2. (8 pts) In contrast to the Heine-Borel Theorem in R", show that the closed unit...
(a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1} is compact in R3. (b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous bi- jective function f between R2 and S2 such that the inverse function f−1 is continuous). Question 1. (2 marks) (a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere is compact in R3. (b) Show that R2 and S2...
Real Analysis II Please do it without using Heine-Borel's theorem and do it only if you're sure Problem: Let E be a closed bounded subset of En and r be any function mapping E to (0,∞). Then there exists finitely many points yi ∈ E, i = 1,...,N such that Here Br(yi)(yi) is the open ball (neighborhood) of radius r(yi) centered at yi. Also, following definitions & theorems should help that E CUBy Definition. A subset S of a topological...
R i 11. Prove the statement by justifying the following steps. Theorem: Suppose f: D continuous on a compact set D. Then f is uniformly continuous on D. (a) Suppose that f is not uniformly continuous on D. Then there exists an for every n EN there exists xn and > 0 such that yn in D with la ,-ynl < 1/n and If(xn)-f(yn)12 E. (b) Apply 4.4.7, every bounded sequence has a convergent subsequence, to obtain a convergent subsequence...
8. Show that Theorem 3.1, the Nested intervals theorem, may be proved as a direct consequence of the Cauchy criterion for convergence (Theorem 3.14). (Hint: Suppose I. = {x: 0, <x<bn} is a nested sequence. Then show that {an} and {b} are Cauchy sequences. Hence they each tend to a limit. Since b.-4, 0, the limits must be the same. Finally, the Sandwiching theorem shows that the limit is in every 1.] Definition. An infinite sequence {n} is called a...
la. (5 pts) Show that A={,:n eN} is neither open nor closed; 1b. (15 pts) Let A= {(x, y) R2: \x - 21 <1, \y-1 <2}. Show that A is open and find A', A and A. Justify your answers.
Convex Optimization Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
1) (2 pts) Write a script to plot the function 15V4r +10 r29 10 r<0 for-5ss 30
(8) The Divergence Theorem for Flux in Space F(x, y, z) =< P, Q, R >=< xz, yz, 222 > S: Bounded by z = 4 – x² - y2 and z = 0 Flux =S} F înds S (8a) Find the Flux of the vector field F through this closed surface. (8) The Divergence Theorem for Flux in Space F(x,y,z) =< P,Q,R >=< xz, yz, 222 > S: Bounded by z = 4 – x2 - y2 and z...
Let C be the closed curve defined by r(t) = costi + sin tj + sin 2tk for 0 <t< 27. (a) (5 pts] Show that this curve C lies on the surface S defined by z = 2xy. (b) (20 pts] By using Stokes' Theorem, evaluate the line integral F. dr C where F(x, y, z) = (y2 + cos x)i + (sin y +22)j + xk
(15 pts) 7) Using the Divergence Theorem, calculate the flux integral JSF dĀ where F(x, y, z) =< 2 + x2,r2 + y2y +> and S is the closed cylinder 22 + y2 = 1 with 03:31.