Class of 2023

Homework, Exams, and Announcements

Prerecorded lectures. The prerecorded lectures do not have the same numbers as the actual one due to a restructuring of the course starting this year and can only be used as a supplementary material when you miss class meetings for whatever reasons. Prerecorded Lectures 1-8 are meant as a review for those students who want to refresh basics facts from the theory of convergence (Calculus and Analysis). The course starts from prerecorded Lecture 9. Some of the facts from Lectures 1-8 will be discussed in due course (when needed) in the actual in-person lectures.

Office Hours:  after class meetings on MWF, 6-7 pm

Assignment 1: HW1 with solutions
Assignment 2: HW2 with solutions
Assignment 3 (Final): HW3 with solutions

The UF teaching evaluation period for this semester is November 26 – December 7. It is done online. You have to log in Evaluation Page using your Gatorlink user name and password. Please do not forget to do so!

Lecture notes

Sergei V. Shabanov, Distributions and Operators for Theoretical Physicists

Chapter 1: Integration in Euclidean spaces
Chapter 2: Distributions
Chapter 3: Applications to basic linear PDEs. (to be posted)

Supplementary material: Some facts from Calculus and Analysis

Video Lectures: There are recorded video lectures from the Covid era. They do not exactly match the actual lectures but can be used as a supplementary material (e.g., if you miss a class meeting). They also contain some material from the prerequisites for the course (Lectures 1-8). They can be suitable to refresh some relevant topics from Calculus and basic analysis. Links to the video lectures are given below the course meetings schedule and topics. You have to login with your UF password to watch these lectures (they are stored in the UF Mediasite).

Topic 0: Numerical series and sequences. The limit of a sequence. The sum of a series. The limit points and Bolzano-Weierstrass theorem. The limit of a non-decreasing sequence. The Cauchy criterion for convergence. The absolute and conditional convergence of a series. The comparison test. The upper and lower limits of a sequence. The root and ratio tests. De Morgan test. Tests for conditionally convergent series. Alternating series. Rearrangements. The Riemann theorem about rearrangements.

Recommended additional literature:
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Chapter 2 ;
W. Rudin, Principles of Mathematical Analysis, Chapter 3.

Video Lectures: 1-4

Topic 00: Functional sequences and series. Pointwise convergence. Power series of a complex variable. Radius of convergence. Uniform convergence. Uniform and absolute convergence. The uniform convergence and continuity of the limit function and/or the sum of a series. The uniform convergence and differentiability of the limit function and/or the sum of a series. The uniform continuity and Riemann integrability of the limit function and/or the sum of the series. An example of a function of a real variable that is continuous everywhere, but nowhere differentiable. The Stone-Weierstrass theorem about a uniform approximation of a continuous function by polynomials on a closed interval.

Recommended additional literature:
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Chapter 3 ;
W. Rudin, Principles of Mathematical Analysis, Chapter 7.

Video Lectures: 5-8

Lecture Topics

Topic 1: Integration in Euclidean spaces

 A review of the Riemann integration theory. Improper Riemann integrals. Absolutely and conditionally convergent integrals. Sets of measure zero in Euclidean spaces. Properties of sets of measure zero in Euclidean spaces. Properties that hold almost everywhere. Measurable functions. Properties of measurable functions. Measurable sets in Euclidean spaces. Construction of the Lebesgue integral as the limit of a sequence of Riemann integrals of piece-wise continuous functions. Independence of the Lebesgue integral of the choice of a sequence of Riemann integral. Properties of the Lebesgue integral. Integrability of the absolute value of an integrable function. Lebesgue dominated convergence theorem. Levy’s theorem. Comparison of Lebesgue and Riemann integrals. Examples. Fubini-Lebesgue theorem. Example of a non-integrable function of two variables with existing (but not equal) iterated integrals. Integrability of power functions over bounded and unbounded regions in Euclidean spaces. Change of variables in Lebesgue integrals. Functions defined by the Lebesgue integral. Theorems about continuity and differentiability of such functions. Examples. Gaussian integrals. Fourier integrals.

