Noesis

Maths

Number, structure, change and the certainty of proof.

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Test bank →Learning path →Deep dives → (484 topics)

Phase 1 — Absolute Foundation

Age 8–12 equivalent · 3–4 months @ 1 hr/day3 chapters
1.1

Arithmetic & Number Systems

22 topics
  1. 1.1.1Natural numbers, whole numbers, integers — definitions and the number line
  2. 1.1.2Place value system — units, tens, hundreds, thousands, lakhs, crores
  3. 1.1.3Addition and subtraction — carrying, borrowing, word problems
  4. 1.1.4Multiplication — tables (1–20), long multiplication, area model
  5. 1.1.5Division — long division, remainder, dividend - divisor - quotient vocabulary
  6. 1.1.6Order of operations — BODMAS - PEMDAS with nested brackets
  7. 1.1.7Factors and multiples — all factors of a number, factor pairs
  8. 1.1.8Prime numbers — Sieve of Eratosthenes, primality testing
  9. 1.1.9Prime factorization — factor trees, ladder method
  10. 1.1.10HCF (GCD) — prime factorization method, Euclidean algorithm
  11. 1.1.11LCM — prime factorization method, relationship HCF × LCM = product
  12. 1.1.12Fractions — proper, improper, mixed numbers
  13. 1.1.13Equivalent fractions, simplifying fractions
  14. 1.1.14Addition, subtraction, multiplication, division of fractions
  15. 1.1.15Decimals — place value, reading and writing
  16. 1.1.16Converting - fractions ↔ decimals ↔ percentages
  17. 1.1.17Operations on decimals — all four operations
  18. 1.1.18Percentages — finding %, % of a quantity, % increase - decrease
  19. 1.1.19Ratio and proportion — equivalent ratios, dividing in a ratio
  20. 1.1.20Unitary method — direct and inverse proportion
  21. 1.1.21Profit, loss, discount, simple interest — basic applications
  22. 1.1.22Absolute value - modulus — definition, number line interpretation
1.2

Basic Geometry

17 topics
  1. 1.2.1Points, lines, line segments, rays — notation and differences
  2. 1.2.2Types of angles — acute, right, obtuse, straight, reflex, complete
  3. 1.2.3Angle measurement — protractor use, angle relationships (complementary, supplementary)
  4. 1.2.4Parallel and perpendicular lines — properties, transversal, alternate - co-interior angles
  5. 1.2.5Triangles — scalene, isosceles, equilateral; acute, right, obtuse
  6. 1.2.6Triangle properties — angle sum = 180°, exterior angle theorem
  7. 1.2.7Quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium, kite
  8. 1.2.8Properties of each quadrilateral — diagonals, angles, symmetry
  9. 1.2.9Circles — centre, radius, diameter, chord, arc, sector, segment
  10. 1.2.10Circumference and area of a circle
  11. 1.2.11Perimeter of polygons — regular and irregular
  12. 1.2.12Area — triangle, parallelogram, trapezium, composite shapes
  13. 1.2.133D shapes — cube, cuboid, cylinder, cone, sphere, prism, pyramid
  14. 1.2.14Surface area and volume of all above 3D shapes
  15. 1.2.15Nets of 3D shapes
  16. 1.2.16Symmetry — line symmetry, rotational symmetry, order of symmetry
  17. 1.2.17Transformations — translation, reflection, rotation, enlargement (basic)
1.3

Basic Data & Probability

7 topics
  1. 1.3.1Data collection — primary vs secondary, tally charts, frequency tables
  2. 1.3.2Bar charts, pictograms, pie charts — drawing and reading
  3. 1.3.3Line graphs and scatter plots — basic interpretation
  4. 1.3.4Mean, median, mode — calculation for raw and grouped data
  5. 1.3.5Range — definition and calculation
  6. 1.3.6Probability basics — sample space, events, P(E) = favourable - total
  7. 1.3.7Complementary events — P(A') = 1 − P(A)

