The goal of this section of the website is to provide free, easy to understand, educational materials for the most important concepts in physics and math. This is my attempt to compile everything I learned as an undergraduate physics major into one easy-to-navigate website. Unfortunately it is not easy to compile 4-5 years of intense work and study into a single website. This takes time, and a lot of hard work. Fortunately my hard work pays off since it makes your work easier when you sit down and try to actually learn this stuff.
The Scientific Method:
“If it disagrees with experiment. It’s WRONG! – Richard Feynman
The Scientific Method is a body of techniques for investigating phenomena where predictions are tested using repeatable controlled experiments. The Scientific Method of inquiry is distinguished from other methods of inquiry in that scientists seek to let reality speak for itself, supporting a theory when a theory’s predictions are confirmed and challenging a theory when its predictions prove false.
The Oxford English Dictionary says that scientific method is: “a method or procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses.”
What does it mean to be scientific?
In order to be scientific we must routinely test our predictions or hypotheses with controlled, repeatable experiments. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of reasoning.
Physics: What is it all about?
Physics is the study of Energy and Motion. How things move, why things move, where they will move under what conditions and when, and so on and so forth. Physics is the most exact science there is, but physics is still only based on models with measurable limits of accuracy. A simplistic model is a good first approximation when it has a 1 sigma confidence level which is about 68%, As more subtle effects (called perturbations) are added to a model, the accuracy level can be increased. A 2 sigma confidence level is 95% accurate and is generally all you need to be sure that your theory is in agreement with experiment since there is always going to be some level or random noise, interference, or error in your measurements.
Physicists use a variety of tools, the most important however are the tools of mathematics. There is a list of essential mathematics which every physicist should have some general working knowledge of. These tools are different from the standard everyday physics equations which students learn in General Physics 1 and 2 (sometimes 3). All physicists should know those standard equations by heart. The tricky stuff comes with finding series approximations for functions and other ways to simplify more complicated models without sacrificing too much accuracy in the model.
Mathematics are important because in physics we don’t just like to qualitatively undestand something, we strive to quantitatively be able to understand exactly how much of everything is involved. And that doesn’t mean doing crazy hard mathematics all the time, it can also mean doing a Back of the Envelope Calculation where we just try to get a rough figure or estimate about something. These types of simple small calculations are an important part of being a physicist, as is all the other stuff.
Undergraduate Level Physics Curriculum:
This section is meant to provide the core curriculum for an undergraduate degree in Physics and Mathematics, condensed down to an overview of the most important information. This is meant to act as a study guide for physics hobbyists as well as students interested in pursuing future careers in science. This is not meant to be a replacement for a college education or your standard college text book covering the relevant subject matter. Additional links and resources are provided on each page. If you find the subject matter interests you, it is highly recommended you buy a college text book on the subject in addition to studying the material provided on this website.
Curriculum Requirements for an Undergraduate Degree in Physics:
Also check the requirements for different colleges such as MIT:
Calculus I – Xeno’s Paradox, Limits, Derivatives, Chain Rule, Power Rule, The Fundamental Theorem of Calculus, etc.
Chemistry I –
General Physics I – Kinematics, Motion, Speed, Velocity, Acceleration, Mass, Forces, Collisions, Momentum, and Ballistic motion.
Calculus II – Integration, By Parts, Chain Rule, Riemann Sums, Partial Derivatives, Integral sums
Chemitry II –
General Physics II – Intro to Circuits, Electricity and Magnetism, Heat and thermodynamics, waves and sound, etc.
Calculus III – Series, Expansions, Functions, Double and Triple integrals, Gaussians
Differential Equations – First, Second, and higher order. Bernoulli Equation, Exact Equations, Homogeneous Equations, Integrating Factors, Series Solutions, Wronskians, Lagrangians, etc.
Math Physics – Vector Calculus, Coordinate systems, Jacobians, Intro to Differential Eq. linear algebra, etc.
Modern Physics – Intro to Quantum Mechanics, Relativity, and “Modern” physics. Laboratory Experiments: Michaelson-Moreley, Millikan Oil Drop,
Linear Algebra – Matrices, Vectors, Arrays, Matrix Mechanics, Determinants, Reduced Row Echelon, Lagrange Multipliers, Eigenvectors, etc.
Experimental Physics I – How to conduct experiments and write papers. Labs include: Simple undamped Pendulum, damped pendulum, (LC Circuits: High Pass Filter, Low Pass Filter, Band Pass Filter), Nuclear Magnetic Resonance, Hyper-Fine Splitting of Rb.
