Dependency map
For every lesson: what you need to know first and which lessons build on it.
| Lesson | Requires | Used by |
|---|---|---|
| Mathematics 0.1 Numbers, powers, logarithms and elementary functions |
— | 0.2, 0.3, 0.5, 1.2 |
| Mathematics 0.2 Equations, inequalities, polynomials and partial fractions |
0.1 | 0.4, 0.5, 1.2, 1.7, 2.1, 4.4, Control engineering 1.1 |
| Mathematics 0.3 Trigonometry and hyperbolic functions |
0.1 | 0.4, 0.5, 1.1, 1.2, 1.7, 2.2, 3.2 |
| Mathematics 0.4 Analytic geometry and conic sections |
0.2, 0.3 | 1.1, 1.7, Physics 5 |
| Mathematics 0.5 Units, dimensional analysis, estimation and measurement uncertainty |
0.1, 0.2, 0.3 | 1.4, 5.2, Physics 1, Physics 10 |
| Mathematics 1.1 Vectors |
0.3, 0.4 | 1.3, 2.1, 3.1, Physics 1 |
| Mathematics 1.2 Sequences, limits and continuity |
0.1, 0.2, 0.3 | 1.3, 1.6 |
| Mathematics 1.3 Derivatives |
1.1, 1.2 | 1.4, Physics 1 |
| Mathematics 1.4 Differential calculus in practice |
0.5, 1.3 | 1.5, 3.1, 6.2, Physics 2, Physics 11 |
| Mathematics 1.5 Integrals |
1.4 | 1.6, 3.3, 4.1, 6.1, Physics 1 |
| Mathematics 1.6 Series and power series |
1.2, 1.5 | 1.7, 4.2, 5.1, Control engineering 3.4 |
| Mathematics 1.7 Complex numbers and phasors |
0.2, 0.3, 0.4, 1.6 | 2.2, 2.3, 4.1, Physics 6, Physics 17, Control engineering 2.3 |
| Mathematics 2.1 Matrices, determinants and linear systems |
0.2, 1.1 | 2.2, 3.3, 5.2 |
| Mathematics 2.2 Linear maps: rotations, change of basis, Lorentz transformation |
0.3, 1.7, 2.1 | 2.3, Physics 7, Flight dynamics 1, Astronautics 2 |
| Mathematics 2.3 Eigenvalues and eigenvectors |
1.7, 2.2 | 3.1, 4.5, Physics 18, Control engineering 4.1, Flight dynamics 1 |
| Mathematics 3.1 Multivariable chain rule, extrema and Lagrange multipliers |
1.1, 1.4, 2.3 | 3.2, 3.4, 4.6, 5.2, 6.1 |
| Mathematics 3.2 Polar, cylindrical and spherical coordinates |
0.3, 3.1 | 3.3, 4.7 |
| Mathematics 3.3 Double and triple integrals |
1.5, 2.1, 3.2 | 3.4, 5.1 |
| Mathematics 3.4 Line integrals, conservative fields and exact differentials |
3.1, 3.3 | 3.5, Physics 9 |
| Mathematics 3.5 Flux, divergence and curl |
3.4 | 3.6, 4.7 |
| Mathematics 3.6 The theorems of Green, Stokes and Gauss |
3.5 | Physics 10, Physics 13 |
| Mathematics 4.1 Ordinary differential equations |
1.5, 1.7 | 4.2, 6.1, 6.2, Physics 6, Control engineering 1.2, Astronautics 1 |
| Mathematics 4.2 Fourier series |
1.6, 4.1 | 4.3, 4.7, Physics 11 |
| Mathematics 4.3 The Fourier transform and the Dirac delta |
4.2 | 4.4 |
| Mathematics 4.4 The Laplace transform |
0.2, 4.3 | 4.5, Control engineering 2.1 |
| Mathematics 4.5 Linear systems and state space |
2.3, 4.4 | 4.6, Control engineering 4.1, Flight dynamics 5, Astronautics 6 |
| Mathematics 4.6 Nonlinear systems, phase plane and stability |
3.1, 4.5 | Control engineering 4.5, Control engineering 4.6, Flight dynamics 2, Flight dynamics 5 |
| Mathematics 4.7 PDEs: the wave, diffusion and Laplace equations |
3.2, 3.5, 4.2 | Physics 16, Physics 19 |
| Mathematics 5.1 Probability and distributions |