Phase 2 — Pre-Algebra & Intermediate

Age 12–15 equivalent · 4–6 months @ 1.5 hrs/day7 chapters
2.1

Algebra — Introduction & Intermediate

24 topics
  1. 2.1.1Variables, constants, coefficients — algebraic expressions
  2. 2.1.2Like and unlike terms — simplification
  3. 2.1.3Addition and subtraction of algebraic expressions
  4. 2.1.4Multiplication of algebraic expressions — monomial × polynomial, polynomial × polynomial
  5. 2.1.5Algebraic identities — (a+b)², (a−b)², (a+b)(a−b), (a+b)³, (a−b)³, (a³+b³), (a³−b³)
  6. 2.1.6Factoring — common factor extraction, grouping, using identities
  7. 2.1.7Linear equations in one variable — solving, transposition method
  8. 2.1.8Word problems using linear equations
  9. 2.1.9Linear equations in two variables — graphical and algebraic solutions
  10. 2.1.10Simultaneous equations — substitution, elimination, cross-multiplication
  11. 2.1.11Inequalities — linear, solving, number line representation
  12. 2.1.12Compound inequalities — AND, OR
  13. 2.1.13Polynomials — degree, types (monomial, binomial, trinomial)
  14. 2.1.14Polynomial long division and synthetic division
  15. 2.1.15Remainder theorem and factor theorem — proof and applications
  16. 2.1.16Quadratic equations — factoring, completing the square
  17. 2.1.17Quadratic formula — derivation by completing the square
  18. 2.1.18Discriminant — nature of roots (real - equal - complex)
  19. 2.1.19Vieta's formulas — sum and product of roots
  20. 2.1.20Formation of quadratic with given roots
  21. 2.1.21Rational expressions — simplification, operations
  22. 2.1.22Radical (surd) expressions — simplification, rationalization
  23. 2.1.23Equations with radicals — squaring both sides, extraneous solutions
  24. 2.1.24Absolute value equations and inequalities
2.2

Functions

11 topics
  1. 2.2.1Concept of a function — input, output, mapping
  2. 2.2.2Domain, codomain, range
  3. 2.2.3Function notation — f(x), g(x)
  4. 2.2.4Vertical line test for functions
  5. 2.2.5Types — constant, linear, quadratic, polynomial, rational, radical, piecewise
  6. 2.2.6Graphs of functions — plotting, reading key features
  7. 2.2.7Transformations — vertical - horizontal shifts, reflections, stretches - compressions
  8. 2.2.8Composition of functions — f(g(x)), g(f(x))
  9. 2.2.9Inverse functions — finding f⁻¹(x), horizontal line test
  10. 2.2.10Even and odd functions — graphical and algebraic tests
  11. 2.2.11Increasing and decreasing functions — intuitive definition
2.3

Coordinate Geometry

14 topics
  1. 2.3.1Cartesian plane — axes, quadrants, ordered pairs
  2. 2.3.2Distance formula — derivation using Pythagoras
  3. 2.3.3Midpoint formula
  4. 2.3.4Section formula — internal and external division
  5. 2.3.5Slope (gradient) — definition, formula, interpretation
  6. 2.3.6Equations of a line — slope-intercept, point-slope, two-point, standard (ax+by+c=0)
  7. 2.3.7Intercepts — x-intercept, y-intercept
  8. 2.3.8Parallel lines — equal slopes
  9. 2.3.9Perpendicular lines — product of slopes = −1
  10. 2.3.10Distance from a point to a line
  11. 2.3.11Area of triangle using coordinate formula
  12. 2.3.12Collinearity of three points
  13. 2.3.13Circle equation — standard form (x−h)² + (y−k)² = r²
  14. 2.3.14General form of circle — converting, finding centre and radius
2.4

Trigonometry — Foundation

7 topics
  1. 2.4.1Pythagorean theorem — proof (by similar triangles, rearrangement), converse
  2. 2.4.2Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cot
  3. 2.4.3SOH-CAH-TOA mnemonic
  4. 2.4.4Trig ratios of standard angles — 0°, 30°, 45°, 60°, 90° (derive, don't memorize blindly)
  5. 2.4.5Complementary angle relationships — sin(90−θ) = cos θ etc.
  6. 2.4.6Reciprocal identities — cosec, sec, cot in terms of sin, cos, tan
  7. 2.4.7Applications — heights and distances problems
2.5

Number Theory (Intermediate)