Recommended additional literature:
V.S. Vladimirov, Equations of Mathematical Phyics, Chapter 1, Section 1;
L. Schwartz, Mathematical Methods for Physical Sciences, Chapter 1;
E.C. Titchmarsh, The Theory of Functions, Chapter 10

Week 1 (08/23-25)
L1. W. Sets and functions in a Euclidean space. Basic terminology. Functional Series and Sequences. Pointwise and uniform convergences on a set. Differentiation of a series (Sections 1.1-3).
L2. F. Riemann integral on a rectangle. Basic properties. Fubini’s theorem. Riemann integrable functions. Volume of a ball in a Euclidean space (Sections 1.4-6).

Week 2 (08/28-09/1)
L3. M. Sets of measure zero in a Euclidean space.  Basics properties. Transformations of sets of measure zero. Smooth surfaces in a Euclidean space as sets of measure zero. Necessary and sufficient condition for Riemann integrability of a function. Riemann integrability of the sum of series. (Sections 1.7-10).
W. Hurricane. No classes.The Campus is closed
L4. F.  Improper Riemann integrals. 2D example. Exhaustion of an integration region. Definition of an improper integral.  Improper integrals of non-negative functions. Absolutely and conditionally convergent integrals (Sections 2.1-4).

Practice problems for Weeks 1 and 2: Section 1.11

Week 3 (09/4-8)
M. Labor Day. No classes.
L5. W. Riemann theorem about alternating series and conditionally convergent improper Riemann integrals. Basic comparison tests for absolute convergence of Riemann integrals (unbounded functions, unbounded integration regions). Gaussian integrals in a Euclidean space (Sections 2.5-7).
L6. F. Piece-wise continuous functions. Measurable sets and functions. Properties of measurable functions. Completeness of the space of measurable functions. On the existence of non-measurable functions and sets. Definition of the Lebesgue integral (Sections 3.1-4).
Practice problems for Week 3: Section 2.8

Week 4 (09/11-15)
L7. M. Basic properties of the Lebesgue integral. Lebesgue vs Riemann integrals. Comparison test for Lebesgue integrability. Lebesgue dominated convergence theorem. Levi’s theorem (Section 4).
L8. W. Lebesgue integral depending on parameters. Fubini’s theorem. Smoothness, integrability, and differentiability of functions defined by Lebesgue integrals. Example: Fourier transform (Section 5).
L9. F. Line and surface integrals. Levi-Civita symbols. Integration over M dimensional surfaces in a Euclidean space.  Flux of a vector field. Divergence theorem. Integration by parts in a Euclidean space. Green’s identities (Section 6).
Practice problems for Week 4: Sections 4.12(1-9); 5.4(1,3-6); 6.7(1-5)

Week 5 (09/18-22)
L10. M. Cauchy line integrals in the complex plane. Residue theorem. Fresnel’s integral. Complex Gaussian integrals (Section 7).
L11.W. Potential-like integrals. Continuity and differentiability. Poisson equation (Section 8).
L12.F. Functions defined by improper Lebesgue integrals. Continuity and differentiability. Abel’s theorem for convergence of improper integrals (Section 9).
Practice problems for Week 5: Sections 7.5(4,5); 8.6; 9.4

Topic 2: The theory of distributions

Part 1. Basic distributions: Linear continuous functionals.  The space of test functions. The hat and bump functions.Topology in the space of test functions. Distributions. Regular and singular distributions. Topology in the space of distributions. Transformations of distributions as the adjoint transformations of linear continuous transformations of the space of test functions. Distributional regularization of singular functions.  Sokhotsky formulas. Multiplication of a distribution by smooth functions. Differentiation of distributions. Basic properties of derivatives of distributions. Classical and distributional derivatives. Differentiation of distributionally convergent series. Trigonometric Foureir series. Poisson summation formula. Regularization of distributions by smooth functions. Change of variables in distributions. Product of distributions via the localization theorem. Algebraic equations with distributions. Integration of distributions.  Direct product of distributions. Convolution of distributions.  Commutativity and continuity of the convolution. Differentiation of the convolution. Non-associativity of the convolution. Existence of the convolution. Special subsets of distributions on which the convolution is associative. The case of regular distributions. The case of distributions with bounded support. The case of one-dimensional distributions with the support bounded from below and their convolution algebra. The case of 4-dimensional distributions with the support in the future light cone. Equations in the convolution algebra. Fundamental solutions for differential operators. Tempered distributions. The structure of tempered distributions with a point  support. The direct product of tempered distributions. The convolution of tempered distributions. The Fourier transform of tempered distributions. The inverse Fourier transform. Product of distributions via the Fourier transform.

