Multivariable calculus!

University of Toronto – MAT237Y1 – LEC5201

Multivariable calculus!

Jean-Baptiste Campesato October 10^th, 2019

Question 1(At home). Review the in-class questions from Sep 17 and from Sep. 24 (the solutions are online).

Question 2. Let𝐴 ⊂ ℝ^𝑛and𝐵 ⊂ ℝ^𝑚.

1. Prove that if𝐴and𝐵are open then𝐴 × 𝐵 ⊂ ℝ^𝑛+𝑚is too.

2. Prove that if𝐴and𝐵are closed then𝐴 × 𝐵 ⊂ ℝ^𝑛+𝑚is too.

Question 3. Let𝐴, 𝐵 ⊂ ℝ^𝑛. Prove the following statements: 1.𝐴 ⊂ 𝐵 ⟹ 𝐴 ⊂ ̊̊ 𝐵 2.𝐴 ⊂ 𝐵 ⟹ 𝐴 ⊂ 𝐵 Question 4. Let𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄ ⊂ ℝ²be defined by

𝑆₁ = [1, 3] × [0, 4] 𝑆₂ = {(𝑥, 𝑦) ∈ ℝ² ∶ (𝑥 + 2)²+ (𝑦 − 2)² < 4}

𝑆₃ = {(𝑥, 𝑦) ∈ ℝ² ∶ 𝑥²+ 𝑦²= 4} 𝑆₄= {(𝑡³, −𝑡²) ∶ 𝑡 ∈ ℝ}

1. Draw𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄.

2. Prove that𝑆₁, 𝑆₂, 𝑆₃, 𝑆₄are path-connected.

3. Prove that𝑆₁∪ 𝑆₂∪ 𝑆₃∪ 𝑆₄is path-connected.

Question 5. Prove that𝑆 = {(𝑥, 𝑦) ∈ ℝ² ∶ 1 < 𝑥²+ 𝑦²≤ 4}is path-connected.

Question 6. Let𝑆 = {x∈ ℝ^𝑛 ∶ ‖x‖ = 1}and𝑓 ∶ 𝑆 → ℝbe a continuous function.

1. (★) Prove that𝑆is path-connected.

2. (★) Prove that there existsx∈ 𝑆 such that𝑓 (x) = 𝑓 (−x). 3. Prove that𝑆is compact.

4. Prove that𝑓 has a min and a max.

Question 7. Are the sets from Question 1 and Question 2 open? closed? bounded? compact?

Question 8. Let𝑓 ∶ ℝ^𝑛 → ℝbe a continuous function.

1. Guess and write the formal definition (i.e.𝜀 − 𝛿-style) of lim

‖x‖→+∞𝑓 (x) = +∞.

2. (★) Prove that if lim

‖x‖→+∞𝑓 (x) = +∞then𝑓 has a min.

Question 9. Let(𝑆_𝑘)_𝑘∈ℕ

≥0be a decreasing sequence of non-empty compact subsets ofℝ^𝑛: 𝑆₀⊃ 𝑆₁ ⊃ 𝑆₂ ⊃ ⋯ ⊃ 𝑆_𝑘⊃ ⋯ 1. (★) Prove that ⋂

𝑘∈ℕ_≥0

𝑆_𝑘≠ ∅

2. Can the above intersection be empty if we replace compact by closed?

3. Can the above intersection be empty if we replace compact by bounded?

2 Reviews

Question 10(Arithmetico–geometric sequences).

Let(𝑢_𝑛)_𝑛∈ℕ

≥0be a real-valued sequence satisfying

∀𝑛 ∈ ℕ_≥0, 𝑢_𝑛+1= 𝛼𝑢_𝑛+ 𝛽 1. We assume that𝛼 = 1. Find an algebraic expression for𝑢_𝑛(in terms of𝑢₀).

2. We assume that𝛼 ≠ 1.

(a) Find𝑟 ∈ ℝsuch that𝑣_𝑛 = 𝑢_𝑛− 𝑟is a geometric sequence.

(b) Deduce an algebraic expression for𝑢_𝑛(in terms of𝑢₀).

Question 11. We define a sequence(u_𝑛)_𝑛≥0inℝ²byu₀= (1, 0)andu_𝑛+1= ¹

2u_𝑛+ (1, 2).

Is it convergent? If so, what is its limit?

Question 12. Let𝑆 = {(𝑥, 𝑦) ∈ ℝ² ∶ 𝑥 ∈ ℚand𝑦 ∉ ℚ}.

1. Without computing its interior or closure, tell whether𝑆is closed and/or open.

2. Compute𝑆̊,𝑆and𝜕𝑆.

We recall that (see the notes from Oct. 1)

• ∀𝑥₁, 𝑥₂ ∈ ℝ, 𝑥₁< 𝑥₂ ⟹ (∃𝑞 ∈ ℚ, 𝑥1 < 𝑞 < 𝑥₂)

• ∀𝑥₁, 𝑥₂ ∈ ℝ, 𝑥₁< 𝑥₂ ⟹ (∃𝑟 ∈ ℝ ⧵ ℚ, 𝑥1< 𝑟 < 𝑥₂)

Question 13. Let𝑓 ∶ (𝑎, 𝑏) → ℝbe a function defined on an open interval and let𝑥₀∈ (𝑎, 𝑏).

1. Recall the definition of “𝑓 is differentiable at𝑥₀” from MAT135/137/157.

2. Recall the definition of “𝑓 is differentiable at𝑥₀” from MAT237.

3. Prove that𝑓 is differentiable at𝑥₀ in the sense of MAT1XX if and only if 𝑓 is differentiable at𝑥₀ in the sense of MAT237.

4. If𝑓 is differentiable at𝑥₀, how are the derivative of𝑓 at𝑥₀and the differential of𝑓 at𝑥₀related?

Question 14. Are the following functions differentiable? If so determine their differentials and gradients.

1. 𝑓 (𝑥, 𝑦) = 𝑒^𝑥𝑦(𝑥 + 𝑦)onℝ². 2. 𝑓 (𝑥, 𝑦, 𝑧) = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥onℝ³.

Question 15. Let𝑓 ∶ ℝ² → ℝbe defined by𝑓 (𝑥, 𝑦) = {

𝑥𝑦²

𝑥²+𝑦² if(𝑥, 𝑦) ≠ (0, 0) 0 otherwise 1. Is𝑓 continuous?

2. Do the partial derivatives of𝑓 exist? Are they continuous?

3. Is𝑓 differentiable?

Question 16. Same questions as above for𝑓 ∶ ℝ²→ ℝbe defined by𝑓 (𝑥, 𝑦) = {

𝑥𝑦^{𝑥2−𝑦2}

𝑥²+𝑦² if(𝑥, 𝑦) ≠ (0, 0)

0 otherwise

Question 17. Prove that the following functionsℝ²→ ℝadmit directional derivatives at(0, 0)along any direction but are not continuous at(0, 0).

1. 𝑓 (𝑥, 𝑦) = {

𝑥²𝑦

𝑥⁴+𝑦² if(𝑥, 𝑦) ≠ (0, 0) 0 otherwise 2. 𝑔(𝑥, 𝑦) =

{

𝑦²ln(|𝑥|) if𝑥 ≠ 0

0 otherwise

Question 18. Let𝑓 ∶ ℝ² → ℝdefined by𝑓 (𝑥, 𝑦) = √𝑥²+ 𝑦².

1. Prove that𝑓 is differentiable at(4, 3)and determine its differential at(4, 3).

2. Give an approximation of√4.05²+ 2.93². (★)≡advanced question.