Friday, April 11, 2025

Front for Mathphys Archive

 "I write not because I know something but to learn something."

"The most important book to me is my own notebook because it is written in the way I understood."

This pinned post contains the list of my lecture notes on mathematics, physics, and related areas (theoretical computer science, mathematical biology, and mathematical finance). They are categorized by subjects in each area for readers' convenience. This post is subject to updates as I continue to add new lecture notes.

Mathematics

Advanced Calculus

Lecture Notes

Problem Sets

References and Further Reading

Ivan S. Sokolnikoff, Advanced Calculus, McGraw-Hill Book Compny, Inc., 1939

Abstract Algebra I: Groups

Lecture Notes

Problem Sets

References and Further Reading

John B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Pearson, 2002
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017

Abstract Algebra II: Rings, Fields, Galois Theory, Modules, Representation Theory

Lecture Notes 

Problem Sets

References and Further Reading

John B. Fraleigh, A First Course in Abstract Algebra, 7th Edition, Pearson, 2002
I. N. Herstein, Abstract Algebra, 3rd Edition, Wiley, 1996
I. N. Herstein, Topics in Algebra, 2nd Edition, John Wiley & Sons, 1975
Ramji Lal, Algebra 1, Groups, Rings, Fields and Arithmetic, Springer, 2017
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017

Algebraic Topology: Homotopy, Homology, Cohomology

Lecture Notes

Homotopy

Homology

Cohomology

Problem Sets

Homotopy

Homology

Cohomology

References and Further Reading

Jean Gallier and Jocelyn Quaintance, A Gentle Introduction to Homology, Cohomology, and Sheaf Cohomology
Mikio Nakahara, Geometry, Topology and Physics, 2nd Edition, Graduate Student Series in Physics, Taylor & Francis, 2003
Charles Nash and Siddhartha Sen, Topology and Geometry for Physicists, Academic Press, Inc., 1983

Calculus

Differential Calculus

Lecture Notes

Calculus 4: How to Calculate Limits I
Calculus 5: How to Calculate Limits II  
Calculus 6: How to Calculate Limits III
Calculus 7: Continuity
Calculus 8: Limits Involving Infinity and Asymptotes 
Calculus 9: Finding the Equation of Tangent Line to a Curve $y=f(x)$
Calculus 10: Derivatives
Calculus 11: Velocity and Acceleration
Calculus 12: Basic Differentiation Formulas  
Calculus 13: Continuity versus Differentiability
Calculus 14: The Product and Quotient Rules
Calculus 15: Derivatives of Trigonometric Functions
Calculus 16: The Chain Rule  
Calculus 17: The Proof of the Chain Rule
Calculus 18: Implicit Differentiation
Calculus 19: Derivatives of Logarithmic and Exponential Functions  
Calculus 20: Related Rates
Calculus 21: Linear Approximations and Differentials
Calculus 22: Maximum and Minimum  
Calculus 23: Mean Value Theorem

Problem Sets

Integral Calculus

Lecture Notes

Problem Sets

Sequences, Series, and Vector-Valued Functions

Lecture Notes

Problem Sets

Functional Analysis/Fourier Analysis

Functional Analysis

Lecture Notes 

Problem Sets

Fourier Analysis

Lecture Notes 

Problem Sets

References and Further Reading

Gerald B. Folland, Fourier Analysis and Its Applications, Pure and Applied Undergraduate Texts, American Mathematical Society, 2009
Sigurdur Helgason, Topics in Harmonic Analysis on Homogeneous Space, Birkhäuser
Einer Hille and Ralph S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, 1957
Erwin Kreyszig, Introductory Functional Analysis with Applications, 1st Edition, Wiley, 1989
Michael Reed and Barry Simon, Methods of Mathematical Physics I: Functional Analysis, Revised and Enlarged Edition, Academic Press, 1980
Michael Reed and Barry Simon, Methods of Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1972
Michael Reed and Barry Simon, Methods of Mathematical Physics III: Scattering Theory, Academic Press, 1972
Michael Reed and Barry Simon, Methods of Mathematical Physics IV: Analysis of Operators, Academic Press, 1978
H. L. Royden, Real Analysis, Second Edition, The Macmillan Company
Walter Rudin, Functional Analysis, McGraw-Hil Book Company, 1973
Peter Szekeres, A Course in Modern Mathematical Physics, Groups, Hilbert Space and Differential Geometry, Cambridge University Press, 2004

