212 Assignments

Math 212 Assignments

Homework 1, due Friday, February 8

6.1.1
6.1.3 (no proofs required)
6.1.4
6.2.3
6.3.5 (proof required)
1. Let \(X\) and \(Y\) be sets of real numbers. Define \[X+Y:=\{x+y:x\in X\text{ and }y\in Y\}.\] Show that if \(\tilde{x}=\sup X\) and \(\tilde{y}=\sup Y\), then \(\tilde{x}+\tilde{y}=\sup (X+Y)\).
2. Let \(X\) be a set of real numbers, and let \(c>0\). Define \(cX=\{cx:x\in X\}\). Prove that if \(\tilde{x}=\sup X\), then \(c\tilde{x}=\sup(cX)\).

Homework 2, due Friday, February 15

6.2.7 (Use problem [A], below, and disregard the comment in the text about letting \(f=0\).)
6.3.4 (hint: MVT)
6.4.3 (hint: consider \(F_2-F_1\), and see the previous hint)
6.5.6 (note: it is not obvious that \(\chi_R\) is integrable)
[A] Let \(X\) be a non-empty subset of \(\mathbb{R}^n\), and let \(f\) and \(g\) be real-valued functions on \(X\). Suppose that \(f(x)\leq g(x)\) for all \(x\in X\) and that \(f\), hence \(g\), is bounded below on \(X\). Prove that \(\inf f(X)\leq \inf g(X)\).
[B] Evaluate the integrals of the following functions using Fubini’s theorem.
1. \(f(x,y) = x^2+y\) on \([0,1]\times[0,2]\)
2. \(g(x,y) = x^2+y\) on \(\{(x,y): x\geq0,\ y\geq0,\ x+y\leq1\}\)
3. \(h(x,y,z) = x\) on \(\{(x,y,z): x\geq0,\ y\geq0,\ z\geq0,\ x+y+z\leq1\}\).

Homework 3, due Friday, February 22

Homework 4, due Friday, March 1

6.7.1
6.7.3
6.7.8
6.7.10
6.7.11
6.7.14 (abc)
Let \(v=(1,0,3)\) and \(w=(2,3,-4)\). Write \(v\otimes w\) as a linear combination of tensor products of the standard basis vectors.

Homework 5, due Friday, March 8