11 topics
  1. 2.5.1Divisibility rules — 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (with proofs where possible)
  2. 2.5.2Integers — operations, number line, absolute value
  3. 2.5.3Rational numbers — definition, decimal expansion (terminating - repeating)
  4. 2.5.4Irrational numbers — √2, π, e — proof that √2 is irrational
  5. 2.5.5Real number system — ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
  6. 2.5.6Modular arithmetic — definition, addition, multiplication, congruence
  7. 2.5.7Euclidean algorithm — GCD computation
  8. 2.5.8Extended Euclidean algorithm
  9. 2.5.9Bézout's identity
  10. 2.5.10Chinese Remainder Theorem (intro)
  11. 2.5.11Fermat's little theorem (statement)
2.6

Matrices & Determinants — Introduction

12 topics
  1. 2.6.1Matrix definition — rows, columns, order, elements
  2. 2.6.2Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric
  3. 2.6.3Matrix operations — addition, subtraction (conditions)
  4. 2.6.4Scalar multiplication
  5. 2.6.5Matrix multiplication — conditions, process, non-commutativity
  6. 2.6.6Transpose — definition, properties
  7. 2.6.7Determinant of 2×2 matrix
  8. 2.6.8Determinant of 3×3 matrix — cofactor expansion
  9. 2.6.9Properties of determinants
  10. 2.6.10Inverse of 2×2 matrix
  11. 2.6.11Solving 2×2 systems using Cramer's rule
  12. 2.6.12Solving systems using matrix inversion
2.7

Statistics & Probability — Intermediate

13 topics
  1. 2.7.1Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped)
  2. 2.7.2Cumulative frequency — ogive, median from graph
  3. 2.7.3Measures of dispersion — variance, standard deviation
  4. 2.7.4Box-and-whisker plots — quartiles, IQR
  5. 2.7.5Probability — classical, empirical, axiomatic (Kolmogorov axioms)
  6. 2.7.6Mutually exclusive events — addition rule
  7. 2.7.7Independent events — multiplication rule
  8. 2.7.8Conditional probability — P(A - B) = P(A∩B) - P(A)
  9. 2.7.9Bayes' theorem — derivation and applications
  10. 2.7.10Permutations — nPr, arrangements with restrictions
  11. 2.7.11Combinations — nCr, Pascal's triangle
  12. 2.7.12Binomial theorem — expansion, general term
  13. 2.7.13Binomial distribution — PMF, mean, variance

Phase 3 — Advanced Pre-University

Age 15–17 equivalent · 6–8 months @ 2 hrs/day6 chapters
3.1

Advanced Trigonometry

22 topics
  1. 3.1.1Unit circle definition of trig functions — all 6 trig functions for any angle
  2. 3.1.2Radian measure — definition, conversion formula degrees ↔ radians
  3. 3.1.3Arc length and sector area using radians
  4. 3.1.4Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)
  5. 3.1.5Reference angles
  6. 3.1.6Graphs of sin x, cos x, tan x — key features, period, amplitude
  7. 3.1.7Graphs of cosec x, sec x, cot x
  8. 3.1.8Transformations of trig graphs — A·sin(Bx + C) + D (amplitude, period, phase, vertical shift)
  9. 3.1.9Pythagorean identities — sin² + cos² = 1, derivations of the other two
  10. 3.1.10Reciprocal identities, quotient identities
  11. 3.1.11Co-function identities
  12. 3.1.12Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs
  13. 3.1.13Double angle formulas — sin 2A, cos 2A (three forms), tan 2A
  14. 3.1.14Half angle formulas — derivations from double angle
  15. 3.1.15Product-to-sum formulas
  16. 3.1.16Sum-to-product formulas
  17. 3.1.17Inverse trig functions — arcsin, arccos, arctan — domain, range, graphs
  18. 3.1.18Solving trig equations — general solutions, solutions in given range
  19. 3.1.19Law of sines — proof and applications (ambiguous case)
  20. 3.1.20Law of cosines — proof and applications
  21. 3.1.21Area of triangle = ½ab·sin C
  22. 3.1.22Heron's formula (derivation using trig)
3.2