Recommended additional literature:
V.S. Vladimirov, Equations of Mathematical Physics.
L. Schwartz, Mathematical Methods for Physical Sciences.
G. Grabb, Distributions and Operators
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume II, Fourier Analysis

Week 6 (September 25-29)
L13.M. Basic idea of a distribution. Dirac delta-function (Section 11).
L14.W. Space of test functions. Topology in the space of test functions (Section 12.1). The hat and bump functions. Approximations of locally integrable functions by smooth functions (Section 12.2). Linear continuous transformations of the space test functions (Section 12.4). The space of test functions as a dense subset in various functional spaces (Section 12.5).
L15.F. Distributions (Section 13.1). Regular distributions (Section 13.2). Du Bois-Reymond lemma and isomorphism between regular distributions and locally integrable functions (Section 13.7)
Practice problems for Week 6: Sections 11.4(1,2,5,6); 12.6(2,3);

Week 7 (October 2-6)
L16.M. Singular distributions. Examples (Section 13.3-5)
L17.W. Topology in space of distributions. Examples of distributional limits (Section 13.8) . Completeness of the space of distributions (Section 13.11).
October 6: Homecoming, no class
Practice problems for Week 7: Sections 13.12(1-4);

Week 8 (October 9-13)
L18.M. Support of a distribution. Distributions equal in an open set (Section 14.1). Extension of a distribution. Distributional regularization of singular functions. Principle value distributions (Section 14.2).
L19.W. Distributional regularization of singular functions by shifting singularities into a complex plane. Sokhotsky’s equation (Section 14.3). Higher-dimensional examples (Section 14.4).
L20.F. The adjoint transformations of distributions (Section 13.9). Multiplication of a distribution by a smooth function (Section 13.10). Distributions independent of some variables (Section 15.1). Differentiation of distributions. Examples (Section 17.1-2).
Practice problems for Week 8: Sections 13.12(1-4); 14.5(5,6,8,10,14,15)

Week 9 (October 16-20)
L21.M. Classical vs distributional derivatives. Distributional derivatives of piecewise smooth functions (Section 17.4). Distributional Green’s formula (Section 17.5)
L22.W. Properties of distributional derivatives (Section 18.1-2). Distributional derivatives of functional series (Section 18.4). Distributional trigonometric Fourier series.
L23.F. Poisson summation formula (Section 18.5). Distributional solutions to differential equations by the Fourier method. Classical vs distributional solutions (Section 18.6).

Practice problems for Week 9: Sections 17.8(1,2,4,8,11); 18.6(3,5,8,9)

Week 10 (October 23-27)
L24.M. Distributions as weak limits  of a sequence of smooth functions (section 15)
L25.W. Change of variables in distributions (Section 16)
L26.F. Division problem. Localization theorem for distributions. Product of distributions by the localization method (section 19)

Practice problems for Week 10: Sections 16.7(1,4-7); 19.5(1,3)

Week 11 (October 30-November 3)
L26.M. Algebraic equations in distributions. Distributions with point support.
L27.W. Integration of distributions. Antiderivative of a distribution. Higher dimensional generalizations
L28.F. Basic differential equations in distributions. Examples: Linear first-order equation, 2D wave equation.

Week 12 (November 6-11)
L30.M. Direct product of distributions. Properties. Commutativity associativity. Differentiation.
L31.W. Convolution of distributions. Existence. Commutativity.  Non-associativity. Sufficient conditions for associativity. Differentiation of the convolution. Equations in the convolution algebra.
F. Veterans Day. No class.

Week 13 (November 13-17)
L32.M. Fundamental solutions for a differential operator. Uniqueness. Green’s function of a differential operator. Example: Distributional solutions to the Poisson equation.
L33.W. Fundamental solution for a 2D wave operator. Distributional solutions. d’Alembert’s formula for classical solution of the Cauchy problem. Fundamental solution for a general 1D linear differential operator. Example: Initial value problem for electrical circuits and oscillators.
L34.F. Fundamental solution for the Helmholtz operator. Sommerfeld radiation conditions. Green’s function for monochromatic wave radiation. Example: radiation of sound waves by a point dipole source.

