MATH 223 Problems Problem Set for Final, Spring 2017

Math 223 - Advanced Calculus, Spring 2017 Review exercises, part 3 Exercise 1. (1) Let f : R2 → R be a function of class Ck. Suppose that f(x, 0) = 0 for all x ∈ R. Show that there exists a Ck−1 function g : R2 → R such that f(x, y) = yg(x, y). (Hint: consider the function g(x, y) = � 1 0 ∂f ∂y(x, ty)dt). (2) Let ρ(x, y) = y − h(x) be of class Ck and put Z = {ρ = 0} ⊂ R2. Let f : R2 → R be a Ck function vanishing on Z (this means that f(v) = 0 for all v ∈ Z). Show that there exists a Ck−1 function g : R2 → R such that f = ρg. (Hint: similar to (1) consider g(x, y) = � 1 0 ∂f ∂y(x, (1 − t)h(x) + ty)dt). (3) Let ρ : R2 → R be a Ck function Z = {ρ = 0}, and suppose that 0 ∈ Z is not a critical point for ρ. Let f be as in (2). Show that in a neighborhood of 0 we have f = ρg for a certain Ck−1 function g. (Hint: use the implicit function theorem - you can assume that the function obtained by using it is Ck). (4) With the notation of (3), show that ∇f(v) is a multiple of ∇ρ(v) for all v ∈ Z. Exercise 2. Let Z = {(x, y) ∈ R2 : y = x2}. Using the Lagrange multipliers method (established in the sixth assignment, Ex. 4), find the local maxima and local minima on Z of the following functions f : R2 → R: (1) f(x, y) = −y; (2) f(x, y) = x + y; (3) f(x, y) = x2 + y2 − 2y + 1. Exercise 3. Using the Lagrange multipliers method, find the local maxima and local minima on Z of the following functions f : R2 → R: (1) f(x, y) = x + y, Z = {(x, y) ∈ R2 : x2 + y2 = 1}. (2) f(x, y) = x2 + y2, Z = {(x, y) ∈ R2 : xy = 1}. Exercise 4. Let U ⊂ Rn be an open domain, and let Ωp(U) be the vector space of p-forms. (1) Define ⋆ : Ω1(U) → Ωn−1(U) in the following way: if ω = �n j=1 fjdxj, ⋆ω = �n j=1(−1)j−1fjdx1 . . . dxj−1dxj+1 . . . dxn. Show that ⋆ is a linear isomorphism. (2) Associate in the usual way a vector field F = (f1, . . . , fn) to the 1-form ωF = � j fjdxj. Assume n = 3, and let F, G be vector fields. Show that ⋆ωF×G = ωFωG (here F × G is the cross product). (3) From the definition, the product ω(⋆ω) is an n-form. Show that � U ω(⋆ω) ≥ 0 for all ω ∈ Ω1(U), and that the equality holds if and only if ω = 0. (4)** Show that ∥ω∥ = � U ω(⋆ω) defines a norm on Ω1(U). Exercise 5. Let ψ : Rn → Rn be a C1 diffeomorphism, and let ω be a p-form on Rn. We say that ω is ψ-invariant if ψ∗ω = ω. (1) Fix a diffeomorphism ψ. Show that the set of ψ-invariant forms in Ωp(Rn) is a vector space. (2) Show that if ω, η are ψ-invariant then ψη and dω are also ψ-invariant. (3) Given θ ∈ [0, 2π], define the (linear) diffeomorphism ψθ : R2 → R2 as ψθ(x, y) = ((cos θ)x − (sin θ)y, (sin θ)x + (cos θ)y) (the rotation of angle θ). Show that the 1-form (x2 + y2)(xdy − ydx) is ψθ-invariant for all θ ∈ [0, 2π]. (Hint: the easiest way is to show that x2 + y2 and xdy − ydx are ψθ-invariant, and to use (2)). (4)* Let ψ be a diffeomorphism satisfying ψ ◦ ψ(v) = v for all v ∈ Rn. Show that any p-form ω can be written as ω = η1 + η2, where ψ∗η1 = η1 and ψ∗η2 = −η2. Exercise 6. Let φ : [−1, 1] × [−1, 1] → R3 be the function