1.6, 3.3 | 5.2, Physics 8, Physics 17 |
| Mathematics 5.2 Measurement statistics and curve fitting |
0.5, 2.1, 3.1, 5.1 | Control engineering 1.2, Control engineering 4.4 |
| Mathematics 6.1 Calculus of variations |
1.5, 3.1, 4.1 | Physics 12, Control engineering 4.3, Control engineering 5.1 |
| Mathematics 6.2 Numerical methods: Python and C++ |
1.4, 4.1 | Control engineering 1.3, Control engineering 3.4, Control engineering 5.1, Flight dynamics 5, Flight dynamics 10, Astronautics 2, Astronautics 4, Astronautics 9 |
| Physics 1 Motion and its causes |
Mathematics 0.5, Mathematics 1.1, Mathematics 1.3, Mathematics 1.5 | 2, 7, Control engineering 1.1 |
| Physics 2 Energy |
Mathematics 1.4, 1 | 3, 6, 13, Flight dynamics 6 |
| Physics 3 Momentum, collisions and the centre of mass |
2 | 4, 8, Astronautics 4 |
| Physics 4 Rotation and rigid bodies |
3 | 5, Flight dynamics 1, Flight dynamics 4 |
| Physics 5 Gravitation and orbits |
Mathematics 0.4, 4 | Astronautics 1 |
| Physics 6 Oscillations, resonance and linear systems |
Mathematics 1.7, Mathematics 4.1, 2 | 11, Control engineering 1.2, Control engineering 2.3 |
| Physics 7 Reference frames and special relativity |
Mathematics 2.2, 1 | — |
| Physics 8 Heat and kinetic theory |
Mathematics 5.1, 3 | 9 |
| Physics 9 Thermodynamics and entropy |
Mathematics 3.4, 8 | 10 |
| Physics 10 Fluids: pressure, flow, viscosity and aerodynamic forces |
Mathematics 0.5, Mathematics 3.6, 9 | Flight dynamics 4, Astronautics 10 |
| Physics 11 Waves and sound |
Mathematics 1.4, Mathematics 4.2, 6 | 12 |
| Physics 12 Optics: least time, interference, diffraction |
Mathematics 6.1, 11 | — |
| Physics 13 The electric field and Gauss's law |
Mathematics 3.6, 2 | 14 |
| Physics 14 Potential, conductors and capacitors |
13 | 15 |
| Physics 15 Current and the magnetic field |
14 | 16 |
| Physics 16 Induction, Maxwell's equations and electromagnetic waves |
Mathematics 4.7, 15 | — |
| Physics 17 Probability amplitudes |
Mathematics 1.7, Mathematics 5.1 | 18 |
| Physics 18 Quantum states, two-state systems and spin |
Mathematics 2.3, 17 | 19 |
| Physics 19 The hydrogen atom and the periodic table |
Mathematics 4.7, 18 | — |
| Control engineering 1.1 Why feedback? Open and closed loops |
Mathematics 0.2, Physics 1 | 1.2 |
| Control engineering 1.2 How systems respond: basic elements and the step response |
Mathematics 4.1, Mathematics 5.2, Physics 6, 1.1 | 1.3 |
| Control engineering 1.3 P and PI control in the loop: why the integral removes the error |
Mathematics 6.2, 1.2 | 2.1 |
| Control engineering 2.1 Block diagrams and the loop transfer function |
Mathematics 4.4, 1.3 | 2.2, 2.3 |
| Control engineering 2.2 Closed-loop poles and zeros, root locus |
2.1 | 2.4, Flight dynamics 8 |
| Control engineering 2.3 Frequency response and Bode plots |
Mathematics 1.7, Physics 6, 2.1 | 2.4 |
| Control engineering 2.4 Loop stability: the −1 point, margins and delay |
2.2, 2.3 | 2.5 |
| Control engineering 2.5 Disturbances, noise and sensitivity: S, T and loop shaping |
2.4 | 3.1, 5.2 |
