Math 642-600 Assignments — Spring 2012

Assignment 1 - Due Friday, 1/27/2012.

• Do the following problems.
  1. Problem 2, page 204 (§ 5.1).
  2. Problem 7, page 204 (§ 5.1).
  3. Hanging chain problem. A chain having uniform linear density ρ hangs between the points (0,0) and (1,0). The total mass m, which is fixed, and the total energy E of the chain are
     
     Assuming that the chain hangs in a shape that minimizes the energy, find the shape of the hanging chain.
  4. Consider the functional J(y) = ∫_a^bF(x,y,y′)dx, where y ∈ C¹ and y(a)=A, y(b)=B are fixed.
     1. Let y extremize J, subject to all of the conditions above. Verify that if η ∈ C¹ and η(a)=η(b)=0, then
        Δ J = ∫_a^b (-g(x) + F_y′(x,y,y′))η′(x)dx, where g(x) = ∫_a^xF_y(u,y,y′)du.
        
     2. Use (a) to show that -g+F_y′(x,y,y′)=c, a constant. From this, it follows that F_y′(x,y,y′)∈ C¹. (Du Bois-Reymond, 1879.)
  5. Smoothing spline. Let J[y] = ∫₀¹ (y′(x))²dx. Let the admissible set for J be all piecewise C¹ curves on [0,1] that satisfy y(k/n) = y_k , for k = 0, 1, 2, ..., n, with the discontinuities in y′ appearing only at the points k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(k/n,y_k), k = 0, ..., n}.
  6. Let x = (x,y) and let J[x] = x²y(x² + y²)^{− 1} if x ≠ 0, J[0] = 0. Show that the Gâteaux derivative exists at at x = 0 for every direction η = (h,k). Also, show that the Fréchet derivative doesn't exist at x = 0.

Assignment 2 - Due Wednesday, 2/8/2012.

• Read sections 5.2.1, 5.2.2, and 5.4.
• Do the following problems.
  1. Problem 11, page 204 (§ 5.1).
  2. Problem 12, page 205 (§ 5.1).
  3. Problem 4, page 205 (§ 5.2).
  4. Problem 6, page 205 (§ 5.2).
  5. Problem 7, page 206 (§ 5.2). (Find the Hamiltonian and write out Hamilton's equations.)
  6. Problem 8, page 206 (§ 5.2). Also, find the Hamiltonian of the system, and write out Hamilton's equations for it. Finding integrals (constants) of the motion should follow immediately from these equations. (Note: There is an error in the expression for L. The (dθ/dt)² sin²θ) should be replaced by (dφ/dt)² sin²θ.)
  7. Find the Legendre transformation H(p) for F(x) = x^T A x, where x is in Rⁿ and A is a symmetric, positive definite n×n matrix.

Assignment 3 - Due Friday, 2/17/2012.

• Read sections 6.1 and 6.2.
• Do the following problems.
  1. Problem 1, page 207 (§ 5.4).
  2. Problem 4, page 208 (§ 5.4).
  3. Problem 6, page 208 (§ 5.4). Take Ω to be a 2D disk centered at 0 and having radius r = a. How does the lowest eigenvalue change with the radius a?
  4. Problem 8, page 208 (§ 5.4).
  5. Use a quadratic polynomial to estimate the lowest eigenvalue for the Sturm-Liouville problem with Lu = − u′′ = λu, u(0) = 0, u′(1)+u(1) = 0.
  6. In class we showed that the Hamiltonian H = (p_r² + r ^{− 2}p_θ²)/(2m) − K/r for a planet in orbit around the sun. In addition, we also found that H = E and p_θ = L are constants of the motion.
     1. Show that p_r = Lr ^{− 2}dr/dθ = −L d(r ^{− 1})/dθ.
     2. Let u = 1/r − mK/L². Show that 2m E/L² + (mK/L²)² = (du/dθ)² + u². Solve this first order differential equation for u in terms of θ.
     3. Use your solution to the previous part to show that the orbit is an ellipse.

Assignment 4 - Due Friday, 2/24/2012.

• Read sections 6.4 and 6.5.
• Do the following problems.
  1. Problem 3, page 274 (§ 6.1).
  2. Problem 4, page 208 (§ 6.1).
  3. Problem 6, page 208 (§ 6.1).
  4. Problem 5(c), page 275 (§ 6.2). (The answer in the back of the book is wrong.)
  5. Problem 6, page 276 (§ 6.2).
  6. Problem 9, page 276 (§ 6.2).
  7. Find all of the Laurent expansions for f(z) = ((z² − 1)(z+2))⁻¹ about z = 0.

Assignment 5 - Due Wednesday, 3/28/2012.

