Exercice Consider the circle S¹, defined by S¹ = {(x,y) = R² | ²+²=1}. Let U₁ = {(x,y) S¹ r>0}, and p1: U₁(-), with 1(x, y) = arctan, U₂ = {(x,y) Us=(x, y) S¹|r<0}, Sy>0}, U₁= {(r,y) € S¹r>0}. 1. Construct functions 92, 93, and 4, such that {(U1₁, 1), (U2, 42), (U3, 43), (U₁, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0
Exercice Consider the circle S¹, defined by S¹ = {(x,y) = R² | ²+²=1}. Let U₁ = {(x,y) S¹ r>0}, and p1: U₁(-), with 1(x, y) = arctan, U₂ = {(x,y) Us=(x, y) S¹|r<0}, Sy>0}, U₁= {(r,y) € S¹r>0}. 1. Construct functions 92, 93, and 4, such that {(U1₁, 1), (U2, 42), (U3, 43), (U₁, Py)} is a smooth atlas. 2. Show theoretically that the torus T2 = S¹ x S¹ is a smooth Manifold. 3. Use the Atlas constructed in question 1 to construct a smooth atlas on T². 4. Let 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.2: Properties Of Division
Problem 53E
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