Exponentials & Logarithms

13 topics
  1. 3.2.1Exponential functions aˣ — graphs, properties, asymptote
  2. 3.2.2Laws of exponents — review with real exponents
  3. 3.2.3The number e — definition as limit of (1+1 - n)ⁿ, natural growth context
  4. 3.2.4Natural exponential function eˣ — graph, derivative preview
  5. 3.2.5Exponential growth and decay models — half-life, doubling time
  6. 3.2.6Logarithm — definition as inverse of exponential
  7. 3.2.7Common log (log₁₀) and natural log (ln x)
  8. 3.2.8Laws of logarithms — product, quotient, power rules — proofs
  9. 3.2.9Change of base formula — proof
  10. 3.2.10Solving exponential equations using logarithms
  11. 3.2.11Solving logarithmic equations
  12. 3.2.12Graphs of logarithmic functions
  13. 3.2.13Logarithmic scale — decibels, Richter, pH
3.3

Sequences & Series

14 topics
  1. 3.3.1Arithmetic progression (AP) — nth term, sum of n terms — derivations
  2. 3.3.2Geometric progression (GP) — nth term, sum of n terms — derivations
  3. 3.3.3Sum of infinite GP — when it converges, proof
  4. 3.3.4Harmonic progression — definition, HM
  5. 3.3.5AM-GM-HM inequalities — proofs
  6. 3.3.6Arithmetic-geometric progression — finding sum
  7. 3.3.7Sigma (Σ) notation — evaluating, telescoping sums
  8. 3.3.8Formulae — Σ1, Σn, Σn², Σn³ — proofs
  9. 3.3.9Mathematical induction — principle, steps, problems
  10. 3.3.10Strong induction
  11. 3.3.11Binomial theorem — statement, proof by induction
  12. 3.3.12Pascal's triangle — combinatorial interpretation
  13. 3.3.13General term of binomial expansion — finding specific terms
  14. 3.3.14Binomial theorem for rational indices (approximate values)
3.4

Conic Sections

12 topics
  1. 3.4.1Definition via focus, directrix, eccentricity e
  2. 3.4.2Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis
  3. 3.4.3Reflective property of parabola (application in telescopes, antennas)
  4. 3.4.4Ellipse — standard forms, semi-major - minor axes, foci, eccentricity, latus rectum
  5. 3.4.5Sum of focal radii = 2a property
  6. 3.4.6Kepler's connection — orbits are ellipses (motivation)
  7. 3.4.7Hyperbola — standard forms, asymptotes, foci, eccentricity
  8. 3.4.8Difference of focal radii = 2a property
  9. 3.4.9Rectangular hyperbola xy = c²
  10. 3.4.10Circle as degenerate conic (e = 0)
  11. 3.4.11General second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 — discriminant classification
  12. 3.4.12Parametric forms of all conics
3.5

Complex Numbers

13 topics
  1. 3.5.1Imaginary unit i = √(−1), i² = −1, powers of i cycle
  2. 3.5.2Complex number a+bi — real part, imaginary part
  3. 3.5.3Argand plane — geometric representation
  4. 3.5.4Modulus - z - and argument arg(z)
  5. 3.5.5Polar form — r(cos θ + i sin θ) = r·cis θ
  6. 3.5.6Euler's formula — e^(iθ) = cos θ + i sin θ (proof via Taylor series)
  7. 3.5.7Exponential form z = re^(iθ)
  8. 3.5.8Algebraic operations — add, subtract, multiply, divide (rectangular and polar)
  9. 3.5.9Complex conjugate — properties, applications in division
  10. 3.5.10De Moivre's theorem — statement, proof, applications
  11. 3.5.11nth roots of complex numbers — finding all n roots
  12. 3.5.12Roots of unity — cube roots, nth roots, geometric interpretation
  13. 3.5.13Applications — solving polynomial equations with complex roots
3.6

3D Geometry

12 topics
  1. 3.6.1Coordinate system in 3D — x, y, z axes, octants
  2. 3.6.2Distance formula in 3D
  3. 3.6.3Section formula in 3D
  4. 3.6.4Direction cosines and direction ratios
  5. 3.6.5Relation between direction cosines - l² + m² + n² = 1
  6. 3.6.6Equation of a line in 3D — vector, symmetric, parametric forms
  7. 3.6.7Angle between two lines
  8. 3.6.8Equation of a plane — normal form, intercept form, general form
  9. 3.6.9Angle between two planes
  10. 3.6.10Angle between line and plane
  11. 3.6.11Distance from a point to a plane
  12. 3.6.12Skew lines — shortest distance

