Math 642-600 Current Assignment — Spring 2011
Assignment 1 - Due Friday, 1/28/2011.
- Do the following problems.
- Problem 2, page 204 (§ 5.1).
- Problem 6, page 204 (§ 5.1).
- Problem 7, page 204 (§ 5.1).
- 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.
- Consider the functional J(y) =
∫abF(x,y,y′)dx, where y ∈
C1 and y(a)=A, y(b)=B are fixed.
- 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 = ∫ab (-g(x) +
Fy′(x,y,y′))η′(x)dx, where g(x) =
∫axFy(u,y,y′)du.
- Use (a) to show that -g+Fy′(x,y,y′)=c, a
constant. From this, it follows that
Fy′(x,y,y′)∈ C1. (Du
Bois-Reymond, 1879.)
- 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.
- Problem 9, page 204 (§ 5.1).
- Problem 11, page 204 (§ 5.1).
- Problem 4, page 205 (§ 5.2).
- Problem 6, page 205 (§ 5.2).
- Problem 7, page 206 (§ 5.2).
- 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.)
- Find the Legendre transformation H(p) for F(x) =
xT 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.
- Section 5.4: 1 (This requires solving the three problems in (a),
(b), and (c).)
- 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.
- 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.
- Section 6.1: 4, 6
- Find the Laurent expansions for f(z) = ((z2 −
1)(z+2))−1 about z = 0.
- Section 6.2: 5(c), 6, 9
- 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.
- 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
- 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) = ∑nbnzn.
Show that the coefficients in the Laurent expansion f(z)g(z) = ∑ cnzn are given by
cn = ∑m ambn−m.
- 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.
- Let L: H→ H be a bounded operator. Show that σ(L)
⊆ {λ ∈ C : |λ| ≤ ||L||}.
- 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 PC below is self adjoint.
PC = (−2πi)−1 ∳C
R(λ)dλ
- Let L = L* satisfy < Lf,f > ≥ 0. Show that σ(L) ⊆
[0,∞).
- Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find
the norm and adjoint L*.
- Section 7.2: 1(b), 2.
Assignment 6 - Due Monday, 4/25/2011.
- Do the following problems.
- Section 7.2: 5, 6(b), 9, 10
- Consider the one dimensional heat equation, ut =
uxx, 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.)
- Let f be in Schwartz space, and let g be C∞
and satisfy
|g(m)(t)| ≤
cm(1+t2)nm
for all nonnegative integers m. Here cm and nm
depend on g and m. Show that fg is in Schwartz space.
- 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.
- Section 10.3: 1, 4, 7, 8, 9, 11
Updated: 5/3/11 (fjn)