Math-and-Physics-How-To
A repository of publicly-accessible resources for learning & self-teaching mathematics & physics up to the graduate level.
Scroll to the bottom of the page to see Onri’s Table of Critical Equations
“You can master mathematics if you practice enough final exam reviews of desired mathematical subjects. Being good at mathematics is a matter of practice”. - O.J.B.
In Case You Want Some Fun Open Access Interactive Tools to Try:
Great Resources for Practicing Computational Methods:
- J Robert Johansson’s Website on Scientific Computing & QuTiP
- QuTiP Tutorials
- Open a New Google Colab Notebook
- Browse Some Fun Data Science Notebooks in Google Colab
Map of Mathematics & Their Prerequisites:
Borrowed from: Learn with the Map of Mathematics, The Bright Side of Mathematics (2023)
https://youtu.be/ljGSId-uHw8?si=xKNup3hOVsWC6uTv&t=200
Pro Tips for Solving Mathematical Problems by (O.J.B.):
| Tip | Explanation/ Details |
|---|---|
| Substitute giveb variables with custom variables (e.g., AbcdEfG) |
Solve the equation using your own variables, then mirror the steps onto the original problem for proportional reasoning. |
| Interpret the equal sign as “converts to” |
Thinking of “=” as “converts to” can facilitate substitutions & manipulations in other mathematical expressions. |
| Think in terms of ratios by default |
Viewing values as ratios can simplify problem-solving & conceptual understanding. |
| Use software tools for conversion to markdown or LaTeX |
Convert equations for better inspection & rendering, ensuring accuracy. |
| Leverage Python & libraries like SymPy |
Write equations in Python for execution & manipulation, aiding clarity & verification. |
| Remember solutions on graphs are line intersections |
Graphical solutions typically correspond to intersection points of lines or curves. |
| Assume invisible exponents of 1 | This assumption helps maintain organization & supports & grouping symbols |
| Use preferred mathematical notations |
Include curly brackets, e-notation, prime/dot notation, & highlight invisible symbols for clarity & precision. |
| Stay aware of term replacements | Recognize when terms are replaced or approximated in mathematical contexts. |
| Consider various methods (axiomatic, first principles, empirical) |
Use diverse approaches, including logical derivations, empirical evidence, & hybrid methodologies for problem-solving. |
| Explore graphical, tabulated, or geometric representations |
Visual or tabular methods can simplify complex mathematical concepts. |
| Practice final exam reviews | Mastery in mathematics comes with regular & extensive practice, particularly of exam-style problems. |
Basic Math Symbols ≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
Geometry Symbols ∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
Algebra Symbols ≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗 | 〉
Set Theory Symbols ∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
Logic Symbols ¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣
Calculus & Analysis Symbols ∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
Greek Letters 𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔
Below is an Example by (O.J.B.) on How to Substitute with Custom Variables:
Note: the W, x, y, z, T variables in the example above are merely substites & do not correspond to any physical variables.