Phase 4 — University Mathematics

Age 17–20 equivalent · 12–18 months @ 2 hrs/day10 chapters
4.1

Calculus I — Limits & Derivatives

33 topics
  1. 4.1.1Intuitive concept of a limit — table of values, graphical
  2. 4.1.2Limit laws — sum, product, quotient, constant multiple
  3. 4.1.3One-sided limits — left-hand, right-hand
  4. 4.1.4Infinite limits and limits at infinity — vertical - horizontal asymptotes
  5. 4.1.5Squeeze theorem (sandwich theorem)
  6. 4.1.6Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e
  7. 4.1.7Continuity — definition, types of discontinuity (removable, jump, infinite)
  8. 4.1.8Intermediate Value Theorem
  9. 4.1.9Epsilon-delta definition of a limit — formal proofs
  10. 4.1.10Derivative from first principles — difference quotient definition
  11. 4.1.11Interpretation — instantaneous rate of change, slope of tangent
  12. 4.1.12Power rule — proof for integer, rational exponents
  13. 4.1.13Sum, difference, constant multiple rules
  14. 4.1.14Product rule — proof
  15. 4.1.15Quotient rule — proof
  16. 4.1.16Chain rule — proof, composite function derivatives
  17. 4.1.17Derivatives of sin x, cos x — proofs from first principles
  18. 4.1.18Derivatives of all six trig functions
  19. 4.1.19Derivatives of eˣ and aˣ — proofs
  20. 4.1.20Derivatives of ln x and logₐ(x)
  21. 4.1.21Derivatives of inverse trig functions — all six
  22. 4.1.22Implicit differentiation — technique, applications
  23. 4.1.23Parametric differentiation — dy - dx, d²y - dx²
  24. 4.1.24Higher-order derivatives — notation, physical meaning
  25. 4.1.25Related rates — setting up and solving
  26. 4.1.26L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms
  27. 4.1.27Mean Value Theorem — proof, Rolle's theorem
  28. 4.1.28Applications — increasing - decreasing, local extrema (first derivative test)
  29. 4.1.29Second derivative test — concavity, inflection points
  30. 4.1.30Curve sketching — systematic approach
  31. 4.1.31Optimization — constrained, unconstrained, real-world problems
  32. 4.1.32Linear approximation and differentials
  33. 4.1.33Newton-Raphson method for root finding
4.2

Calculus II — Integration

18 topics
  1. 4.2.1Antiderivative — definition, family of solutions (+C)
  2. 4.2.2Basic integration rules — power, trig, exponential, log
  3. 4.2.3Riemann sums — left, right, midpoint; formal definition of definite integral
  4. 4.2.4Fundamental Theorem of Calculus — Part 1 and Part 2 — full proofs
  5. 4.2.5Net change theorem
  6. 4.2.6U-substitution — technique, change of limits for definite integrals
  7. 4.2.7Integration by parts — derivation from product rule, LIATE mnemonic
  8. 4.2.8Trigonometric integrals — sinᵐ·cosⁿ cases, tan and sec cases
  9. 4.2.9Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases
  10. 4.2.10Partial fractions — linear, repeated, irreducible quadratic factors
  11. 4.2.11Improper integrals — Type I (infinite limits), Type II (discontinuous integrand)
  12. 4.2.12Convergence tests for improper integrals — comparison
  13. 4.2.13Area between curves — horizontal and vertical slices
  14. 4.2.14Volume of revolution — disk method, washer method
  15. 4.2.15Volume of revolution — shell method
  16. 4.2.16Arc length formula — derivation
  17. 4.2.17Surface area of revolution
  18. 4.2.18Average value of a function
4.3

Calculus III — Sequences & Series

19 topics
  1. 4.3.1Sequences — convergence, divergence, boundedness, monotonicity
  2. 4.3.2Squeeze theorem for sequences
  3. 4.3.3Series — partial sums, convergence definition
  4. 4.3.4Geometric series — convergence condition, proof
  5. 4.3.5Telescoping series
  6. 4.3.6Divergence test (necessary but not sufficient)
  7. 4.3.7Integral test — proof, p-series
  8. 4.3.8Direct comparison test
  9. 4.3.9Limit comparison test
  10. 4.3.10Alternating series test — Leibniz test, proof
  11. 4.3.11Absolute vs conditional convergence
  12. 4.3.12Ratio test — proof, limitations
  13. 4.3.13Root test
  14. 4.3.14Power series — centre, radius of convergence, interval of convergence
  15. 4.3.15Term-by-term differentiation and integration of power series
  16. 4.3.16Taylor series — derivation from power series
  17. 4.3.17Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all
  18. 4.3.18Taylor's remainder theorem — error estimation
  19. 4.3.19Applications — approximation, evaluating limits
4.4

