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.
- 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
Borrowed from: Learn with the Map of Mathematics, The Bright Side of Mathematics (2023)
https://youtu.be/ljGSId-uHw8?si=xKNup3hOVsWC6uTv&t=200
| Tip | Explanation/ Details |
|---|---|
| Substitute given 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. |
| Term or Concept | Description or Note |
|---|---|
| Another way of saying abstract | “Indeterminate” or “boundless". |
| Light | Generally refers to the electromagnetic field or electromagnetic radiation. When quantized, it usually refers to photons but can also mean quantized modes, coherent and squeezed states, polaritons, or plasmons and surface plasmon polaritons. |
| Optics | A branch of physics concerned with the generation, propagation, manipulation, and detection of electromagnetic radiation (especially in and around the visible range), as well as its interactions with matter, including phenomena like reflection, refraction, diffraction, interference, and polarization. There are sub-disciplines or sub-classifications of optics such as electron, ion, and quantum optics. |
| Plastic | Often used metaphorically to refer to something moldable or flexible; in a physical context, it can mean a polymer material or exhibit plastic (irreversible) deformation. |
| Radiation | The emission or transmission of energy through space or a medium in the form of electromagnetic waves (e.g., radio waves, visible light, X-rays, gamma rays) or subatomic particles (e.g., alpha, beta, neutrons). In a broader sense, it can also refer to acoustic waves, though in physics “radiation” typically implies electromagnetic or particle radiation. Many people equate “radiation” solely with ionizing radiation, which is harmful in large doses and includes X-rays, gamma rays, and high-energy particles that can ionize atoms. This is a common misunderstanding because non-ionizing radiation (like visible light, microwaves, radio waves) is also “radiation,” just not ionizing. |
| Electric field | Generally refers to the force per unit charge in a region due to a static or dynamic charge distribution. It can also be generated or excited by time-varying magnetic fields (as in electromagnetic waves) or by charge redistribution effects caused by incident radiation (e.g., in the photoelectric effect). Note: In formal treatments, the photoelectric effect is more often framed in a quantum context rather than purely classical terms. |
| Pure electric fields can be justified as being “pure” | If they originate from idealized situations: a point charge, a parallel plate capacitor, uniformly charged conductors, electrostatic lenses (vacuum tube-based focusing), and electric dipole fields. |
| Platform(s) | Can be a physical or conceptual foundation upon which systems, processes, or experiments are built. |
| Linearity | 1:1 ratio in the response by default, with a consistent or zero rate of change. |
| Non-linearity | Any relationship where the ratio or response exhibits a non-zero rate of change; not a simple 1:1 linear relationship. |
| Theory | Often confused with “hypothesis” or “mathematical justification.” In science, a theory is a well-substantiated explanation of some aspect of the natural world. |
| Magnetic | Refers to magnetism. By default, all matter made of atoms is at least diamagnetic. If it is not diamagnetic, its magnetism (paramagnetism, ferromagnetism, etc.) depends on or is determined by the behavior of its electrons. When describing the spin direction of a single electron, it is referred to as the magnetic dipole moment, while a magnetic domain refers to a bulk region of collective spins or uniformly aligned spins. |
| By default, it may be better to think of values in terms of ratios or slopes. | A guiding principle in analyzing systems or problems: scaling relationships, slopes, and derivatives often reveal more insight than absolute numbers. |
| Possible alternative name for microwave photon detector | Microwave photon radiation detector. |
| Everything is a transmission line, and everything is a capacitor (including self-capacitance). | A broad conceptual notion in electronics and physics, emphasizing that all structures can be modeled as having transmission line properties and inherent capacitance. |
| Electron spins, including all spin 1/2 particles, are physical realizations, out of many, of the abstract, mathematical spinor representations in nature. Interestingly, this is one example where an abstract mathematical object has experimentally measurable effects or direct experimental consequences. Spins in the technical sense generally refer to the description of an intrinsic angular momentum, meaning that it is purely quantum mechanical. In mathematics and physics, a spinor* is a type of object used to describe particles with half-integer spin (spin -1/2, spin +1/2, etc.). These objects transform in a particular way under rotations (technically under the group SU(2), which is the double cover of the rotation group SO(3)). Here, a 2π rotation changes the phase of a spinor by −1, meaning it does not return to its original state but instead acquires a sign flip. In quantum theory, classifying all possible particle types comes down to looking at irreducible representations of the Poincaré group (in special relativity) or the Galilean group (in non relativistic mechanics). Spin 1/2 emerges naturally when you look at certain irreducible representations—namely, those described by spinors. Any spin- fermion—such as quarks, protons (composite, but effectively spin 1/2 in total), and neutrinos—can also be described by spinors or "spinor formalism". Additional note: the spin of protons arises from a complex interplay of quark spins, gluon angular momentum, and orbital motion. Its total spin behaves like a fundamental fermion, but its substructure is different from an elementary particle. |
| *Spinors are two-component objects, they do return to the same quantum state after a 4π rotation but not after 2π. |
| Key terms: irreducible representations, experimental consequences |
| Method | Mechanism | Key Considerations | Applications |
|---|---|---|---|
| Photoelectric Effect | Light ejects electrons from a surface, creating charge imbalance | Requires photon energy > work function; works best in vacuum | Spacecraft charging, photoemission devices |
| Controlled Charge Separation | Electrons are emitted and collected on a secondary surface via photoemission, field emission, or thermionic emission | Requires an electron collector; prevents neutralization | Photoemission-based capacitors, charge storage |
| Vacuum Conditions | Electrons travel freely, leading to sustained electric fields | No surrounding medium to neutralize charge | Electron beam devices, vacuum tube applications |
| Voltage Bias Application | A potential difference guides photoelectrons to a specific region | Ensures continuous charge separation | Controlled electron beams, energy harvesting |
| UV Radiation-Induced Charging | High-energy UV photons eject electrons via the photoelectric effect | Effective in space or high-intensity UV environments | Spacecraft charging, UV-sensitive detectors |
| Solar Wind Charging | Plasma interactions induce surface charging and electron displacement | Occurs naturally in space; depends on plasma density and material properties | Spacecraft potential buildup, lunar dust levitation |
| Photoemission Cathodes | Use of a light-activated electron source in a circuit | Requires efficient cathode material | Photocathodes for electron guns, free-electron lasers |
| Term | Definition |
|---|---|
| Arbitrary | Refers to chosen values or units that maintain internal consistency without relying on an external, standardized reference. Example: arbitrary units used in graphs and charts. |
| Arbitrary Units (a.u.) | Used in graphs and charts, they represent a consistent measure but do not correspond to a standardized physical unit. They are meaningful within the given context but are not directly comparable to a universal scale. |
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
Note: the W, x, y, z, T variables in the example above are merely substites & do not correspond to any physical variables.
| 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 |
| Poisson's Equation for Beginners | 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) |
| 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 |
Markdown script for the rendered tables above are available: click here