| Control engineering 3.1 The PID controller: P, I and D action and steady-state error |
2.5 | 3.2, 4.1 |
| Control engineering 3.2 Tuning: a model from a test, tuning rules and cascade control |
3.1 | 3.3, Flight dynamics 9 |
| Control engineering 3.3 Nonlinear reality: saturation, windup, feedforward and gain scheduling |
3.2 | 3.4, 4.6, Flight dynamics 8, Flight dynamics 9, Flight dynamics 11, Astronautics 13 |
| Control engineering 3.4 Digital control: sampling, the Z-transform and the controller in code |
Mathematics 1.6, Mathematics 6.2, 3.3 | 4.4 |
| Control engineering 4.1 State feedback, pole placement and controllability |
Mathematics 2.3, Mathematics 4.5, 3.1 | 4.2, 4.3 |
| Control engineering 4.2 State observers and observability |
4.1 | 4.4 |
| Control engineering 4.3 LQR: a cost instead of poles |
Mathematics 6.1, 4.1 | 4.6, 5.1, 5.2, Flight dynamics 12 |
| Control engineering 4.4 The Kalman filter |
Mathematics 5.2, 3.4, 4.2 | 4.5, 5.2 |
| Control engineering 4.5 The extended Kalman filter and single-axis sensor fusion |
Mathematics 4.6, 4.4 | Flight dynamics 3 |
| Control engineering 4.6 Lyapunov stability and nonlinear control |
Mathematics 4.6, 3.3, 4.3 | — |
| Control engineering 5.1 Pontryagin's maximum principle, time-optimal control and the shooting method |
Mathematics 6.1, Mathematics 6.2, 4.3 | Flight dynamics 12 |
| Control engineering 5.2 LQG and robustness |
2.5, 4.3, 4.4 | — |
| Flight dynamics 1 Attitude in 3D: rotation matrices, Euler angles, axis–angle and quaternions |
Mathematics 2.2, Mathematics 2.3, Physics 4 | 2, 3, 10 |
| Flight dynamics 2 The rigid body in 3D: the inertia tensor, Euler's equations and the intermediate axis |
Mathematics 4.6, 1 | 7, 10, 11, Astronautics 11 |
| Flight dynamics 3 Sensor fusion in 3D: attitude estimation with quaternions |
Control engineering 4.5, 1 | Astronautics 12 |
| Flight dynamics 4 Static stability and trim: why an aircraft has a tail |
Physics 4, Physics 10 | 5 |
| Flight dynamics 5 Flight in the plane of symmetry: the longitudinal model and stability derivatives |
Mathematics 4.5, Mathematics 4.6, Mathematics 6.2, 4 | 6 |
| Flight dynamics 6 Longitudinal modes: the phugoid and the short period |
Physics 2, 5 | 7, 9 |
| Flight dynamics 7 Lateral motion: roll, spiral and Dutch roll |
2, 6 | 8 |
| Flight dynamics 8 Stability augmentation: yaw and pitch dampers |
Control engineering 2.2, Control engineering 3.3, 7 | 9 |
| Flight dynamics 9 The autopilot: inner and outer loops |
Control engineering 3.2, Control engineering 3.3, 6, 8 | 10 |
| Flight dynamics 10 Six-degree-of-freedom simulation: putting the aircraft together |
Mathematics 6.2, 1, 2, 9 | — |
| Flight dynamics 11 The rocket: attitude and thrust vectoring |
Control engineering 3.3, 2, Astronautics 4 | 12 |
| Flight dynamics 12 Guidance: the nominal trajectory and tracking deviations |
Control engineering 4.3, Control engineering 5.1, 11, Astronautics 4 | — |
| Astronautics 1 The two-body problem and the orbit equation |
Mathematics 4.1, Physics 5 | 2, 4, 10, 11 |
