MATH1116: Advanced Mathematics and Applications Assignment - Mathematics Assignment Help

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Question 1. Integrability (Show Working) 25 points Suppose that f is a 2-variable real-valued function defined on a rectangle D, that is, f : [a, b] × [c, d] −→ R, with D = [a, b] × [c, d]. Also suppose that D0 is another rectangle that is a subset of D, so that D0 = [a 0 , b0 ] × [c 0 , d0 ] with a ≤ a 0 < b0 ≤ b and c ≤ c 0 < d0 ≤ d. Prove that if f is Riemann-Darboux integrable on D, then f is Riemann-Darboux integrable on D0 . [Hint: one approach is to use both the ‘if’ and the ‘only if’ parts of the test for integrability given in Analysis Lecture 2.]

Question 2. Upper Sums and Riemann Sums (Show Working) 25 points Suppose that f : [a, b] × [c, d] −→ R be a bounded function, and that P is a partition of [a, b] × [c, d]. Prove that the upper sum U(f, P) of f over P is the supremum of the set of all Riemann sums of f over P. Riemann sums, different to upper and lower sums, will be defined in Analysis Lecture 2. [Note: of course, a mirror image result is that L(f, P) is the infimum of the set of all Riemann sums of f over P, but you’re only asked to write out the proof of the upper sum result for this question.]

Question 3. Integrating Over Other Domains (Show Working) 25 points In the next few weeks, in lectures we will prove that a continuous multivariable function defined on a rectangular domain must be Riemann-Darboux integrable. You should note that we defined the Riemann-Darboux integral only for functions on domains that are n-dimensional rectangles. We have also stated without proof, and in exercises and assessments we have used without proof, a theorem concerning the integrability of multivariable functions defined on domains that are more general than just rectangles. While we will not prove the more general theorem, the purpose of this assignment problem is to familiarise you with some of the techniques involved in its proof. The intention is for you to prove the following very specific theorem (overleaf), while relying on some of the theorems we are still to prove: MATH1116, Assignment 1 4 Let T be the closed triangular region in R 2 with vertices at (0, 0), (1, 0) and (1, 1). Suppose that f : T −→ R is continuous. Define ˜f : [0, 1] × [0, 1] −→ R by ˜f(x, y) = ( f(x, y) if (x, y) ∈ T ; 0 if (x, y) ∈/ T . Then the Riemann integral Z [0,1]×[0,1] ˜f exists, and so we can define Z T f := Z [0,1]×[0,1] ˜f .

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