Math 642-600 Assignments — Spring 2018

Assignment 1 - Due Monday, 2/5/2018.

• 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. The point is that one need not assume that y′′ exists.)
  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 x_k = k/n. Use the previous problem to show that the minimizer for J is a linear spline passing through all of the points {(x_k,y_k), k = 0, ..., n}. (Hint: y(x) is in C¹[x_k, x_k+1]).)

Assignment 2 - Due Monday, 2/12/2018.

• Do the following problems.
  1. Problem 5, page 204 (§ 5.1).
  2. Problem 9, page 204 (§ 5.1).
  3. Problem 11, page 204 (§ 5.1).
  4. Show that the functional $J(y):=\int_0^1( xy^2+y'^2)dx$ is Fréchet differentiable, if the the $C^1[0,1]$ norm is used.
  5. The metric of a surface in $\mathbb R^3$ having parametric form $x=x(u,v),y=y(u,v), z=z(u,v)$ is defined by $ds^2=E(u,v)du^2 + 2F(u,v)dudv +G(u,v)dv^2$. Consider the torus (surface of a "doughnut") obtained by rotating a circle with center $(R,0,0)$ and radius $r < R $ about the $z$ axis. Find the metric for it and use it to obtain the 2nd order differential equations for the geodesics.
  6. Problem 6, page 205 (§ 5.2).
  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 Monday, 2/19/2018.

• Do the following problems.
  1. Suppose that the kinetic energy for a system is given by $T(\underline{q}, \underline{q'})= \sum_{i,j=1}^n A_{i,j}(\underline{q})q_i'q_j'$, where $\underline{q}=(q_1,q_2,\ldots,q_n)$ and the matrix $A$ is positive definite. (In matrix form, $T=\underline{q}'^{\,T}A\underline{q}'$.) Use your solution to problem 7 in assignment 2 to find $T$ in terms of $\underline{q}$ and $\underline{p}=(p_1,p_2,\ldots,p_n)$. Show that the Hamiltonian has the form $H(\underline{q},\underline{p}) = T(\underline{q},\underline{p}) +V(\underline{q})$.
  2. Problem 7, page 206 (§ 5.2). (See Fig. 5.4, p. 205 for a diagram.)
  3. Problem 8, page 206 (§ 5.2). (Note: There is an error in the expression for L. The (dθ/dt)² sin²θ) should be replaced by (dφ/dt)² sin²θ.)
  4. For a planet having mass $m$ in orbit around the sun (mass=$M$), the potential is $V(r)=-mMG/r$, where $G$ is the gravitational constant. In addition, the polar form of its Lagrangian is $ L=m\dot r^{2}/2 + mr^2 \dot \theta^{2}/2+ mMGr^{-1}$. ($\dot r=dr/dt$, and $\dot \theta=d\theta/dt$).
     1. Show that the Hamiltonian $H=\frac{1}{2m} (p_r^2+r^{-2}p_\theta^2) - mMGr^{-1}$.
     2. Show that the angular momentum $p_\theta=\ell$ is a constant of the motion.
     3. Show that the Hamiltonian is a constant of the motion, $H=E$, where $E$ is the total energy of the system. (To keep the planet in a finite orbit, we need to have $E<0$.)
     4. Show that $p_r = \ell r^{-2} \frac{dr}{d\theta} = - \ell \frac{d}{d\theta}(r^{-1})$.
     5. Let $u = r^{-1}- m^2MG\ell^{-2}$. Show that $2m E\ell^{-2} + m^4M^2G^2\ell^{-4} = \big(\frac{du}{d\theta}\big)^2 + u^2$. To simplify notation, let $\rho=(m^2MG\ell^{-2})^{-1}$ and $\gamma=2m\rho^2 E \ell^{-2}$. This puts the equations above in the form $u=r^{-1} -\rho^{-1}$ and $\rho^{-2}(1+\gamma)=\big(\frac{du}{d\theta}\big)^2 + u^2$. (We assume that $-1<\gamma<0$, so that $0< 1+\gamma <1$.) Solve this differential equation for u in terms of $\theta$ and obtain $r=r(\theta)$.
     6. Use your solution to the previous part to establish Kepler's laws of planetary motion:
        1. The orbit of a planet is an ellipse, with the sun at one focus.
        2. The radius vector from the sun to the planet sweeps out equal areas in equal times.
        3. The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. The proportionality constant is the same for all planets.

