Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y²=1}. Let U₁ = {(r,y) € S¹ r>0}, and 91: U₁ → (-₁), with 1(x, y) = arctan, S¹ | 1 <0}, U₂= {(1,y) Us=(r,y) S¹y>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, such that {(U₁,91), (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.
Exercice Consider the circle S¹, defined by S¹ = {(x, y) = R² | 2² + y²=1}. Let U₁ = {(r,y) € S¹ r>0}, and 91: U₁ → (-₁), with 1(x, y) = arctan, S¹ | 1 <0}, U₂= {(1,y) Us=(r,y) S¹y>0}, U₁= {(r,y) € S¹|r>0}. 1. Construct functions 92, 93, and 94, such that {(U₁,91), (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.
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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