All The Physics You Need, Curated by (O.J.B.):
| Topic | Source |
|---|---|
| How to Succeed at Physics Without Really Trying | Physics with Elliot |
| The Most Important Math Formula for Understanding Physics | Physics with Elliot |
| The Single Basic Concept Found in (Almost) All Fundamental Physics Equations | Parth G |
| To Master Physics, First Master the Harmonic Oscillator | Physics with Elliot |
| To Master Physics, First Master the Rotating Coordinate System | Dialect |
| 5 Methods for Differential Equations | Physics with Elliot |
| The Wave Equation | Parth G |
| Eigenvalue Equation | Parth G |
| Solving the Schrödinger Equation | Parth G |
| Matrices | Parth G |
| Lagrangian & Hamiltonian Mechanics | Physics with Elliot |
| The Kronecker Delta | Alexander Fufaev |
| Maxwell’s Equations Explained | Parth G |
| Animated Physics Lectures | ZAP Physics |
| More Animated Physics Lectures | Alexander Fufaev |
| Even More Animated Physics Lectures | Dr. Elliot Schneider |
| Physical Sciences & Engineering | Dr. Jordan Edmunds |
| Maths of Quantum Mechanics Playlist | Quantum Sense |
| Quantum Harmonic Oscillators | Pretty Much Physics |
| Dirac Equation Playlist | Pretty Much Physics |
| Quantum Information Science Playlists | Prof. Artur Ekert |
| Griffiths Quantum Mechanics Playlist | Nick Heumann |
| Quantum Physics I | MIT OCW |
| Quantum Physics II | MIT OCW |
| Quantum Physics III | MIT OCW |
| Physical Chemistry | MIT OCW |
| Physical Chemistry | Prof. Derricotte |
| Quantum Chemistry | Trent M. Parker |
| Quantum Transport | Prof. Sergey Frolov |
| Quantum Many-Body Physics | Prof. Luis Gregório Dias |
| Quantum Matter | Prof. Steven Simon |
| Quantum Optics | Prof. Carlos Navarrete-Benlloch |
| Topological Quantum Matter | Weizmann Institute of Science |
| Quantum Field Theory Playlist | Nick Heumann |
| Relativistic Quantum Field Theory Playlist | MIT OCW |
| Important Notes & Physics Etiquettes | Physics with Elliot |
| Math Notes for Quantum Information Science | Introduction to Quantum Information Science |
| Time-Dependent Quantum Mechanics & Spectroscopy Notes | UChicago |
| Solid-State Physics | Prof. M. S. Dresselhaus |
| Transport in Semiconductor Mesoscopic Devices | David K. Ferry (Book 1) / David K. Ferry (Book 2) |
Additional Physics & Mathematics Resources:
| Topic | Source |
|---|---|
| Introduction to Mathematical Reasoning Playlist | Knop’s Course |
| General Mathematical Playlists | Faculty of Khan |
| Geometric Algebra - Why | Parker Glynn-Adey |
| Geometric Algebra - Why 2 | Bivector |
| Zero to Geo[metric Algebra] | sudgylacmoe |
| Differential Geometric Algebra | Crucial Flow Research |
| Advanced Mathematics | The Bright Side of Mathematics |
| Spinors Playlist | eigenchris |
| Weinberg’s Lectures on Quantum Mechanics Playlist | Physics Daemon |
| Thermodynamics & Statistical Physics Playlist | Pazzy Boardman |
| Statistical Mechanics & Thermodynamics Playlist | Physics Daemon |
| Solid State Devices Playlist | nanohubtechtalks |
| QuTech360 Seminars | QuTech Academy |
| Quantum Playlists | Nick Heumann University |
General Comments on Analytical Calculations & Approaches
| # | Reason | Explanation |
|---|---|---|
| 1 | Oversimplified Assumptions | Analytical solutions often rely on simplifying assumptions (e.g., linearity, idealized boundary conditions) that may not reflect real-world behavior. |
| 2 | Inability to Capture Complex Phenomena | Many systems involve nonlinearities, chaos, or coupled interactions that are too complex for closed-form solutions. |
| 3 | High Dimensionality and Computational Complexity | Large systems with many interacting variables can make deriving analytical solutions infeasible or impractical. |
| 4 | Sensitivity to Parameter Variations | Real systems can be highly sensitive to small parameter changes, making analytical solutions unreliable in practice. |
| 5 | Limited Practical Interpretability | Even if an analytical solution exists, it may be too complex to provide intuitive physical insights. |
| 6 | Measurement and Data Limitations | Applying an analytical formula requires accurate input parameters, which may not always be available due to measurement noise or uncertainty. |
| 7 | Evolving or Adaptive Systems | Systems that change over time or adapt dynamically cannot always be captured using static analytical calculations. |
Table of Critical Equations to Know:
Markdown script for the rendered tables above are available: click here