Electricity and Magnetism I – Electrostatics, Coulomb’s Law, Gauss’s Law, Ampere’s Law, Dielectrics, conductors, insulators, Divergences, Gradients, vector field plots,
Classical Mechanics I – Simple Harmonic Oscillators, Basic Gravity, Kepler’s Laws, All the problems from Physics 1 and 2 except this time WITH Air Resistance and much more Calculus…
Experimental Physics II – Coupled Harmonic Oscillators, Photospectroscopy, X-Ray crystallography, Doppler, Compton, Rutherford, Hertz, etc.
Electricity and Magnetism II – Electrodynamics, Faraday’s Law, Induction, wave equation for light, Stokes’ Theorem, Curl, Pointing Vectors, Radios, magnets, magnetization, Maxwell’s Equations, Conservation Laws.
Classical Mechanics II, or Thermal Physics, or Statistical Physics – Statistical Mechanics, Heat Engines, Calorimetry, macro and micro-states, thermal gradients, differentials, Euler’s Equation, Lagrangians, Hamiltonians and Hamilton’s Principle, Lagrange’s equations with undetermined multipliers, etc.
Special Relativity – Lorentz transformations, time dilation, twin paradox, space-time, etc.
Quantum I – Heisenberg, Schrodinger, Normalization, Wave Functions, Time-Dependent and Independant equations and solutions.
General Relativity – Gravity, Einstein’s Field Equations, Tensors and Space-Time Metrics, etc.
Quantum II – Hilbert Spaces, Hyperfine splitting, Spin-Spin interaction, etc.
By the end of their third semester of study, all physicists should have obtained the following basic level math and physic skills.
- Solve a quadratic equation for its roots.
- Solve any n equations for n unknowns.
- Specify the straight line that passes through any two points in a plane.
- Use trigonometric functions and simple trig identities.
- Know and use the areas, circumference, or volume of basic shapes.
- Perform basic differentiation and integration accurately, for instance product and chain rules, basic u substitution, trigonometric substitution integrals.
- List and explain Newton?s Three Laws and Kepler’s Three Laws.
- Give definitions (as vectors if necessary) for force, momentum, angular momentum, kinetic energy, Hooke’s Law, torque, angular variables, centripetal acceleration, and position and velocity under constant acceleration.
- Define the potential energies associated with gravitational interactions between any two objects with mass, electrical interactions between any two charges, springs, and gravity near the surface of the Earth.
- Identify and solve problems requiring the use of Newton’s 2nd Law.
- Identify and solve problems requiring the use of energy, momentum, or angular momentum conservation.
- Define and use the force on a charged particle due to the electric or magnetic field.
- Determine the electric field, magnetic field, and electric potential in simple situations.
- Understand basic circuit theory.
- Know and be able to explain the basic right hand rules as they apply to electricity and magnetism.
By the end of their sixth semester of study, all physicists should have obtained the following intermediate level math and physics skills.
- Integrate using u and basic trigonometric substitutions.
- Integrate by parts.
- Perform surface and volume integrals on basic geometric shapes.
- Sketch the graph of any function f(x) – identifying features such as singularities, local extrema, and asymptotic behavior.
- Take the dot or cross product of any two vectors.
- Take the gradient of a function, the divergence of a vector, and the curl of a vector. Draw pictures defining what the gradient, divergence and curl mean physically.
- Solve linear, 1st order ODEs by separation of variables.
- Solve linear, 2nd order homogeneous ODEs.
- Perform simple matrix algebra: multiplication, determinants, inverse, transpose, row reduction, etc.
- Perform Taylor series expansions and use the binomial theorem.
- Explain the basic principle behind the Fourier transform and Fourier series.
- Know key physics constants such as c, G, Boltzman’s constant, Avogadro’s number, the masses of the proton and electron, the fundamental electric charge, and Planck’s constant.
- Apply solutions to differential equations to physics by evaluating initial conditions, in particular to problems involving the simple harmonic oscillator or air resistance.
- Understand basic experimental design – for example, know how to design experiments that would test Physics I or II concepts (conservation laws, Newton’s Laws, SHM, Ohm’s Law, Snell’s Law, etc.)
- Understand error propagation as it applies to basic experiments.
- List and explain Maxwell’s equations in both integral and differential forms and draw pictures associated with them.
- Understand the basic underlying principles of special relativity.
- Derive the time delay equation in special relativity.
- Understand the basic relevance of quantum mechanics and where it applies in nature.
- Possess a beginning understanding of the calculus of variations and the Euler-Lagrange equations.