Functions of a Complex Variable 

Lecture Notes

Problem Sets

References and Further Reading

Lars Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979
James Brown and Ruel Churchill, Complex Variables and Applications, 8th Edition, McGraw-Hill, 2008
John B. Conway, Functions of One Complex Variable I, 2nd Edition, Graduate Texts in Mathematics, Springer, 1978

Linear Algebra

Lecture Notes

Problem Sets

References and Further Reading

Sheldon Axler, Linear Algebra Done Right, Third Edition, Springer, 2015
Ramji Lal, Algebra 2, Linear Algebra, Galois Theory, Representation Theory, Group Extensions and Schur Multiplier, Springer, 2017
Serge Lang, Introduction to Linear Algebra, Second Edition, Springer, 1986
Serge Lang, Linear Algebra, Third Edition, Springer, 2004

Number Theory and Cryptography

Software Tools for Computation

Study of number theory is often accompanied by heavy computations for which computers can be effectively used. From time to time, I will be using Maxima, an open source computer algebra system (abbreviated as CAS, a software package for symbolic computation) and Python. Sage (CoCalc) is also a good and powerful open source CAS but I will not be using it here. Main reason is that I find it restrictive as it requires a web-based interface (a browser) and an internet connection. Unless you run your own CoCalc server (if you are going to use it just for yourself, why would you run such a server?), you need to have an access to a CoCalc server that is run by someone else. I personally prefer a crude and low tech computing environment that does not require a particular interface and a lot of resources. Maxima and Python fit into such preference of mine. Maxima and Python do have interfaces but they also can be run interface-free in a command shell.

MAXIMA

What is Maxima?: MIT has developed a computer algebra system, called Macsyma, from 1968 to 1982 as part of Project MAC. They turned over a copy of the Macsyma source code to the Department of Energy (DOE). That version is known as DOE Macsyma. It had been maintained by William Schelter at the University of Texas from 1982 until his death in 2001. In 1998, Schelter obtained permission from the DOE to release the DOE Macsyma source code under the GNU Public License, and in 2000 he initiated the Maxima project at SourceForge to maintain and continue developing DOE Macsyma, now called Maxima.

 
Download and Install Maxima: 1. For those who are using Windows, an instruction can be found here and also here.
2. For those who are using MacOS, Maxima can be download from here and an installation instruction can be found here.
3. If you are using Ubuntu Linux OS, simply run the following command in command shell:
sudo apt install maxima && sudo apt install wxmaxima
You also need to install Gnuplot by runnig the command:

sudo apt install gnuplot

4. If you are using FreeBSD Unix, as root run:

# pkg install maxima && pkg install wxmaxima

and also install Gnuplot by running

# pkg install gnuplot

5. For all other operating systems, refer to the instruction at Maxima Downloads page here.

Maxima documentation page is an important source on how to use Maxima for your computational needs and purposes. Another important source on using Maxima, especially for doing mathematics is The MaximaList. The html version of Maxima 5.29.0 manual on Number Theory is available online here. 

 PYTHON

You can find download/installtion instruction for your OS at Python.org. Make sure that you install Python 3.

