Consider the smooth function I f: R2 R3, (s, t) ((2+ cos(s)) (cos(t), sin(t)), sin(s)). It...

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Consider the smooth function I f: R2 R3, (s, t) ((2+ cos(s)) (cos(t), sin(t)), sin(s)). It can be shown that S = f(R2) CR3 is a smooth surface diffeomorphic to T₁ = S¹× S¹. (Indeed, the function : S¹× S¹ →S, (es, eit) f(s, t) is a diffeomorphism.) Let : R2 → S be the abbreviation of f. 2.1. For (s, t) R², let n(st) ER³ be the unit outward pointing normal vector to S at f(s, t), so that n(s,t)v=0 for all v€ Tf(s,t)S. Using the fact that Ĵ is a local diffeomorphism, find a formula for (st) in terms of s and t. 2.2. Using 2.1, we can define the function g: SS², f(s, t)n(at) which can be shown to be smooth. Determine the critical points of g, the regular values of g and g¹(a) for a € S2 any regular value of g. Justify your answers. → Consider the composition gof: R2 S2, use the chain rule and the fact that fis a local diffeomorphism. Consider the smooth function I f: R2 R3, (s, t) ((2+ cos(s)) (cos(t), sin(t)), sin(s)). It can be shown that S = f(R2) CR3 is a smooth surface diffeomorphic to T₁ = S¹× S¹. (Indeed, the function : S¹× S¹ →S, (es, eit) f(s, t) is a diffeomorphism.) Let : R2 → S be the abbreviation of f. 2.1. For (s, t) R², let n(st) ER³ be the unit outward pointing normal vector to S at f(s, t), so that n(s,t)v=0 for all v€ Tf(s,t)S. Using the fact that Ĵ is a local diffeomorphism, find a formula for (st) in terms of s and t. 2.2. Using 2.1, we can define the function g: SS², f(s, t)n(at) which can be shown to be smooth. Determine the critical points of g, the regular values of g and g¹(a) for a € S2 any regular value of g. Justify your answers. → Consider the composition gof: R2 S2, use the chain rule and the fact that fis a local diffeomorphism.