• Read sections 7.1 and 7.2.
• Do the following problems.
  1. Show that Γ(z) given in equation (6.23) is analytic in z for Re(z) > 0. In addition, again for Re(z) > 0, show that Γ′(z) is given by the formula on the top of p. 260.
  2. Problem 3(a), page 279 (§ 6.5).
  3. Problem 5(a), page 279 (§ 6.5).
  4. Problem 6(a), page 279 (§ 6.5).
  5. Problem 8, page 280 (§ 6.5).
  6. A planet moving around the Sun in an elliptical orbit, with eccentricity 0 < ε < 1 and period P has time and angle related in the following way. Let τ = (2π/P)(t − t_p), where t_p is the time when the planet at perihelion -- i.e., it is nearest the Sun. Let θ be the usual polar angle and let u be an angle related to θ via
     (1 − ε)^1/2 tan(u/2) = (1 + ε)^1/2 tan(θ/2).
     It turns out that τ = u − ε sin(u). All three variables θ, u, and τ are measured in radians. They are called the true, eccentric, and mean anomalies, respectively. (Anomaly is another word for angle.)
     1. Show that one may uniquely solve τ = u − ε sin(u) for u = u(τ), that u is an odd function of τ, and that g(τ) = u(τ) − τ is a 2π periodic function of τ.
     2. Because g is odd and 2π periodic, it can be represented by a Fourier sine series,
        g(τ) = ∑_n b_n sin(nτ).
        Show that b_n = (2/n)J_n(nε), n = 1, 2, ..., where J_n is the n^th order Bessel function of the first kind. Thus, we have that
        u = τ + ∑_n (2/n)J_n(nε) sin(nτ).
        

Assignment 6 - Due Wednesday, 4/4/2012.

• Read sections 7.1 and 7.2.
• Do the following problems.
  1. Problem 9, page 280 (§ 6.5).
  2. Problem 13, page 280 (§ 6.5).
  3. Problem 19, page 281 (§ 6.5).
  4. Problem 2(a,b), page 328 (§ 7.1). For each operator in 2(a), 2(b), find the norm and adjoint L*.
  5. Problem 4, page 329 (§ 7.1).
  6. Let L: H→ H be a bounded operator. Show that σ(L) ⊆ {λ ∈ C : |λ| ≤ ||L||}.

Assignment 7 - Due Friday, 4/13/2012.

• Read sections 7.2.1, 7.2.2, 10.1
• Do the following problems.
  1. Suppose that L is a densely defined self-adjoint operator on a Hilbert space H, with domain D(L).
     1. Show that L is a closed operator. (Use what you know about adjoints.)
     2. You are given this theorem:
        Let T:H → H be a one-to-one closed linear operator with dense domain D(T) and range R(T) = H, then T⁻¹: H → D(T) is bounded.
        Use this to show that if Im(λ) ≠ 0, then the resolvent R_λ = (L−λI )⁻¹ is bounded and satisfies ||R_λ|| ≤ |Im(λ)|⁻¹.
     3. Show that the spectrum σ(L) is real and that the residual spectrum σ_r(L) = ∅. (This means that σ(L) = σ_d(L)∪σ_c(L).)
  2. Let L = L* satisfy < Lf,f > ≥ 0. Show that σ(L) ⊆ [0,∞).
  3. Problem 1(b), page 329 (§ 7.2).
  4. Problem 5, page (§ 7.2).
  5. Problem 6(b,c), page 329 (§ 7.2).
  6. Problem 7, page 329 (§ 7.2).
  7. Problem 9, page 329 (§ 7.2).

Assignment 8 - Due Friday, 4/27/2012.

• Read sections 10.1-10.3
• Do the following problems.
  1. Problem 11(a), page 330 (§ 7.2).
  2. Consider the one dimensional heat equation, u_t = u_xx, with u(x,0) = f(x), where − ∞ < x < ∞ and 0 ≤ t <∞. By taking the Fourier transform in x, show that the solution u(x,t) is given by
     u(x,t) = ∫_R K(x−y,t)f(y)dy, K(ξ,t) = e^−ξ2/4t (4πt)^{− 1/2}.
     (The function K(x−y,t) is the one-dimensional heat kernel.)
  3. Let f be in Schwartz space, and let g be C^∞ and satisfy
     |g^(m)(t)| ≤ c_m(1+t²)^n_m
     for all nonnegative integers m. Here c_m and n_m depend on g and m. Show that fg is in Schwartz space. Explain how to define the product g(x)T(x), where T is a tempered distribution.
  4. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and e^ibxT(x).
  5. Let T(x) := (1 - |x|)₊, which is a "hat" function. Explain why this is a tempered distribution. Find T′ and T′′, along with their Fourier transforms. Use these to find the Fourier transform of T.
  6. Problem 1, page 7 in the notes, Tempered Distributions, by Professor Joel Feldman (University of British Columbia).

Assignment 9 - Not to be handed in.

• Read sections 10.1-10.3
• Do the following problems.
  1. Problem 7, page 464 (§10.3)
  2. Problem 9, page 465 (§10.3)
  3. Problem 11, page 465 (§10.3)