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Unformatted text preview: MAT237Y – Multivariable Calculus Summer 2009 Assignment # 1 with corrections. Questions are worth 10 marks each. Due Tuesday, May 19, at 6:10pm sharp. 1. (a) Using the dot product, show that for x, y ∈ Rn , the formula 2x holds. (b) The norm on Rn can be deﬁned in terms of the dot product by √ the formula x = x • x. Show that the reverse is true. That is, ﬁnd a formula for x • y involving the norms of vectors ( x , y , x + y , and x − y for example), and without using coordinates. 2. (a) If A ⊂ Rm , and Bn ⊂ Rm for n ∈ N, show that
∞ ∞ 2 +2 y 2 = x+y 2 + x−y 2 A\ Bn =
n=1 n=1 (A \ Bn ). (b) Show if the subsets An ⊂ Rm are open for n ∈ N, then the countable union ∞ An is open. n=1 3. Construct a set S ⊂ R such that • S, • int S , • S, • int S , and • int(S ) are all distinct (that is, no two of them are equal). No late assignments. 1 ...
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