LongAssignment_2

# LongAssignment_2 - Oﬃce Problem 2 - Math 139 Due midnight...

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Unformatted text preview: Oﬃce Problem 2 - Math 139 Due midnight Dec. 10th Instructor Oﬃce Web page Mauro Maggioni 293 Physics Bldg. www.math.duke.edu/˜ mauro/teaching.html In this Oﬃce Problem you will be constructing the Riemann integral for functions f : Rn → R with n ≥ 1. (i) You will start by doing Project 4 in Chapter 4 (page 161) of the textbook, which takes you through the construction of the Riemann integral on 2-dimensional rectangles. (ii) After this, you can extend the deﬁnition of integral to higher dimensions: describe how you would deﬁne the Riemann integral on n-dimensional rectangles, i.e. sets in the form R = [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ], and discuss what parts of Project 4 would need to be modiﬁed, if any, in order to carry out this construction. (iii) Finally, we may extend the deﬁnition of these higher-dimensional Riemann integrals to more general domains than rectangles. Explain in detail how you would deﬁne the Riemann integral on a 2-dimensional unit ball, i.e. on the set {(x, y ) ∈ R2 : x2 + y 2 ≤ 1} ⊂ R2 : how would the reasonings in Project 4 change for such a domain? (iv) State a conjecture on which types of domains (besides rectangles and balls) to which you think you could extend the deﬁnition of Riemann integrals. I will be meeting with you individually to answer your questions about this long assignment: contact me by e-mail with times that work for you and we will arrange to meet. ...
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## This note was uploaded on 01/16/2012 for the course MATH 139 taught by Professor Pardon,w during the Fall '08 term at Duke.

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