How to Learn Math and Physics
This post summarizes John Baez's advice on How to Learn Math and Physics and all the recommended resources to get started.
John Baez's advice summarized:
- Patience and Persistence: Learning math and physics is a gradual process that requires patience and persistence. It takes time to build a solid foundation of knowledge before delving into more advanced topics.
- Start with the basics: To understand theoretical physics, it's essential to have a solid grounding in mathematics and the fundamentals of physics. It's recommended to begin with simpler concepts and gradually work your way up to more complex topics.
- While it's okay to be interested in exciting topics like quantum mechanics or string theory, it's important to remember that these theories are still highly speculative and may not yet have enough evidence to support them.
- In order to truly learn math and physics, it's necessary to not only read books but also do calculations and experiments. This means working on homework problems, creating your own research projects, and potentially taking courses to gain hands-on experience.
- Read lots of books: There is no substitute for reading books to learn advanced math and physics. Websites may not provide the in-depth material needed to learn technical subjects. Click here for free math books → Free Math Books.
- Do lots of calculations: Learning math and physics requires lots of practice and calculations. Textbooks are full of homework problems that can be helpful.
- Take courses: If possible, taking courses in math and physics is one of the best ways to learn. Courses offer the opportunity to hear lectures, meet students and professors, and do things that you wouldn't be able to do otherwise.
- Ask questions and explain things to people: It's important to ask questions and explain concepts to others in order to solidify your own understanding. Finding a study partner or joining an online community can be a great way to do this. If you can't find local like-minded people you can join forums like→ Physics Forums to discuss math and physics topics. To ask math and physics related questions you can check out these websites→ Physics Stack Exchange, Physics Overflow, Math Stack Exchange, Math Overflow.
- Admit your mistakes: It's essential to admit when you're wrong and avoid clinging to a theory that has been disproven. Avoid looking foolish by being willing to acknowledge your mistakes and uncertainties.This helps you to maintain credibility and learn from your errors.
How to Learn Physics 🤓
There are 5 topics that every physicist should learn:
1. Classical Mechanics
📚 Recommended book(s):
Classical Mechanics 3RD EDITION by Herbert Goldstein
2. Statistical Mechanics
📚 Recommended book(s):
Fundamentals of Statistical and Thermal Physics by Frederick Reif
3. Electromagnetism
📚 Recommended book(s):
Classical Electrodynamics Third Edition 3rd Editionby John David Jackson
4. Special Relativity
📚 Recommended book(s):
Spacetime and Geometry: An Introduction to General Relativity 1st Edition by Sean M. Carroll
Spacetime Physics 2nd Edition
5. Quantum Mechanics
📚 Recommended book(s):
Quantum Mechanics and the Particles of Nature: An Outline for Mathematicians 1st Edition
Quantum Mechanics, Volume 1-3 by Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë
**in roughly that order. Once you know these, you'll be ready for General Relativity (which gets applied to cosmology) and Quantum Field Theory (which gets applied to particle physics):
General Relativity
Before tackling the details:
📚 Recommended book(s):
Black Holes and Time Warps: Einstein's Outrageous Legacy by Kip S. Thorne
Space, Time and Gravity: The Theory of the Big Bang and Black Holes 2nd Edition by Robert M. Wald
General Relativity from A to B
For when you get serious:
📚 Recommended book(s):
Introducing Einstein's Relativity: A Deeper Understanding by Ray d'Inverno, James Vickers
Gravity: An Introduction to Einstein's General Relativity 1st Edition by James B. Hartle
A First Course in General Relativity 3rd Edition by Bernard Schutz
Cosmology
📚 Recommended book(s):
Cosmology: The Science of the Universe 2nd Edition
Principles of Cosmology and Gravitation by M. Berry
Cosmological Physics by John A. Peacock
Quantum Field Theory
Before tackling the details:
📚 Recommended book(s):
QED: The Strange Theory of Light and Matter by Richard P. Feynman
For when you get serious:
📚 Recommended book(s):
An Introduction To Quantum Field Theory by Michael E. Peskin, Daniel V. Schroeder
Quantum Field Theory in a Nutshell by A. Zee
Warren Siegel, Fields, available for free on → arXiv
Quantum Field Theory by Mark Srednicki
Aspects of Symmetry: Selected Erice Lectures Reprint Edition by Sidney Coleman
Local Quantum Physics: Fields, Particles, Algebras by Rudolf Haag
Quantum Field Theory for Mathematicians by Robin Ticciati
Particle Physics
📚 Recommended book(s):
Quarks, Leptons & Gauge Fields by Kerson Huang
Leptons and Quarks by L. B. Okun
Particle Physics and Introduction to Field Theory by T.D.. Lee
The Weak Interaction in Nuclear, Particle, and Astrophysics 1st Edition by K. Grotz
While studying general relativity and quantum field theory, you should take a break now and then and dip into this book: it's a wonderful guided tour of the world of math and physics:
📚 Recommended book(s):
The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose
Some books on more advanced topics...