Multivariable Calculus

34 topics
  1. 4.4.1Functions of several variables — graphs, level curves, level surfaces
  2. 4.4.2Limits and continuity in 2D — path-dependence issue
  3. 4.4.3Partial derivatives — notation, calculation, geometric meaning
  4. 4.4.4Clairaut's theorem — mixed partials are equal (under conditions)
  5. 4.4.5Tangent planes and linear approximations to surfaces
  6. 4.4.6Differentiability in multiple variables
  7. 4.4.7Chain rule for multivariable functions — all cases
  8. 4.4.8Directional derivative — definition, formula
  9. 4.4.9Gradient vector ∇f — definition, properties
  10. 4.4.10Gradient as direction of steepest ascent
  11. 4.4.11Gradient perpendicular to level curves - surfaces
  12. 4.4.12Critical points — finding, classifying
  13. 4.4.13Second derivative test — Hessian determinant
  14. 4.4.14Absolute extrema on closed bounded regions
  15. 4.4.15Lagrange multipliers — one and two constraints
  16. 4.4.16Double integrals over rectangles — Fubini's theorem
  17. 4.4.17Double integrals over general regions — Type I and II
  18. 4.4.18Changing order of integration
  19. 4.4.19Double integrals in polar coordinates — Jacobian r
  20. 4.4.20Triple integrals in Cartesian, cylindrical, spherical coordinates
  21. 4.4.21Change of variables — general Jacobian
  22. 4.4.22Applications — mass, centre of mass, moments of inertia
  23. 4.4.23Vector fields — definition, visualization
  24. 4.4.24Divergence — definition, physical meaning (flux density)
  25. 4.4.25Curl — definition, physical meaning (rotation)
  26. 4.4.26Conservative vector fields — potential functions
  27. 4.4.27Line integrals — scalar and vector, work done
  28. 4.4.28Fundamental theorem for line integrals
  29. 4.4.29Green's theorem — proof sketch, both forms
  30. 4.4.30Parametric surfaces — tangent planes, surface area
  31. 4.4.31Surface integrals — scalar and vector (flux)
  32. 4.4.32Stokes' theorem — statement, curl-circulation connection
  33. 4.4.33Divergence theorem (Gauss's theorem) — statement, flux-divergence connection
  34. 4.4.34Unification — all three theorems as generalized Stokes
4.5

Linear Algebra (Full)

44 topics
  1. 4.5.1Vectors in ℝⁿ — operations, geometric interpretation
  2. 4.5.2Dot product — formula, cosine formula, Cauchy-Schwarz inequality proof
  3. 4.5.3Cross product — formula, geometric meaning (area), right-hand rule
  4. 4.5.4Projection of vectors
  5. 4.5.5Lines and planes in 3D — vector equations
  6. 4.5.6Matrices — review, operations, types
  7. 4.5.7Matrix multiplication — definition, associativity, non-commutativity
  8. 4.5.8Systems of linear equations — matrix form Ax = b
  9. 4.5.9Gaussian elimination — forward elimination, back substitution
  10. 4.5.10Row echelon form and reduced row echelon form
  11. 4.5.11Pivot positions, free variables
  12. 4.5.12Rank of a matrix — definition, row rank = column rank theorem
  13. 4.5.13Null space (kernel) and column space (image) — basis, dimension
  14. 4.5.14Rank-nullity theorem — proof
  15. 4.5.15Linear independence — formal definition, testing
  16. 4.5.16Span — definition
  17. 4.5.17Basis — definition, uniqueness of representation
  18. 4.5.18Dimension — basis cardinality
  19. 4.5.19Coordinate vectors — change of basis
  20. 4.5.20Change of basis matrix
  21. 4.5.21Determinants — cofactor expansion along any row - column
  22. 4.5.22Properties — row operations, multiplicativity
  23. 4.5.23Geometric interpretation — signed volume
  24. 4.5.24Cramer's rule
  25. 4.5.25Invertible matrix theorem — 12+ equivalent conditions
  26. 4.5.26LU decomposition — algorithm, applications
  27. 4.5.27Linear transformations — definition, kernel, image
  28. 4.5.28Matrix representation of linear transformations
  29. 4.5.29Eigenvalues and eigenvectors — characteristic polynomial
  30. 4.5.30Finding eigenspaces
  31. 4.5.31Diagonalization — conditions, procedure
  32. 4.5.32Complex eigenvalues — rotation-scaling interpretation
  33. 4.5.33Inner product spaces — dot product generalization
  34. 4.5.34Orthogonal sets and orthonormal basis
  35. 4.5.35Gram-Schmidt orthogonalization — algorithm
  36. 4.5.36QR decomposition
  37. 4.5.37Orthogonal matrices — properties, det = ±1
  38. 4.5.38Symmetric matrices — spectral theorem (real eigenvalues, orthogonal eigenvectors)
  39. 4.5.39Quadratic forms — positive definite, negative definite, indefinite
  40. 4.5.40Singular Value Decomposition (SVD) — full derivation
  41. 4.5.41Least squares — normal equations, QR approach
  42. 4.5.42Pseudoinverse
  43. 4.5.43Abstract vector spaces — axioms, examples beyond ℝⁿ
  44. 4.5.44Subspaces — four fundamental subspaces of a matrix
4.6