1. For Windows system, select and download Python Windows installer, for example python-3.3.0.msi and run it.
2. For MacOS, select and download .dmg Mac Installer Disk Image and run it.
3. For Ubuntu Linux OS, run in command shell, for example

sudo apt install python3.3 && sudo apt install idle3

IDLE is a covenient interface for Python. I love iPython (Interactive Python Shell). It allows you to run .py files in command shell. To install ipython run

sudo apt install ipython3

4. For FreeBSD Unix, as root run:

# pkg install python

This command will install Python 3 and IDLE 3 as well. To install ipython, as root run:

# pkg install py37-ipython

Lecture Notes

Problem Sets

References and Further Reading

Elementary Number Theory

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Oxford at the Clarendon Press, 1975
Neal Koblitz, A Course in Number Theory and Cryptogtraphy, Graduate Texts in Mathematics 114, Springer-Verlag, 1994
Franz Lemmermeyer, Numbers and Curves, Springer-Verlag, 2001
Manfred Schroeder, Number Theory in Science and Communication with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Fifth Edition, Springer-Verlag, 2009
Simon Singh, Fermat's Last Theorem, The Story of a Riddle that Confounded the World's Greatest Minds for 358 Years, Fourth Estate
André Weil, Number Theory for Beginners, Springer-Verlag, 1979

Cryptography

Sara Arias-de-Reyna and Gabor Wiese, Algebraic Curves and Applications to Cryptography
Steven Galbraith, Mathemathematics of Public Key Cryptography
Neal Koblitz, A Course in Number Theory and Cryptogtraphy, Graduate Texts in Mathematics 114, Springer-Verlag, 1994
Gabor Wiese, Théorie des nombres et applications à la cryptographie

Elliptic Curves

J.W.S. Cassels, Lectures on Elliptic Curves, Cambridge University Press, 1991
Dale Husemöller, Elliptic Curves, Graduate Texts in Mathematics 111, Springer, 2002
Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics 97, Springer-Verlag, 1984
J.S. Milne, Elliptic curves, 1996
Bjorn Poonen, Elliptic Curves, 2001
Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1985

Probability

Lecture Notes

Problem Sets

References and Further Reading

Sheldon Ross, A First Course in Probability, Fifth Edition, Prentice Hall, 1997 

Real Analysis I

Lecture Notes

Problem Sets

References and Further Reading

Walter Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-Hill, Inc., 1976

Real Analysis II: Measure Theory

Lecture Notes

Problem Sets

References and Further Reading

H. L. Royden, Real Analysis, Second Edition, The Macmillan Company

Quantum Mechanics for Mathematicians

Lecture Notes

Problem Sets

References and Further Reading

L. D. Faddeev and O. A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students, American Mathematical Society, 2009
N.P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer, 1998

Special Topics in Mathematics

Lecture Notes

These lecture notes include my discussions on some special topics in mathematics that do not necessarily belong to particular courses.

Problem Sets

References and Further Reading

Stochastic Differential Equations

Lecture Notes

Problem Sets

References and Further Reading

Lawrence C. Evans, An Introduction to Stochastic Differential Equations, Version 1.2
Don S. Lemons, An Introduction to Stochastic Processes in Physics, Containing "On the Theory of Brownian Motion" by Paul Langevin, translated by Anthony Gythiel, The Johns Hopkins University Press, 2002
Bernt Øksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Corrected Printing, Springer-Verlag, 2000
Philip E. Protter, Stochastic Integration and Differential Equations, Second Edition, Springer, 2004
Sheldon Ross, A First Course in Probability, Fifth Edition, Prentice Hall, 1997

Physics

Classical Mechanics 

Lecture Notes

Problem Sets

References and Further Reading

A. P. French, Newtonian Mechanics, The M.I.T. Introductory Physics Series, Thomas Nelson and Sons LTD, 1971
Walter Greiner, Classical Mechanics, Point Particles and Relativity, Springer, 2003
Walter Greiner, Classical Mechanics, Systems of Particles and Hamiltonian Dynamics, Springer, 2000
Walter D. Knight and Malvin A Ruderman, Mechanics, Charles Kittel, Berkeley Physics Course Volume 1, Second Edition, McGraw-Hill, 1973
L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics Volume 1, Third Edition, Elsevier, 1976
Jakob Schwichtenberg, No-Nonsense Classical Mechanics, A Student Friendly Introduction, No-Nonsense Books, 2019

Mathematical Physics 

Lecture Notes

Problem Sets

References and Further Reading

George Arfken, Mathematical Methods for Physicists, Third Edition, Academic Press, INC., 1985

Special Topics in Physics

Lecture Notes

These lecture notes include my discussions on some special topics in physics that do not necessarily belong to particular courses.

