Mathematics and Physics from the Ground Up

A free course for science and engineering students: from a pre-course to Maxwell's equations and the hydrogen atom. Every lesson has simulations, worked examples, exercises with answers and a printable version.

Mathematics

From a pre-course to partial differential equations. Every lesson starts with a question from physics or engineering and ends with exercises and answers.

0 Pre-course 5
I Vectors and single-variable calculus 7
II Linear algebra 3
III Multivariable calculus 6
IV Differential equations and integral transforms 7
V Probability and data 2
VI Toolkit 2

Physics

Physics in the spirit of the Feynman Lectures — from a phenomenon and a question to the law, with simulations and exercises.

I Mechanics 7
II Heat and fluids 3
III Waves and optics 2
IV Electromagnetism 4
V Quantum physics 3

Control engineering

From feedback seen in the time domain, before any transforms, through the frequency domain and the PID controller in practice, to state space, LQR and the Kalman filter. Running example: a drone that has to hold its altitude. Part V is a bridge to A. Tewari's book “Advanced Control of Aircraft, Spacecraft and Rockets”. Continued in the Flight dynamics and Astronautics courses. In control engineering the imaginary unit is written j.

I Feedback in the time domain 3
II The s and frequency domains: tools for the loop 5
III The PID controller in practice 4
IV State space, estimation and optimal control 6
V A bridge to Tewari 2

Flight dynamics

Attitude and the rigid body in 3D, aircraft stability and modes, dampers and the autopilot, a full six-degree-of-freedom simulation and, at the end, rockets: thrust vectoring and guidance. Running example: a light aircraft at 55 m/s (data: Tewari, example 3.3). Builds on physics (chapter 10) and control engineering (parts I–IV).

I Attitude and the rigid body in 3D 3
II The aircraft 7
III The rocket 2

Astronautics

Orbits, getting a rocket to orbit, manoeuvres, rendezvous and interplanetary travel, and finally spacecraft attitude. Builds on mechanics (physics 1–5), differential equations and numerical methods; the last three lessons build on control engineering and flight dynamics.

I Orbits 3
II Launch and manoeuvres 3
III Interplanetary travel 4
IV Spacecraft attitude 3

Matura in mathematics

Preparation for the Polish matura exam in mathematics, basic and extended level, following the 2024 core curriculum (sections I–XIII). Polish only.

0 Introduction 1
I Real numbers 5
II Algebraic expressions 2
V Functions 7
III Equations and inequalities 6
IV Systems of equations 2
VI Sequences 4
VII Trigonometry 5
VIII Plane geometry 6
IX Analytic geometry 4
X Solid geometry 3
XI Combinatorics 2
XII Probability and statistics 4
XIII Differential calculus 4
XIV Dress rehearsal 2