Midterm_Solutions

# Midterm_Solutions - Problem 1 It is easy to see that the...

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Problem 1. It is easy to see that the tangent vector to the curve ( r ( t ) ( t )) in polar coordi- nates has components ˙ r and r ˙ θ . So, the length of a curve is l = Z b a q r ( t )) 2 + ( r ( t ) ˙ θ ( t )) 2 dt. Using this formula we compute length of the curve (2sin πt,πt ): l = Z 1 0 p 4 π 2 cos 2 πt + 4 π 2 sin 2 πtdt = 2 π. Problem 2. Let P = { x 0 ,...,x n } be a partition of [ a,b ]. Then we can construct partition Q = { y 1 ...,y n } of [ c,d ] such that x i = φ ( y i ). Also, for any partition of [ c,d ] we can construct corresponding partition of [ a,b ]. Since g = f φ , values of f on [ x i - 1 ,x i ] are the same as values of g on [ y i - 1 ,y i ]. So, we have U ( Q,g,β ) = U ( P,f,α ) , L ( Q,g,β ) = L ( P,f,α ) . The statement follows from these equalities and Theorem 6.6 from [Rudin]. Problem 3. We deﬁne d ( E,F ) = m (( F E ) \ ( F E )). Let d ( E,F ) = 0 and d ( F,G ) = 0. First, we see that
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