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- Prove that topological space E is not homeomorphic to the spaceY = {(x, y) ∈ E^2 : y = ± x} (E represents R equipped with Euclidean distance, E^2 represents R^2 equipped with euclidean distance)Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the…1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?
- A. Let H be the set of all points (x, y) in ℝ2 such that x2 + 3y2 = 12. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are homotopy equivalent. (E^2 represents R^2 equipped with euclidean topology)Give the set of limit points A0 of a singleton A = {(5, 2)} on the plane R2 with the discretemetric.
- A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?A. Let H be the set of all points (x, y) in ℝ2 such that x2 + xy + 3y2 = 3. Show that H is a closed subset of ℝ2 (considered with the Euclidean metric). Is H bounded?Does there exist a surface x(u, v) such that E=G=1,F=0 and L=1,M=−1,N=0?