defined as φ(s, t) = (s + t, s − t, st), and put Σ = φ([−1, 1] × [−1, 1]). (1) Show that Σ is a regular surface and compute the normal vector at any point. (2) Let ω be the 2-form ω = −dx2dx3 −x1dx1dx2. Compute � Σ ω (with the orientation on Σ given by φ). (3) Define the vector field F as F(x1, x2, x3) = (−1, 0, −x1). Check that � Σ F = � Σ ω. Exercise 7. Let f : R3 → R be a C1 function, and suppose that f(0) = 0. Show that there exist continuous functions g1, g2, g3 : R3 → R such that f = x1g1 +x2g2 +x3g3. (Hint: fixed (x1, x2, x3) ∈ R3 define the function ℓ : R → R as ℓ(t) = f(tx1, tx2, tx3). Check that f(x1, x2, x3) = � 1 0 ℓ′(t)dt. Find ℓ′(t) by using the chain rule.) Exercise 8. Recall that a regular curve is a C1 function γ : [a, b] → Rn such that γ′(t) ̸= 0 for t ∈ [a, b]. (1) Let h : R → R be C1, and define Z = {(x, y) ∈ R2 : y = h(x)}. Show that there exists a regular curve γ : R → R2 such that γ(R) = Z. (2) Let ρ : R2 → R be C1, and let Z = {(x, y) ∈ R2 : ρ(x, y) = 0}. Assume that 0 ∈ Z and ∇ρ(0) ̸= 0. Show that there exists ϵ > 0 and a regular curve γ : [−ϵ, ϵ] → R2 such that γ([−ϵ, ϵ]) ⊂ Z. (Hint: use the implicit function theorem and (1)). (3)* Let Z = {(x, y) ∈ R2 : x2 = y3}. Show that there exists no regular curve γ : R → R2 such that γ(R) = Z. Exercise 9. Let ρ be a C1 real function in Rn, let Z = {ρ = 0} and suppose that ∇ρ does not vanish on Z. Let γ : R → Rn be a C1 curve such that γ(t) ∈ Z for all t ∈ R. Show that γ′(t) is orthogonal (w.r.t. the usual Euclidean product) to ∇ρ(γ(t)) for all t ∈ R. Let us recall some notation: in Rn, we denote by ∇ the operator ∇ = ( ∂ ∂x1, . . . , ∂ ∂xn). If f is a C1 function, the vector field ∇f = ( ∂f ∂x1, . . . , ∂f ∂xn) is called the gradient of f, while if G = (g1, . . . , gn) is a C1 vector field, the function ∇ · G = ∂g1 ∂x1 + . . . + ∂gn ∂xn (also written simply ∇G) is called the divergence of G. The composition ∇2 = ∇◦∇, also denoted by ∆, is the Laplace operator or Laplacian; for any C2 function f one has ∆f = ∂2f ∂x2 1 + . . . + ∂2f ∂x2n. Exercise 10. (1) Let g : Rn → R be a continuous function. Suppose that � B g = 0 for any ball B ⊂ Rn. Show that g = 0. (2) A C2 function h : Rn → R is called harmonic if ∆h = 0. Let now f : Rn → R be a C2 function, and suppose that we have � ∂B ∇f = 0 for any ball B ⊂ Rn. Show that f is harmonic. Exercise 11. Let f, g : R3 → R be functions of class C2, and let U be a domain of R3 whose boundary bU = Σ is a regular surface. Show that � U f∆g + � U ∇f · ∇g = � Σ f∇g. Exercise 12. We say that a function f : Rn → R has compact support if there exists R > 0 such that f(v) = 0 for all v ∈ Rn such that ∥v∥ > R (in other words f is zero outside a large enough ball). We also say that a p-form ω = � i1,...,ip fi1...ipdxi1 . . . dxip has compact support if all the functions fi1...ip have compact support. (1) Suppose that ω ∈ Ω1(R2) has compact support. Show that there exists R > 0 such that � BR(0) dω = 0. (2)* Suppose that f, g : R3 → R have compact support. Show that for R > 0 large enough we have � BR(0) f∆g = � BR(0) g∆f.