Basic Tools
Physics is like carpentry: Using the right tool makes the job easier, not more difficult. David J. Griffiths in his book "Quantum Mechanics"
Although this is a travel guide to physics, you'll find here, of course, lots of pages about mathematics.
Mathematics is the "language of nature" and physics isn't possible without it. However, in contrast to the usual Wikipedia pages about math topics, our approach can be summarized as follows: "Mathematics is a toolbox and we pick only those tools that are useful in physics".
Especially, this means that for every mathematical concept explained here, there is a "Why is it useful?" section that explains where and how the concept is useful in physics.
- Algebra
- Calculus
- Configuration Space
- Differential Equations
- Dimensional Analysis
- Euler's Formula
- Exponential Function
- Fourier Transform
- Hilbert Space
- Imaginary Numbers
- Lagrange Multiplier
- Linear Algebra
- Logarithm
- Phase Space
- Pi / $\pi$
- Pythagorean Theorem
- Random Walks
- Symbols
- Symmetry
- Tensor Calculus
- Trigonometric Functions
- Variational Calculus
- Vector Calculus
Essential Books
- Mathematical Tools for Physics Physics by James Nearing
- How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
- Street-Fighting Mathematics by Mahajan
- The Art of Insight in Science and Engineering by Mahajan
- Advanced Math for Young Students: A First Course in Algebra by Philip Keller
- No bullshit guide to math and physics by Ivan Savov
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A Course of Mathematics for Students of Physics by Paul Bamberg and Shlomo Sternberg
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Geometric Algebra for Physicists by Chris Doran, Anthony Lasenby
- New Foundations for Classical Mechanics by David Hestenes
Essays
- How to Develop a Mindset for Math
- Developing Your Intuition For Math
- Learning math? Think like a cartoonist.
- Learning To Learn: Intuition Isn’t Optional,
- Learning To Learn: Embrace Analogies,
- Learning To Learn: Pencil, Then Ink,
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Teaching and Learning “What is Mathematics” by Günter M. Ziegler and Andreas Loos
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Learning, Remembering, and Thinking by John Denker
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13 Thinking Tools Robert and Michele Root-Bernstein
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Models and Theories by David Hestenes
- Modeling Software for learning and doing physics by David Hestenes
- Primer on Geometric Algebra
Further Recommendations
- How to Read and Do Proofs by Solow
- The Keys to Advanced Mathematics: Recurrent Themes in Abstract Reasoning by Solow
- How To Solve It by Polya
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Many great explanations of "pre-college" concepts can be found at https://brilliant.org/
- Professor Anthony Lasenby: A New Language for Physics
Quotes
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.