The Interpretation of Quantum Mechanics:
📚 Recommended book(s):
Interpretation of Quantum Mechanics by Roland Omnes
This is a reasonable treatment of an important but incredibly controversial topic. Warning: there's no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems — to understand the theory, you need to be able to use it (and vice versa). If you don't heed this advice, you'll fall prey to all sorts of nonsense that's floating around out there.
The Mathematical Foundations of Quantum Physics:
📚 Recommended book(s):
The Mathematical Foundations of Quantum Mechanics by George Mackey
Loop Quantum Gravity and Spin Foams:
📚 Recommended book(s):
Quantum Gravity by Carlo Rovelli
String Theory:
📚 Recommended book(s):
A First Course in String Theory, 2nd Edition 2nd Edition by Barton Zwiebach
String Theory and M-Theory: A Modern Introduction by Katrin Becker, Melanie Becker, John H. Schwarz
Superstring Theory: Volume 1 & 2 by Michael B. Green, John H. Schwarz, Edward Witten
String Theory Volume 1 & 2 by Joseph Polchinski
How to Learn Math 🤓
Math is a much more diverse subject than physics, there are lots of branches you can learn without needing to know other branches first — though you only deeply understand a subject after you see how it relates to all the others!
After basic schooling, the customary track through math starts with a bit of 👇
Finite Mathematics (combinatorics)
📚 Recommended book(s):
Proofs that Really Count by Arthur T. Benjamin, Jennifer J. Quinn
Probability Theory
📚 Recommended book(s):
Introduction to Probability Revised Edition by Charles M. Grinstead, J. Laurie Snell
**To dig deeper into math you need Calculus and Linear Algebra, which are interconnected 👇
Calculus
📚 Recommended book(s):
Calculus Made Easy by Silvanus P. Thompson, Martin Gardner — Free ebook version available here.
Calculus 3rd Edition by Gilbert Strang
Multivariable Calculus
📚 Recommended book(s):
Mathematical Tools for Physics by James Nearing — Free ebook version available here.
Multivariable Calculus by George Cain and James Herod — Available for free here.
Linear Algebra
📚 Recommended book(s):
Linear Algebra by Elizabeth S. Meckes and Mark Meckes
These books are probably easier, and they're available for free online:
Elementary Linear Algebra by Keith Matthews
Linear Algebra by Jim Hefferon
A First Course in Linear Algebra by Robert A. Beezer
Ordinary & Partial Differential Equations
📚 Recommended free book(s):
Notes on Differential Equations by Bob Terrell
Mathematical Tools for Physics by James Nearing (Check out the sections on Ordinary and Partial Differential Equations and Fourier Series)
Then it's good to learn these 👇
Set Theory and Logic
📚 Recommended book(s):
Elements of Set Theory by Herbert B. Enderton
A Mathematical Introduction to Logic by Herbert B. Enderton
Sets for Mathematics 1st Edition by F. William Lawvere
Complex Analysis
📚 Recommended book(s):
Complex Analysis by George Cain — Available for free here.
Complex Variables and Applications by James Ward Brown and Ruel V. Churchill
Complex Analysis by Serge Lang
Real Analysis
📚 Recommended book(s):
Methods of Real Analysis by Richard R. Goldberg
Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick
Topology
📚 Recommended book(s):
Counterexamples in Topology by Lynn Arthur Steen, J. Arthur Seebach Jr.
