这是一份UCB的MATH 224A课程的数学考试代考成功案例
methematical methods in the physical siences
Problem 1.

1.

(a) The integral representation for the function $J_{2}(r)$ is
$$
J_{2}(r)=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{i r \sin (\theta)} e^{2 i \theta} d \theta
$$
Using this form, and any identities about Dirac delta functions you find useful, find the Fourier transform of this function, $\tilde{J}{2}(k)$.

(b) The functions $f(x)$ and $g(x)$ are defined by

and $g(x)=f(-x)$. Find the function $h(x)$, defined by $h(x)=\int_{-\infty}^{\infty} f(x-y) g(y) d y .$

Proof .

Problem 2.

  1. (a) Find the Green’s function for the equation
    $$
    y^{\prime \prime}-\frac{2}{\sin ^{2} x} y=f(x) \quad 0<x<\pi / 2
    $$
    with the boundary conditions $y(0)=y(\pi / 2)=0 .$ The solutions to the homogeneous equation are $\cot x$ and $1-x \cot x$.
    (b) Use the Green’s function to find the particular solution to this differential equation that satisfies the given boundary conditions, if $f(x)=\sin ^{2}(x)$. You do not need to evaluate any integrals that you end up with.

Proof .

Problem 3.

3.

A hollow sphere of radius 5 is cut out of an infinite metal block. The temperature on the surface of the sphere is maintained at $u(r=5, \theta)=$ $100 \sin ^{2} \theta .$ Find the temperature profile $u(r, \theta)$ in the metal block when the system reaches steady state.
Legendre polynomials:

$P_{0}(x)=1$

$P_{1}(x)=x$

$P_{2}(x)=\left(3 x^{2}-1\right) / 2$

$P_{3}(x)=\left(5 x^{3}-3 x\right) / 2$


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Math 224a: Mathematical Methods for the Physical Sciences. Fall 2021.

InstructorNikhil Srivastava, email: firstname at math.obvious.edu

Lectures: TTh 2:10-3:30pm, Hildebrand B56.

Office Hours: T: 3:40-4:40pm, Th 5-6pm (1035 Evans)

Textbooks. Much of the material will be drawn from the following two books. Both are available to Cal students for free on Oskicat or Springerlink.
Robert Richtmyer, Principles of Advanced Mathematical Physics, Volume I
Reed and Simon, Functional Analysis, Vol I
Jonathan Dimock, Quantum mechanics and quantum field theory [electronic resource] : a mathematical primer
Folland, Intro to PDE, 2e
Trefethen, Approximation Theory and Approximation Practice
I will also draw on several other resources and frequently post lecture notes on this webpage.

Announcements

  • (9/4) Office hours this week: Th 5-6 and F 4-5pm.
  • (9/24) Office thours this week: W 4-5 instead of Th.
  • (10/3) Lecture today is cancelled due to sickness.
  • (11/12) Office hours this week: F 4-5pm instead of T 3:40-4:40.

Syllabus The course will survey methods for solving the fundamental problems of mathematical physics. The overall purpose of the course will be to develop a functional analytic framework for understanding and approximating solutions of differential equations, with an emphasis on physical examples. The content can broadly be divided into three parts:

  1. Functional Analysis. (3 weeks) Lp spaces, Hilbert spaces, distributions, Schwartz functions, Fourier transform, potential theory.
  2. Spectral Theory. (7 weeks) Linear operators, adjoint, spectrum and resolvent, spectral theorem for bounded s.a. operators, Fredholm alternative, Green’s functions, Sturm-Liouville theory.
  3. Orthogonal Polynomials. (4 weeks) Classical orthogonal polynomials; approximation and interpolation theory.

Class Schedule

#DateTopicsReadingsNotesRemarks
1Th 8/29Lebesgue integral, monotone and dominated convergence, completeness of L1RS I.3lec1
2T 9/3L2, Hilbert spaces, separability, orthonormal bases.RS II.1,II.3lec2
3Th 9/5Weierstrass thm, separability of L2, projections, dual space, Riesz-Frechet thmRS II.2-II.3lec3
4T 9/10norm, adjoint, positivity, square rootRS VI.1-VI.2, VI.4lec4
5Th 9/12range and kernel, polar decomposition, compact operatorsRS VI.4-5lec5
6T 9/17spectral thm for compact operatorsRS VI.5lec6
7Th 9/19consequences of spectral thm, trace class operators, Fredholm alternativeRS VI.6lec7
8T 9/24basic ODE theorylec8
9Th 9/26Green’s function, completness of eigenfunctions, regular SL theorylec9
10T 9/31group worklec10
Th 10/3no lecture
11T 10/8oscillation theorylec11
Th 10/10power outage
12T 10/15resolvent, spectrumRS VI.3lec12
13Th 10/17uniform boundedness, spectral radius, multiplication operatorsRS I.4, VI.3, VI.1lec13
14T 10/22cts functional calculus, spectral theoremRS VII.1-2lec14
15Th 10/24group worklec15
16T 10/29Fourier transformDimock 1.1.4lec16
17Th 10/31unbounded operatorsDimock 1.2-1.3.3lec17RS VIII.1-2 for more detail
18T 11/5unbdd spectral theorem, physical applicationsDimock 1.3.3, 4.1-4.4lec18
19Th 11/7tempered distributionsRS V.3lec19
20T 11/12Fourier transform of a distribution, wave equationlec20see also Folland Ch 0
21Th 11/14Malgrange-Ehrenpreis theoremsee Folland PDE Ch 1F
22T 11/19orthogonal polynomials, 3 term recurrence, Jacobi coeffs
23Th 11/21Gauss quadrature, separation theorem
24T 11/26pseudospectral methodslec24guest lecture by Prof. Wilkening
Th 11/28thanksgiving
25T 12/3Chebyshev polyonomials and series, rates of convergenceTrefethen 3,7,8lec25see Trefethen Ch 7-8.
Th 12/5lecture moved to RRR week
26T 12/10Chebyshev interpolation, Hermite integral formulaTrefethen 4, 11makeup lecture for 12/5
27Th 12/12Potential Theory, Lebesgue ConstantsTrefethen 12,13,15makeup lecture for power outage

Homework. Will be due every two weeks, on Thursday in class. HW assignments will be updated (i.e., problems may be added) until upto a week before they are due. Please write clearly or type your solutions using Latex. Collaboration is allowed but you must list your collaborators in your writeup.

  1. HW1, due 9/12.
  2. HW2, due 9/26.
  3. HW3, due 10/10.
  4. HW4, due 10/31.
  5. HW5, due 11/21.
  6. HW6, due 12/17.

Grading. 100% homework. The bottom assignment will be dropped.