Problem Sets

References and Further Reading

Stochastic Processes in Physics

Lecture Notes

Problem Sets

References and Further Reading

Don S. Lemons, An Introduction to Stochastic Processes in Physics, Containing "On the Theory of Brownian Motion" by Paul Langevin, translated by Anthony Gythiel, The Johns Hopkins University Press, 2002
Sheldon Ross, A First Course in Probability, Fifth Edition, Prentice Hall, 1997 

Theory of Special Relativity

Lecture Notes

Problem Sets

References and Further Reading

A. P. French, Special Relativity, The M.I.T. Introductory Physics Series, W. W. Norton & Company Inc, 1968
Gregory L. Naber, The Geometry of Minkowski Spacetime, An Introduction to the Mathematics of the Special Theory of Relativity, Springer-Verlag, 1992
Hans Stephani, Relativity, An Introduction to Special and General Relativity, Third Edition, Cambridge University Presss, 2004

Theory of General Relativity

Lecture Notes

Problem Sets

References and Further Reading

L. P. Hughston and K. P. Tod, An Introduction to General Relativity, Cambridge University Press, 1990
Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravitation, W. H. Freeman and Company, 1973
Eric Poisson, A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge University Press, 2004
Hans Stephani, Relativity, An Introduction to Special and General Relativity, Third Edition, Cambridge University Press, 2004
John Stewart, Advanced General Relativity, Cambridge University Press, 1991
Norbert Straumann, General Relativity and Relativistic Astrophysics, Springer-Verlag, 1984
Robert M. Wald, General Relativity, The University of Chicago Press, 1984 

Wormhole Physics

Lecture Notes

Problem Sets

References and Further Reading

Jerry B. Griffiths and Jiří Podolský, Exact Space-Times in Einstein's General Relativity, Cambridge University Press, 2009
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, 1973

Not for the faint of heart.

L. P. Hughston and K. P. Tod, An Introduction to General Relativity, Cambridge University Press, 1990
Klaas Landsman, Foundations of General Relativity
Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, Gravitation, W. H. Freeman and Company, 1973

A pretty big book with more than 1300 pages. This book is dubbed the bible of general relativity.

Michael S. Morris and Kip S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching geneal relativity, Am. J. Phys. 56 (5), 395-412, May 1988
Michael S. Morris, Kip S. Thorne, and Ulvi Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Physical Review Letters, Volume 61, Number 13, pp. 1446-1449, 26 September 1988
A. Papapetrou, Lectures on General Relativity, D. Reidel Publishing Company, 1974

This book also contains a good bit of materials on differential geometry.

Eric Poisson, An advance course in general relativity, Draft, January 2002
Eric Poisson, A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge University Press, 2004
Hans Stephani, Relativity, An Introduction to Special and General Relativity, Third Edition, Cambridge University Press, 2004
Hans Stephani, Dietrich Kramer, Malcolm Maccallum and Cornelius Hoenselaers, Exact Solutions of Einstein's Field Equations, Second Edition, Cambridge University Press, 2003
John Stewart, Advanced General Relativity, Cambridge University Press, 1991
Norbert Straumann, General Relativity and Relativistic Astrophysics, Springer-Verlag, 1984
Matt Visser, Lorentzian Wormholes, From Einstein to Hawking, AIP Series in Computational and Applied Mathematical Physics, 2008
Robert M. Wald, General Relativity, The University of Chicago Press, 1984

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Biology

Finance


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