Assignment 4 - Due Monday, 2/26/2018.

• Read section 6.2.
• Do the following problems.
  1. Problem 4, page 205 (§ 5.2). (Attaching a mass at the midpoint $x_c$ amounts to making changing the density from $\rho$ to $\rho+ m\delta(x-x_c)$.)
  2. Problem 6, page 205 (§ 5.2). Find the Hamiltonian in terms of $p=m\dot x$ and $q=x$. Find a first order ode that describes the system. (Hint: use conservation of energy.)
  3. Problem 1, page 207 (§ 5.4). (Hint: Use Minimax Principle.)
  4. 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.
  5. 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 a? Explain.

Assignment 5 - Due Wednesday, 3/7/2018.

• Read sections 6.2 and 6.4.
• Do the following problems.
  1. Do problems 4, 5, 11, 12, 15 in the Exercises for Complex Variables

  2. Section 6.1: 4, 6

  3. Section 6.2: 6, 9, 10, 11

  4. Section 6.4: 1

  5. Find all of the Laurent expansions about $z=0$ for $f(z) = \frac{1}{(z^2-1)(z+2)}$.

  6. Let $C_0$ and $C_1$ be simple closed curves, with $C_0$ contained in the interior of $C_1$. Suppose that $f(z)$ is analytic on the two curves and in the region between them, and that there is a simple closed curve $C$ in the region between $C_0$ and $C_1$. Show that all of the coefficients of the Laurent expansion for $f$ are given by $a_n=\frac{1}{2\pi i}\int_{C} \frac{f(\zeta)}{(\zeta - z)^{n+1}}d\zeta$.

  7. (Isolation of zeros.) Suppose that $f(z)$ is analytic in the disk $|z-z_0| < R$ and that $f(z_0)=0$. Show that if $f(z)$ is not identically 0, then there exists $0<\rho< R$ such that $f(z)\ne 0$ for all $z$ in $0<|z-z_0| \le\rho$. Use this to show that $\sin^2(z)+\cos^2(z) = 1$ for all $z\in \mathbb C$, given that it holds for all real $z$.

Assignment 6 - Due Wednesday, 3/21/2018.

• Read section 6.5.1, 6.5.2 and 7.1.
• Do the following problems.
  1. Section 6.4: 3, 7 (assume $a$ and $k$ are positive), 9, 20, 25.

  2. Suppose that $f(z)$ has a pole of order $m>1$ at $z=z_)$. Show that $\mathrm{Res}_{z_0}f= \frac{1}{(m-1)!}\frac{d^{m-1}}{z^{m-1}}\big\{(z-z_0)^mf\big\}(z_0)$. Use this to find the integral $\int_{-\infty}^\infty \frac{e^{i\omega x}}{(1+x^2)^2}dx$.

  3. Suppose that $f(z)$ is analytic in and on a positively oriented simple closed curve $C$, except for isolated singularities at $z_1,z_2,\ldots,z_n$ inside the curve, and one simple pole at a point $z_0$ on the boundary. If $z_0$ is at a corner of the curve and if the exterior angle at the corner is $\alpha$, show that, in the sense of principal values, \[ \oint_Cf(z)dz= 2\pi i\sum_{j=1}^n \mathrm{Res}_{z_j}(f) + i\alpha \,\mathrm{Res_{z_0}}(f). \]

  4. Let $\xi \in \mathbb R$ and consider the integral $G(\xi)=\int_{-\infty}^\infty e^{-x^2}e^{i\xi x}dx$. Show that $G(\xi)= \sqrt{\pi}e^{-\xi^2/4}$, given that $G(0)=\sqrt{\pi}$.

  5. The Gamma function has simple poles at $z=0,-1, -2, \ldots$. Show that the residue at $z=-n$ is $\frac{(-1)^n}{n!}$. (Hint: use $\Gamma(z+1) = z\Gamma(z)$.)

Assignment 7 - Due Wednesday, 4/11/2018.

• Read section 7.2.1.
• Do the following problems.
  1. Section 6.5: 3(a), 7, 8(b), 9 (Just do $J_{3/2}$.), 13, 22
  2. Show that the Wronskian $W(J_{-\nu}(z),J_\nu(z))=\frac{2\sin(\pi \nu)}{z}$.
  3. Section 7.1: 1

Assignment 8 - Due Wednesday, 4/25/2018.

