Math 642-600 Current Assignment — Spring 2011

Assignment 1 - Due Friday, 1/28/2011.

• Do the following problems.
  1. Problem 2, page 204 (§ 5.1).
  2. Problem 6, page 204 (§ 5.1).
  3. Problem 7, page 204 (§ 5.1).
  4. 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.
  5. 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.)
  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, 2/9/2011.

• Read sections 5.2.1, 5.2.2, and 5.4.
• Do the following problems.
  1. Problem 9, page 204 (§ 5.1).
  2. Problem 11, page 204 (§ 5.1).
  3. Problem 4, page 205 (§ 5.2).
  4. Problem 6, page 205 (§ 5.2).
  5. Problem 7, page 206 (§ 5.2).
  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, 2/25/2011.

• Read sections 6.1, 6.2, and 6.4
• Do the following problems.
  1. Section 5.4: 1 (This requires solving the three problems in (a), (b), and (c).)
  2. Section 5.4: 6. 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? Explain.
  3. 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.
  4. Section 6.1: 4, 6
  5. Find the Laurent expansions for f(z) = ((z² − 1)(z+2))⁻¹ about z = 0.
  6. Section 6.2: 5(c), 6, 9
  7. Section 6.4: 2, 4, 7

Assignment 4 - Due Friday, 3/11/2011.

• Read sections 6.5, 7.1 and 7.2
• Do the following problems.
  1. Let p > 0. Use contour integration together with the rectangular contour with corners −R, R, R+ip/2, −R+ip/2 traced counterclockwise to show that
     ∫_−∞^∞ e^−x2+ipxdx = e^−p2/4√π
  2. Suppose that f(z) and g(z) have an isolated singularity at z = 0, and also that they have Laurent expansions given by
     f(z) = ∑_nzⁿ and g(z) = ∑_nb_nzⁿ. Show that the coefficients in the Laurent expansion f(z)g(z) = ∑ c_nzⁿ are given by
     c_n = ∑_m a_mb_n−m.
  3. Section 6.5: 5, 6a, 8b (Hint: use problem 2 above.), 9, 19

Assignment 5 - Due Friday, 4/8/2011.

• Read sections 7.1, 7.2.1, 7.3
• Do the following problems.
  1. Let L: H→ H be a bounded operator. Show that σ(L) ⊆ {λ ∈ C : |λ| ≤ ||L||}.
  2. Let L=L* and suppose that [0,1] is an isolated component of σ(L) (i.e., there is an open set S containing [0,1] such that S - [0,1] is in ρ(L)). Let C ⊂ ρ(L) be a positively oriented simple closed curve containing [0,1] in its interior. Show that the projection P_C below is self adjoint.
     P_C = (−2πi)⁻¹ ∳_C R(λ)dλ
  3. Let L = L* satisfy < Lf,f > ≥ 0. Show that σ(L) ⊆ [0,∞).
  4. Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find the norm and adjoint L*.
  5. Section 7.2: 1(b), 2.

Assignment 6 - Due Monday, 4/25/2011.

• Do the following problems.
  1. Section 7.2: 5, 6(b), 9, 10
  2. Consider the one dimensional heat equation, u_t = u_xx, with u(x,0) = f(x), where − ∞ < x < ∞ and 0 ≤ t x <∞. 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.
  4. Consider the continuous piecewise linear function f(x) is 0 outside of [−2,2] and passes through the points (− 2,0), (−1,2), (0,-1), (1,2), and (2,0). Find the Fourier transform of f this way. First, find the distributional derivative f'', which is a linear combination of δ functions. Then, find the Fourier transform f''. Finally, use Fourier transform properties to obtain the Fourier transform of f.

Suggested Problems - Not to be turned in.

• Read sections 10.1-10.3
• Do the following problems.
  1. Section 10.3: 1, 4, 7, 8, 9, 11