University of Toronto – MAT237Y1 – LEC5201
Multivariable calculus
Reading week questions
Jean-Baptiste Campesato February 21
st, 2020
Here are a few questions if you want to relax by doing mathematics after a few exhausting days of reading week!
Exercise 1. Let𝑈 ⊂ ℝ𝑛be an open set and𝑓 ∶ 𝑈 → ℝ𝑝be a𝐶1function. Let𝐾 ⊂ ℝ𝑛be a compact subset such that 𝐾 ⊂ 𝑈. We want to prove that𝑓|𝐾 ∶ 𝐾 → ℝ𝑝is Lipschitz.
1. Just for this question, we also assume that𝐾is convex. Prove that the conclusion holds.
(Hint: you may use the result fromhttp://www.math.toronto.edu/campesat/ens/1920/IFT-MVT.pdf) Now, we come back to the general statement and the compact set𝐾is no longer supposed to be convex.
We suppose by contradiction that𝑓|𝐾is not Lipschitz.
2. Prove that there exist two sequences(𝑥𝑛)𝑛∈ℕand(𝑦𝑛)𝑛∈ℕwith terms in𝐾 which are respectively convergent to some𝑥 ∈ 𝐾and𝑦 ∈ 𝐾such that∀𝑛 ∈ ℕ, ‖𝑓 (𝑦𝑛) − 𝑓 (𝑥𝑛)‖ > 𝑛‖𝑦𝑛− 𝑥𝑛‖
3. Prove that necessarily𝑥 = 𝑦.
(Hint: use that𝑓𝐾is continuous on a compact…)
4. Prove that there exist𝑟 > 0and𝑁 ∈ ℕsuch that𝐵(𝑥, 𝑟) ⊂ 𝑈and if𝑛 ≥ 𝑁then𝑥𝑛, 𝑦𝑛∈ 𝐵(𝑥, 𝑟).
5. Find a contradiction with Question 1 and conclude.
Exercise 2. Let𝑈 , 𝑉 ⊂ ℝ𝑛be two open sets.
LetΦ ∶ 𝑈 → 𝑉 be a homeomorphism which is also𝐶1(i.e. we assume thatΦis bijective,𝐶1and thatΦ−1is𝐶0, but we don’t assume thatΦ−1is𝐶1).
Let𝑇 ⊂ ℝ𝑛be a Jordan measurable set (i.e.𝑇 is bounded and𝜕𝑇 has zero content) such that𝑇 ⊂ 𝑈.
We want to prove thatΦ(𝑇 )is Jordan measurable (i.e.Φ(𝑇 )is bounded and𝜕(Φ(𝑇 ))has zero content).
1. Prove that𝑇 and𝜕𝑇 are compact.
2. Prove thatΦ(𝑇 )is bounded.
3. We want to prove thatΦ(𝜕𝑇 )has zero content
(a) Prove that a rectangle𝑅 ⊂ ℝ𝑛may be covered by finitely many squares𝑆1, … , 𝑆𝑞such that∑ 𝜈(𝑆𝑖) ≤ 2𝜈(𝑅).
A square is a rectangle whose edges have same length, i.e. a set of the form𝑆 = [𝑎1, 𝑏1] × ⋯ × [𝑎𝑛, 𝑏𝑛]where𝑏𝑖− 𝑎𝑖 = 𝑟 for some𝑟 > 0or equivalently of the form𝑆 = {𝑥 ∈ ℝ𝑛 ∶ |𝑥𝑖− 𝑎𝑖| ≤ 𝑟/2}for some center𝑎 ∈ ℝ𝑛and side length 𝑟 > 0.
(b) Let𝑓 ∶ 𝐴 → ℝ𝑛be Lipschitz with constant𝐶where𝐴 ⊂ ℝ𝑛. Let𝑆 ⊂ ℝ𝑛be a square of side length𝑟and center𝑎 ∈ 𝐴. Prove that𝑓 (𝐴 ∩ 𝑆)is included in a square of volume𝐶𝑛√𝑛𝑛𝑟𝑛.
Hint: you can start by proving that‖(𝑥1, … , 𝑥𝑛)‖ ≤ √𝑛max(|𝑥1|, … , |𝑥𝑛|).
Notice that we only assume that the center of𝑆is in𝐴, we don’t assume that𝑆 ⊂ 𝐴.
(c) Conclude.
Hint: use the result from Exercise1 4. Prove that𝜕(Φ(𝑇 )) = Φ(𝜕𝑇 ).
2 Reading week questions 5. Conclude.
Exercise 3. Let𝑈 ⊂ ℝ𝑛be open,𝜎 ∶ 𝑈 → ℝ𝑝be𝐶1where𝑛 < 𝑝and a rectangle𝑅 = [𝑎1, 𝑏1] × ⋯ × [𝑎𝑛, 𝑏𝑛] ⊂ 𝑈. We want to prove that{𝜎(𝑡) ∶ 𝑡 ∈ 𝑅}has zero content.
Let𝑃 be the partition of𝑅defined by subdividing each[𝑎𝑖, 𝑏𝑖]into𝑁subintervals of the same length.
1. Prove that for𝑖 = 1, … , 𝑝, there exists𝐶𝑖such that∀𝑆subrectangle, ∀𝑥, 𝑦 ∈ 𝑆, |𝜎𝑖(𝑦) − 𝜎𝑖(𝑥)| ≤ 𝐶𝑖‖𝑦 − 𝑥‖.
2. Prove that if𝑆is a subrectangle then𝜎(𝑆)is included in a rectangle of volume 𝐶
𝑁𝑝 where𝐶is a constant.
3. Conclude.
In the above exercises you proved the following theorems:
Theorem 1. Let𝑈 ⊂ ℝ𝑛be an open set,𝑓 ∶ 𝑈 → ℝ𝑝be a𝐶1function and𝐾 ⊂ 𝑈a compact set.
Then𝑓|𝐾∶ 𝐾 → ℝ𝑝is Lipschitz.
Remark. Beware, it is false that a𝐶1function is Lipschitz (e.g.𝑓 ∶ ℝ → ℝdefined by𝑓 (𝑥) = 𝑥2). However, as proved in the above theorem, a𝐶1function is locally Lipschitz.
The following result is important to justify that the integral in the change of variables formula is well-defined.
Theorem 2. Let𝑈 , 𝑉 ⊂ ℝ𝑛 be two open sets,Φ ∶ 𝑈 → 𝑉 be a homeomorphism which is also𝐶1 and𝑇 ⊂ ℝ𝑛a Jordan measurable set such that𝑇 ⊂ 𝑈.
ThenΦ(𝑇 )is Jordan measurable.
Theorem 3. Let𝑈 ⊂ ℝ𝑛be open,𝜎 ∶ 𝑈 → ℝ𝑝be𝐶1where𝑛 < 𝑝and𝑅 ⊂ ℝ𝑛be a rectangle such that𝑅 ⊂ 𝑈.
Then{𝜎(𝑡) ∶ 𝑡 ∈ 𝑅}has zero content.