Abstract Algebra
📚 Recommended book(s):
Galois Theory, 3rd edition by Ian Stewart
**not in order. Proofs become important at this stage. You'll need to know a bit of set theory and logic to understand what a proof is, but you don't need calculus to get started on:
Number Theory
Elementary Books:
📚 Recommended book(s):
Number Theory by George E. Andews
Friendly Introduction to Number Theory by Joseph Silverman
An Illustrated Theory of Numbers by Martin H. Weissman
Advanced Books:
📚 Recommended book(s):
A Classical Introduction to Modern Number Theory, 2nd Edition by Kenneth Ireland and Keith Rosen
Algebraic Number Theory by Jürgen Neukirch
More Books on Mathematical Physics
Here's a good place to start:
A Course of Mathematics for Students of Physics by Paul Bamberg and Shlomo Sternberg
It's also good to get ahold of these books and keep referring to them as needed:
Mathematical Physics by Robert Geroch
Analysis, Manifolds and Physics, Part 1 by Yvonne Choquet-Bruhat, Cecile DeWitt-Morette
Mathematics for Theoretical Physics by Jean Claude Dutailly — Available for free here.
Group Theory in Physics:
📚 Recommended book(s):
Group Theory and Physics by Shlomo Sternberg
Lie Groups for Physicists by Robert Hermann
Unitary Group Representations in Physics, Probability, and Number Theory by George Mackey
Lie Groups, Lie Algebras and their representations — in rough order of increasing sophistication:
📚 Recommended book(s):
Lie Groups, Lie Algebras, and Representations by Brian Hall
Representation Theory: A First Course by William Fulton, Joe Harris
Lectures on Lie Groups by J. Frank Adams
Geometry and Topology for Physicists — in rough order of increasing sophistication:
📚 Recommended book(s):
Topology, Geometry and Gauge Fields: Foundations by Gregory L. Naber
Modern Differential Geometry for Physicists by Chris Isham
Differential Forms with Applications to the Physical Sciences by Harley Flanders
Topology and Geometry for Physicists by Charles Nash and Siddhartha Sen
Geometry, Topology, and Physics by Mikio Nakahara
Differential Topology and Quantum Field Theory by Charles Nash
Geometry and Topology, straight up:
📚 Recommended book(s):
Differential Topology by Victor Guillemin and Alan Pollack
Algebraic Topology:
📚 Recommended book(s):
Algebraic Topology by Allen Hatcher
A Concise Course in Algebraic Topology by Peter May
Geometrical Aspects of Classical Mechanics:
📚 Recommended book(s):
Mathematical Methods of Classical Mechanics by Vladimir I. Arnol'd
Analysis and it's Applications to Quantum Physics:
📚 Recommended book(s):
And moving on to pure mathematics 👇
Knot theory:
📚 Recommended book(s):
Knots and Physics by Louis Kauffman
Knots and Links by Dale Rolfsen
Homological algebra:
📚 Recommended book(s):
An Introduction to Homological Algebra by Joseph Rotman
An Introduction to Homological Algebra by Charles Weibel
Combinatorics:
📚 Recommended book(s):
Generating Functionology 2nd Edition by Herbert S. Wilf
Enumerative Combinatorics, Volume 2: 1st Edition by Richard P. Stanley
Algebraic geometry:
Start with this introduction:
📚 Recommended book(s):
Then try these:
📚 Recommended book(s):
Basic Algebraic Geometry, 1-2 Volumes, 3rd Edition by Igor R. Shafarevich
The Geometry of Schemes by David Eisenbud and Joseph Harris
Principles of Algebraic Geometry 1st Edition by Phillip Griffiths, Joseph Harris
Category theory:
📚 Recommended book(s):
An Invitation to Applied Category Theory: Seven Sketches in Compositionality 1st Edition by Brendan Fong and David Spivak — Available for free here. (Also check out the website with videos and John Baez online course based on this book).
Basic Category Theory by Tom Leinster — Ebook available for free here.
Category Theory in Context by Emily Riehl — Available for free here.
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