Class of 2024
Homework, Exams, and Announcements
The lectures will follow Lecture Notes posted below. Assignments will be taken from Exercise Sections of the Notes. However, not all material presented in the Notes will be covered in class. Reading the Notes and especially examples in them is strongly recommended. The Lecture Notes are self-sufficient.
Office Hours: after class meetings on MWF, 6-7 pm
Assignment 1: 3.7.2; 3.7.5(i,ii); 4.8.1; 4.8.2; 6.15.2; 6.15.7; 8.7.2; 9.4.5; EC 11.4.1 Solutions
Assignment 2: 13.4.7(i,ii); 14.5.5; 15.10.2(iii); 15.10.4; 15.10.9; 16.6.7; 16.6.10; 18.6.2(ii); 20.5.4; 21.8.7; 22.5.1; 23.6.1; EC: any 2 problems Solutions
Assignment 3 (Final): 24.6.2(i); 25.6.1(i); 26.8.5(ii); 28.5.2(iv); 29.6.7; 30.6.4 (1 pt); 31.4.1; 32.12.4 (1 pt); 32.12.6 (iv); 33.9.1; 37.6.2; 39.7.2; EC: any 11 problems make 100% 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: Calculus with distributions
Chapter 4: Convolution and the Fourier transform
Chapter 5: Green’s functions for differential operators
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. Fundamental theorem of calculus for the Lebesgue integral. Absolutely continuous functions. Integration by parts. Functions defined by the Lebesgue integral. Fubini theorem. Theorems about continuity and differentiability of such functions. Examples. Potential-like integrals. Poisson equation. 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 (08/22-23)
L1. F: Review of Riemann integration theory in Euclidean spaces. Volume.
Week 2 (08/26-30)
L2. 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.
L3. W: 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
L4. F: 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).
Week 3 (09/2-6)
M. Labor Day. No classes.
L5. W. Gaussian integrals in a Euclidean space
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
Week 4 (09/9-13)
L7. M. Basic properties of the Lebesgue integral. Lebesgue vs Riemann integrals. Comparison test for Lebesgue integrability. Lebesgue dominated convergence theorem. Levi’s theorem
L8. W. Lebesgue integral depending on parameters. Fubini’s theorem.
L9. F. Smoothness, integrability, and differentiability of functions defined by Lebesgue integrals. Example: Fourier transform
Week 5 (09/16-20)
L10. M. Line and surface integrals. Levi-Civita symbols. Integration over M dimensional surfaces in a Euclidean space.
L11. W. Integration by parts in Euclidean spaces. Divergence and Green’s theorem.
L12. F. Cauchy line integrals in the complex plane. Fresnel’s integral. Complex Gaussian integrals
Week 6 (09/23-27)
L13.M. Residue theorem. Functions defined by improper Lebesgue integrals. Abel’s theorem for convergence of improper integrals
L14. W. Potential-like integrals. Dirac delta function. Intuitive ideas from physics. Example: A instant force making a finite momentum change.
Topic 2: Distributions
Dirac delta function. Examples in physics. How to interpret it in a mathematically sound way. 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.Singular functions as distributions. Extension of a distribution. Principal value regularization of a singular function. Existence of a distributional regularization of a singular function. Sokhotsky’s distributions. Transformations of distributions as the adjoint transformations of linear continuous transformations of the space of test functions.Examples: Linear change of variables in a distribution. Periodic distributions. Distributions independent of some of the variables. Multiplication of a distribution by smooth functions.Regularization of a distribution by a smooth function.
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
L15.F. Mass density of a point-like particle as the Dirac delta function. Basis idea of differentiation of a distribution. Distributions as linear and continuous functionals on a space of test functions.
Week 6 (09/30-10/4)
L16. M. Space of test functions. Topology in the space of test functions. The hat and bump functions. Approximation theorems.
L17. W. Distributions. Regular distributions. Du Bois-Reymond lemma and isomorphism between regular distributions and locally integrable functions.
L18. F. Singular distributions. Examples: Delta functions, principal value distributions, spherical delta functions, surface delta functions.
Week 8 (October 7-11)
L18.M. Topology in space of distributions. Examples of distributional limits. Completeness of the space of distributions. Examples.
L19.F. Distributions equal on an open set Extension of a distribution. Extension of a distribution. Distributional regularization of singular functions. Principal value regularization.
Week 9 (October 14-18)
L19.M. Distributional regularization of singular functions by shifting singularities into a complex plane. Sokhotsky’s equation.
L20.W. The adjoint transformations of distributions. Multiplication of a distribution by a smooth function. Linear change of variables in a distribution. Distributions independent of some variables.
L21.F. Convolution of test function with a distribution. Regularization of a distribution.General change of variables in a distribution.
Week 10 (October 21-25)
Topic 3: Calculus with Distributions
L21.M. Differentiation of distributions as an adjoint transformation. Classical vs distributional derivatives. Distributional derivatives of piecewise smooth functions. Distributional Green’s formula.
L22.W. Properties of distributional derivatives. Distributional derivatives of functional series. Distributional trigonometric Fourier series.
L23.F. Poisson summation formula. Distributional solutions to differential equations by the Fourier method. Classical vs distributional solutions.
Week 11 (October 28-November 1)
L24.M. Green’s function of a differential operator on a circle. Solution to the boundary value problem for a differential operator on a circle via the convolution of periodic distributions
L25.W. Asymptotic expansions of distributions. Analog of Taylor’s theorem for distributions
L26.F. Non-existence theorem about an associative and commutative product on the space of distributions. Localization theorem for distributions.
Week 12 (November 4-8)
L27.M. Product of distributions via the localization theorem
L28.W. Integration of distributions. Structure theorem for distributions. Distributions with a point support
L29.F. Basic algebraic and ordinary differential equations with distributions.
Week 13 (November 11-15)
Topic 4: Convolution and Fourier transform of distributions
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.
Week 14 (November 18-22)
L32.M. Schwartz space of test functions. Temperate distributions. Differentiation. Change of variables. Convolution.
L33.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.
L34.W. Fourier transform of distributions with bounded support. Example: Single layer distribution. Fourier transform of the convolution. Product of distributions via the Fourier transform.
Week 15 (November 25-29): Fall Break, no classes
Week 16 (December 2-4)
Topic 5: Fundamental solutions for differential operators
L35.M. Fundamental solutions for differential operators. Green’s function. Examples: Poisson and Helmhotz equations
L37.W. Cauchy problem for the heat and wave equations. Solutions via causal Green’s functions for the corresponding operators.
TO BE UPDATED
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.
October 6: Homecoming, no class
Practice problems for Week 7: Sections 13.12(1-4);
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)
W-F. Thanksgiving. No classes.
Week 15 (November 27-December 1)
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)