| Astronautics 2 Orbital elements, Kepler's equation and the ground track |
Mathematics 2.2, Mathematics 6.2, 1 | 3, 5 |
| Astronautics 3 Perturbations: Earth's oblateness, Sun-synchronous orbits and drag |
2 | — |
| Astronautics 4 Rocket to orbit: the rocket equation, staging and the gravity turn |
Mathematics 6.2, Physics 3, 1 | Flight dynamics 11, Flight dynamics 12, 5 |
| Astronautics 5 Orbital manoeuvres: the Hohmann transfer and plane changes |
2, 4 | 6, 7 |
| Astronautics 6 Relative motion and orbital rendezvous |
Mathematics 4.5, 5 | — |
| Astronautics 7 Spheres of influence and interplanetary transfers |
5 | 8, 9 |
| Astronautics 8 Gravity assists |
7 | — |
| Astronautics 9 Lambert's problem and porkchop plots |
Mathematics 6.2, 7 | — |
| Astronautics 10 Atmospheric entry |
Physics 10, 1 | — |
| Astronautics 11 A spacecraft without control: spin, dual spin and gravity gradient |
Flight dynamics 2, 1 | 12 |
| Astronautics 12 Attitude determination: sensors, TRIAD and filtering |
Flight dynamics 3, 11 | 13 |
| Astronautics 13 Attitude control: reaction wheels, magnetorquers and thrusters |
Control engineering 3.3, 12 | — |
| Matura in mathematics 0.1 How the maths matura works: the paper, scoring, the formula booklet |
— | — |
| Matura in mathematics 1.1 Operations, intervals and absolute value |
— | 1.2, 1.5, 5.1 |
| Matura in mathematics 1.2 Powers and roots |
1.1 | 1.3, 1.4, 2.1, 7.1 |
| Matura in mathematics 1.3 Logarithms |
1.2 | 5.6 |
| Matura in mathematics 1.4 Percentages, compound interest, deposits and loans |
1.2 | 6.3 |
| Matura in mathematics 1.5 Divisibility and first proofs |
1.1 | — |
| Matura in mathematics 2.1 Special products and rational expressions |
1.2 | 2.2, 3.1, 11.2 |
| Matura in mathematics 2.2 Polynomials: division, factoring, integer roots |
2.1 | 3.3 |
| Matura in mathematics 5.1 Functions and graphs: reading properties |
1.1 | 5.2, 5.3, 5.5, 5.6, 6.1 |
| Matura in mathematics 5.2 The linear function |
5.1 | 9.1 |
| Matura in mathematics 5.3 The quadratic function: three forms and the graph |
5.1 | 5.4, 5.7, 3.2 |
| Matura in mathematics 5.4 The quadratic on an interval and optimisation |
5.3 | 13.4, 14.1 |
| Matura in mathematics 5.5 Inverse proportionality |
5.1 | — |
| Matura in mathematics 5.6 Exponential and logarithmic functions |
1.3, 5.1 | — |
| Matura in mathematics 5.7 Graph transformations, composition, proving monotonicity |
5.3 | 7.3, 13.1 |
| Matura in mathematics 3.1 Linear equations and inequalities, equivalent transformations |
2.1 | 3.2, 4.1 |
| Matura in mathematics 3.2 Quadratic equations and inequalities, factored polynomial equations |
5.3, 3.1 | 3.3, 3.5, 3.6, 4.2 |
| Matura in mathematics 3.3 Polynomial inequalities and biquadratic equations |
2.2, 3.2 | 3.4 |
| Matura in mathematics 3.4 Rational expressions, equations and inequalities |
3.3 | — |
| Matura in mathematics 3.5 Equations and inequalities with absolute value |
3.2 | — |
| Matura in mathematics 3.6 Vieta's formulas and equations with a parameter |
3.2 | — |
| Matura in mathematics 4.1 Linear systems and word problems |