• Read section 7.2.1 and 7.2.3.
• Do the following problems.
  1. Let $L$ be a bounded linear operator on $\mathcal H$; that is, $L \in {\mathcal B}(\mathcal H)$. Show that $\rho(L)$ contains the exterior of the disk $\{\lambda \in {\mathbb C} \colon |\lambda| \le ||L||\}$. (Consequently, $\sigma(L)$ is contained in the disk.)

  2. Section 7.1: 2(a), 2(b). For each operator in 2(a), 2(b), find the norm and adjoint $L^\ast$. (You may use problem 1 above.)

  3. Let $L = L^\ast$ be defined on a domain $D$. In class, we showed that $\sigma_r(L) = \emptyset$ and that $\sigma_d(L) \subset \mathbb R$. Show that $\sigma_c(L) \subseteq \mathbb R$. Thus, $\sigma(L)$ is a subset of the reals.

  4. Let $L=A$ be a self-adjoint, $n\times n$ matrix. Suppose that $A$ has $r$ distinct eigenvalues, arranged in the order $\lambda_1 < \lambda_2 < \cdots < \lambda_r$. In addition, let the orthogonal projections onto the eigenspaces be $Q_1,\ldots,Q_r$. Suppose $a < b$, and neither is an eigenvalue of $A$; let $C_\varepsilon$ be a positively oriented rectangular contour, with corners $a-i\varepsilon$, $b-i\varepsilon$, $b+i\varepsilon$, and $a+i\varepsilon$. Show that \[ \lim_{\varepsilon\downarrow 0}\,-\frac{1}{2\pi i} \oint_{C_\varepsilon} (A-\lambda I)^{-1}d\lambda = E_b-E_a, \] where $E_\lambda := \sum_{j=1}^r H(\lambda - \lambda_j)Q_j$ and $H(t)=\begin{cases} 1 & t\ge 0\\0 & t<0\end{cases}$.

  5. Find the Green's function for the operator $Lu = -u''$, with $D_L=\{u,u''\in L^2[0,\infty): u'(0)=0\}$. Use it and Stone's formula for the spectral family of $L$ to derive the Fourier cosine transform pair: \[ F(\xi)= \frac{2}{\pi}\int_0^\infty f(x)\cos(\xi x)dx \ \text{and}\ f(x)= \int_{0}^\infty F(\xi)\cos(\xi x)d\xi \]

Assignment 9 - Not to be handed in.

• Suggested problems (Fourier transform convention: $\hat f(\xi) = \int_{-\infty}^\infty f(x)e^{-i\xi x}dx$.)
  1. Prove the following theorems:
     1. If $f,g \in L^1(\mathbb R) \cap L^2(\mathbb R)$, then $f\ast g\in L^1(\mathbb R) \cap L^\infty(\mathbb R)$.

  2. Section 7.2: 6(a,b,c).

  3. Consider the one dimensional heat equation, $u_t = u_{xx}$, with $u(x,0) = f(x)$, where $-\infty < x < \infty$ and $0 < t < \infty$. In 7.2.6(c) you found that, after scaling, ${\mathcal F}[e^{-x^2}] =\sqrt{\pi}e^{-\xi^2/4}$. By taking the Fourier transform in $x$, show that the solution $u(x,t)$ is given by \[ u(x,t) = \int_{\mathbb R}K(x-y,t)f(y)dy,\ K(\xi,t) = (4\pi t)^{-1/2} e^{-\xi^2/4t}. \] (The function $K(\xi,t)$ is the one-dimensional heat kernel.)

  4. Let T be a tempered distribution. Find the Fourier transforms for T′, xT(x), T(x−a), and e^ibxT(x).

  5. Let $f$ be in Schwartz space, $\mathcal S$, and let $g$ be $C^\infty$ and satisfy $|g^{(m)}(x)| \le c_m(1+x^2)^{n_m}$ for all nonnegative integers m. Here $c_m$ and $n_m$ depend on $g$ and $m$. Show that $fg$ is in Schwartz space. Use this result to explain how to define the product $g(x)T(x)$, where $T$ is a tempered distribution.