Define φ : Q[x] → Q by φ() = . (a) Prove that φ is a ring homomorphism. (b) Find the kernel of φ. and" + ...a12 + ao We were unable to transcribe this image
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
(5) Let Ф, : Z5[x] Zg denote the evaluation homomorphism at r Zg Find a nonzero polynomial of smallest degree which in kernel of all φ-for r 0,1,2,3,4 (5) Let Ф, : Z5[x] Zg denote the evaluation homomorphism at r Zg Find a nonzero polynomial of smallest degree which in kernel of all φ-for r 0,1,2,3,4
Find the gradient ∇φ of the following: a) φ = (r2/a2)e-r/a (using spherical coordinates) b) φ = 2√(x2+y2+z2) (using both cartesian and spherical coordinates, after converting)
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f). (7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
Answer to (a) is image = Z2 • {0,2} (where • is the external direct product). And the kernel is {e,r^2} (where r is the rotation). Answer to (c) is isomorphic to Z2 • Z2. Please show work. I’m given answers but need to see how to get there. Thanks (20 poiants) Amer aocat (a) (5 points) Identify the kernel and image of the homomorphism from D, to Z2 Z1 (the infinite cyclic group) given by the rules p(r) (1,0...
6. Show that the followings define metrics on R2: For r = (11, 12), y = (y1, y2) ER, the company = 139-un +100 - 247 91.42.), y =(1,9) ER di(x,y) = |21 - y1| + |22 - y2), doo (I, y) = max{\21 – yı], \12 - y2|}.
x = f(x) 5. Determine the flow φι: R2-+ R2 for the nonlinear system (1) with -zi f(x) = | and show that the set S = {x E R2 | x2 =-x/4) is invariant with respect to the flow φ x = f(x) 5. Determine the flow φι: R2-+ R2 for the nonlinear system (1) with -zi f(x) = | and show that the set S = {x E R2 | x2 =-x/4) is invariant with respect to the...
2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아 C R2 has volume zero in R2 and the set B-{(x, ψ (x)) : x E R} C R2 has measure zero in IK. 2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아...