3.1 | 4.2, 9.1 |
| Matura in mathematics 4.2 Systems with a quadratic equation |
3.2, 4.1 | 9.3 |
| Matura in mathematics 6.1 Sequences: general term, recursion, monotonicity |
5.1 | 6.2, 6.3 |
| Matura in mathematics 6.2 Arithmetic sequences |
6.1 | — |
| Matura in mathematics 6.3 Geometric sequences |
1.4, 6.1 | 6.4, 14.1 |
| Matura in mathematics 6.4 Limits of sequences and geometric series |
6.3 | 13.1 |
| Matura in mathematics 7.1 Trigonometry of angles from 0° to 180° |
1.2 | 7.2, 7.3, 8.1, 10.1 |
| Matura in mathematics 7.2 The law of cosines and the area of a triangle |
7.1 | 7.5, 8.2 |
| Matura in mathematics 7.3 Radians, graphs, periodicity and reduction formulas |
5.7, 7.1 | 7.4 |
| Matura in mathematics 7.4 Angle-sum formulas and trigonometric equations |
7.3 | — |
| Matura in mathematics 7.5 The law of sines and solving triangles |
7.2 | — |
| Matura in mathematics 8.1 Circles: chords, tangents, inscribed and central angles, sectors |
7.1 | 8.5 |
| Matura in mathematics 8.2 Triangles: types and special points |
7.2 | 8.3, 8.4 |
| Matura in mathematics 8.3 Quadrilaterals and regular polygons |
8.2 | — |
| Matura in mathematics 8.4 Thales' theorem and similarity |
8.2 | 8.5 |
| Matura in mathematics 8.5 Trigonometry in figures and geometric proofs |
8.1, 8.4 | 8.6, 14.1 |
| Matura in mathematics 8.6 Cyclic and tangential quadrilaterals, converse of Thales' theorem |
8.5 | — |
| Matura in mathematics 9.1 Lines in the plane |
5.2, 4.1 | 9.2, 9.4 |
| Matura in mathematics 9.2 Distance, the equation of a circle and symmetries |
9.1 | 9.3, 14.1 |
| Matura in mathematics 9.3 Line and circle, two circles, tangents and perpendicular lines |
4.2, 9.2 | — |
| Matura in mathematics 9.4 Vectors |
9.1 | — |
| Matura in mathematics 10.1 Lines, planes and angles in space |
7.1 | 10.2 |
| Matura in mathematics 10.2 Volumes and surface areas, similar solids |
10.1 | 10.3, 14.1 |
| Matura in mathematics 10.3 Cross-sections and the three perpendiculars theorem |
10.2 | — |
| Matura in mathematics 11.1 The multiplication and addition rules |
— | 11.2, 12.1 |
| Matura in mathematics 11.2 Permutations, variations, combinations and binomial coefficients |
2.1, 11.1 | 12.4 |
| Matura in mathematics 12.1 Classical probability |
11.1 | 12.3 |
| Matura in mathematics 12.2 Statistics: means, median, mode |
— | 14.1 |
| Matura in mathematics 12.3 Conditional and total probability, Bayes' formula |
12.1 | 12.4 |
| Matura in mathematics 12.4 Bernoulli trials |
11.2, 12.3 | — |
| Matura in mathematics 13.1 Limits and continuity of functions, the Darboux property |
5.7, 6.4 | 13.2 |
| Matura in mathematics 13.2 The derivative: definition, meaning and computation |
13.1 | 13.3 |
| Matura in mathematics 13.3 Monotonicity and extrema |
13.2 | 13.4 |
| Matura in mathematics 13.4 Optimisation with the derivative |
5.4, 13.3 | 14.2 |
| Matura in mathematics 14.1 Mock exam: basic level |
5.4, 6.3, 8.5, 9.2, 10.2, 12.2 | 14.2 |
| Matura in mathematics 14.2 Mock exam: extended level |
13.4, 14.1 | — |