Hint: If not, show that the restriction of the map -:D" S-1->|||f||to a D" is homotopic to the identity map.)

Given: n≥2 and f:Dn→Rn is smooth, ∥fx-x∥<1 for all x∈Sn-1. To Prove: There is some x∈Dnso that…

Answered: 11. Letn 22 and f: D"R" be smooth.…

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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1a
1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
2. A map f: Rm → R" is called an open map if for every U open in Rm, its image f(U) is open in R". (a) Find a function f: R → R that is continuous but not open. (b) Find a function ƒ : R² → R that is both continuous and open.
how would you prove this
1C
Cove Ocas1= ocasa<1
2. Let T: P₁ (R) → R2 be the map given by T(f) = (f(1), f" (1)) where P4 (R) is the set of polynomials of degree 4 or less and f"(1) is the evaluation of the third derivative of f at 1, i.e.. (a) Show that T is a linear map. d.r.³ a=1
(c) How many primitive roots does 19 have in total? (d) Find all primitive roots of 19 and list them. The list should only contains the positive numbers less than 19.
Definition. Let f: X→Y and ACX. The direct image of A under f is the set f(A) = {y €Y | 3a € A such that f(a) = y}. Remark. Another way to define the direct image of A under f is as the set ƒ(A) = {f(a) | a € A}. Problem 2. Do the following for the function f: R²R given by f(x, y) = x² + y². (a) For A = {(-1,0), (0, 1), (1,0), (2, 1)}, list the elements of f(A). (b) What is f(R²)? Is it all of R or something else? Problem 3. Let f: X→Y, and A and B be subsets of X. Prove the following statements. (a) If ACB, then f(A) ≤ f(B). (b) f(ANB) ≤ f(A) nf(B).
Attached.

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11. Letn 22 and f: D"R" be smooth. Suppose f(x)-x< 1 for all x € S-1. Prove that there f is some x E D" so that f(x) = 0. (Hint: If not, show that the restriction of the map - o aD" is homotopic to the identity map.) : DS-1