Study Materials

Complete algebra reference reorganized into 12 topic-based parts — exponents & radicals through systems of equations & matrices. Every concept with MATLAB commands, LaTeX syntax boxes, and worked examples.

Associativity, commutativity, identity, inverse, and distributivity — deep explanations with 10 examples each, covering when the laws break (matrices, subtraction) and the connection to abstract algebra.

Clean, step-by-step derivation of $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ by completing the square, with the discriminant explained.

25 proofs across 8 collections: factoring identities (GCF, difference of squares, cubes), polynomial theorems (Division Algorithm, Remainder, Factor, Rational Root), quadratic formula, Cartesian proofs, conic sections, symmetry, logarithmic proofs ($e$ is irrational), and matrix row reduction (REF/RREF).

The foundational five: negative × negative = positive, sum & difference of cubes, distance formula from Pythagorean theorem, $\sqrt{2}$ is irrational, and logarithm properties from exponent rules.

60 original problems across 11 topic sections — exponents, radicals, factoring, equations, inequalities, absolute value, functions, conics, polynomials, logarithms, and systems. Every solution includes LaTeX code and MATLAB verification commands.

Classical mechanics through modern physics — kinematics, Newton's laws, energy, momentum, rotation, gravitation, oscillations, waves, fluids, thermodynamics, electromagnetism, optics, special relativity, and quantum mechanics. Every formula with LaTeX and MATLAB.

DC/AC circuits, phasors, impedance, power (real/reactive/apparent/3-phase), capacitors, inductors, RLC transients, Maxwell's equations, signals & systems (Fourier, Laplace, transfer functions), control theory (PID, state-space), semiconductors, op-amps, digital logic, and communications.

Atomic structure, stoichiometry, gas laws, thermodynamics, kinetics (Arrhenius), equilibrium, acids/bases/pH (Henderson-Hasselbalch), solubility, colligative properties, electrochemistry (Nernst), and nuclear chemistry. Every formula with LaTeX and MATLAB.

Complete reference for the SLAM engine: KF, EKF, ESKF, IEKF, UKF variants, IMU kinematic model, LiDAR measurement model, fusion architecture, state-of-the-art systems comparison, and academic resources.

From Einstein's two postulates to the Lorentz transformations, time dilation, length contraction, and $E = mc^2$ — every step derived with algebra and the Pythagorean theorem. Includes relativistic velocity addition, momentum, the energy–momentum relation $E^2 = (pc)^2 + (mc^2)^2$, spacetime intervals, and Minkowski geometry. With MATLAB explorations including the muon lifetime example.

From the Equivalence Principle through curved spacetime to Einstein's field equations $G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}$. Covers the metric tensor, geodesics, Christoffel symbols, Riemann curvature, Ricci tensor, Einstein tensor, stress–energy tensor, the Newtonian limit, Schwarzschild black holes, 8 experimental confirmations, the cosmological constant, and why quantum gravity is needed — ending at the last equation.

Comprehensive command reference organized by learning progression — basic arithmetic, exponents, fractions, Greek letters, sets, matrices, functions, and complete equation examples for both Algebra I/II and Calculus I. Copy-paste ready for Obsidian and Overleaf.