Question 1. (a) Consider R with lower limit topology Tab and R² with the producttopology 7, = 7]a,b] X 7[a,b[ generated by the upper and lower limit topologies. Is thefunction defined as g : R → R² with the rule g(x) = (x, 2) an embedding? T(b) In (a) if the product topology T is considered as T7 = T[a,b[ × 7[a,b[, which of thefollowing results can be obtained and why?Explain your answers.(p): R is homeomorphic to R²(g): Rx {2} is a To-space

Check whether the function g is injective and continuous. Injectivity: Suppose g(x1) = g(x2) for…

Answered: Question 1. (a) Consider R with lower…

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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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)
5. Let W be the space of polynomials p(x) with real coefficients of degree < 10, satisfying p(0) = p(1) = p'(1) = 0. The dimension of W is (a) 1 (b) 7 (c) 8 (d) 9 (е) 11
2. Suppose f is holomorphic in an open set UCC and a E U. Show that for any integer n > 0 there is a unique holomorphic function h:U → C such that f(z) = f(a)+f'(a)(z-a)+ f"(a)²- a)² 2! (z – a)" +f(m) (a). | +(z-a)"+'h(z), ze U. n! n+1
3) Let u be the usual topology on the real line R and t be the upper limit topology on R. Furthermore let f: R → R be defined by S6) = {x+2 if x 1. f (x) (x + 2 Show that f is not u-u continuous. Show that f is t-T continuous. Also Draw its Graph.
15. Let f: A → B and g: C → D. Define fx g = {(a, c), (b, d): (a, b) = f and (c,d) = g}. (a) Prove that f × g is a mapping from A × C to B x D. (b) For (a, c) = A x C, write (f× g)(a, c) in terms of f(a) and g(c).
If Two Entire Functions Agree On A Segment Of The Real Axis, Must They Agree On C? Use Uniqueness Theorem Suppose f(z) is analytic in a domain D and that {zn} is a sequence of distinct points converging to a point zo in D. If f(zn) = 0 for each n, then f(z) = 0 throughout D. Identity Theorem Suppose {zn} is a sequence of points having a limit point in a domain D. If f(2) and g(2) are analytic in D with f(zn) = g(zn) for each n, then f(2) = g(2) throughout D.
The cyclic rule is a sometimes useful identity that applies to the situation in which three variables; x, y, and z are related to one another by a function, z = z(x, y). The rule is: (az/ax)(ax/ay)(ay/əz) = -1 (cyclic rule) Show that the cyclic rule holds for. x² + 2y²+ (1/2)z = 12
Let φ1 = 1, φ2 = x, φ3 = x2 on the interval 0 ≤x ≤1. Find the following inner products:(a) < φ1,φ2 >(b) < φ2,φ3 >(c) Distance between φ1 and φ2
Show that (1) => (2) and (2) => (1). (1) X is connected. (2) There is no continuous function f : X → {-1,1}, where {-1,1} is equipped with discrete topology. Note: The function need not be surjective.

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