Week 14 (November 13-17)
L35.M.  Schwartz space of test functions. Temperate distributions. Schwartz structure theorem for temperate distributions. Differentiation. Change of variables. Convolution.
W-F. Thanksgiving. No classes.

Week 15 (November 27-December 1)
L36.M. Fourier transform on the Schwartz space of test functions. Fourier transform of tempered distributions. Inverse Fourier transform of distributions. Examples: Dirac delta function. Step function. Sokhotski’s distributions. Basic properties.
L37.W.  Fourier transform of distributions with bounded support. Example: Single layer distribution.  Fourier transform of the convolution. Product of distributions via the Fourier transform.
L39.F. Fundamental solutions for linear differential operators via the Fourier transform. Examples: 3D Laplace, 3D Helmholtz, and 4D wave operators. Advanced and retarded Green’s functions, and the  Feynman propagator for the 4D wave operator.

Week 16 (December 4-6)
L40.M. Retarded (causal) Green’s functions for the wave, heat, and Schroedinger operators. Distributional solution to the Cauchy problem.
L41.W. Reserved for a further discussion of the Cauchy problem.

Note: L39-41 offer a very basic introduction to applications of distributions to PDEs in physics and engineering. The first part of the spring semester is devoted to studies of distributional solutions to PDEs with applications to wave propagation in spacetime (Maxwell’s equations, Klein-Gordon-Fock equation) and probability amplitude propagation in quantum mechanics (Schroedinger and Dirac equations).

Supplementary Video Lectures for Chapter 1

Lecture 1; Numerical sequences
Lecture 2; Cauchy sequences in Euclidean spaces
Lecture 3: Series. Absolute convergence.
Lecture 4: Series. Conditional convergence.
Lecture 5; Functional series and sequences
Lecture 6; Uniform convergence
Lecture 7: Uniform convergence and continuity;
Lecture 8: Uniform convergence and differentiation
Lecture 9, The Riemann integral. A review.
Lecture 10. The Riemann integrability and sets of measure zero.
Lecture 11. Improper Riemann integrals. Absolutely convergent integrals.
Lecture 12,  Conditionally convergent integrals. Abel’s theorem
Lecture 13, Gaussian integrals in Euclidean spaces.
Lecture 14, Residue theorem and improper integrals.
Lecture 15, Lebesgue integral. Definition.
Lecture 17, Functions defined by Lebesgue integrals. Fubini’s theorem
Lecture 18, Lecture 19,  Potential-like integrals.
Lecture 20, Functions defined by conditionally convergent integrals

Supplementary Video Lectures for Chapter 2

Lecture 21, Basic idea of distributions. Section 21. Test functions.
Lecture 22, Distributions as linear continuous functional on the space of test functions
Lecture 23, Regular and singular distributions, delta-sequences.
Lecture 24, Du Bois-Reymond Lemma. Support of a distribution. Completeness of the space of distributions.
Lecture 25, Sokhotsky’s equations. Multiplication of a distribution by smooth functions.
Lecture 26, Change of variables in a distribution.
Lecture 27, Differentiation of distributions
Lecture 28, Properties of distributional derivatives
Lecture 29, Distributions defined by Fourier series. Poisson summation formula.
Lecture 31, Direct product of distributions.
Lecture 30,  Integration of a distribution. Basic distributional ordinary differential equation
Lecture 32, Convolution of distributions.
Lecture 33, Existence of the convolution.
Lecture 34, Fundamental solution for a linear differential operator. One-variable operator with constant coefficients.
Lecture 35, Distributional solutions to Poisson equation. On uniqueness of a fundamental solution.
Lecture 36,  Convolution algebra and the Cauchy problem for ODEs
Lecture 37, Regularization of distributions. Distributional and classical (smooth) solutions to PDEs
Lecture 38, Tempered distributions
Lecture 39,  Fourier transform of distributions
Lecture 40,  Inverse Fourier transform of distributions
Lecture 41, Fundamental solutions via the Fourier transform. Examples: Laplace and Helmholtz operators
Lecture 42b, Generalized Initial Value Problem for linear ODEs
Lecture 42c, Generalized Cauchy problem for PDEs (Wave, Heat, Transfer, Schroedinger equations). Its solution via Fourier transform.
Supplementary Lecture, Fundamental solutions for the wave and Helmholtz operators (solutions to some problems in the lecture notes)