Ordinary Differential Equations

33 topics
  1. 4.6.1Classification — order, degree, linear vs nonlinear, autonomous vs non-autonomous
  2. 4.6.2Direction fields and Euler's method — visual - numerical intuition first
  3. 4.6.3Separable ODEs — technique, implicit solutions
  4. 4.6.4First-order linear ODEs — integrating factor method (derivation)
  5. 4.6.5Bernoulli equations — substitution
  6. 4.6.6Exact equations — exactness condition, finding potential function
  7. 4.6.7Integrating factors for non-exact equations
  8. 4.6.8Existence and uniqueness theorem — Picard-Lindelöf (statement)
  9. 4.6.9Second-order linear ODEs — superposition principle, general theory
  10. 4.6.10Homogeneous with constant coefficients — characteristic equation
  11. 4.6.11Case 1 - two distinct real roots
  12. 4.6.12Case 2 - repeated real root — reduction of order
  13. 4.6.13Case 3 - complex conjugate roots — Euler's formula connection
  14. 4.6.14Non-homogeneous — method of undetermined coefficients (annihilator method)
  15. 4.6.15Non-homogeneous — variation of parameters
  16. 4.6.16Cauchy-Euler (Equidimensional) equation
  17. 4.6.17Power series solutions — ordinary points
  18. 4.6.18Frobenius method — regular singular points
  19. 4.6.19Bessel's equation and Bessel functions (intro, physical relevance)
  20. 4.6.20Legendre's equation and Legendre polynomials (intro)
  21. 4.6.21Systems of first-order linear ODEs — matrix method
  22. 4.6.22Phase plane analysis — trajectories, critical points
  23. 4.6.23Stability of equilibria — stable, unstable, saddle, spiral, centre
  24. 4.6.24Linearization of nonlinear systems
  25. 4.6.25Laplace transform — definition, region of convergence
  26. 4.6.26Transforms of standard functions — proofs
  27. 4.6.27Properties — linearity, first - second shift theorems, scaling
  28. 4.6.28Laplace of derivatives — key property for solving ODEs
  29. 4.6.29Inverse Laplace transform — partial fractions, tables
  30. 4.6.30Solving ODEs with Laplace (including discontinuous forcing)
  31. 4.6.31Heaviside step function and Dirac delta function
  32. 4.6.32Convolution theorem — proof, applications
  33. 4.6.33Impulse response and transfer function (GNC connection)
4.7

Partial Differential Equations

21 topics
  1. 4.7.1Classification — elliptic, parabolic, hyperbolic (discriminant test)
  2. 4.7.2Initial value problems (IVP) vs boundary value problems (BVP)
  3. 4.7.3Fourier series — motivation from periodic functions
  4. 4.7.4Dirichlet conditions for convergence
  5. 4.7.5Full Fourier series — coefficients derivation
  6. 4.7.6Half-range sine and cosine series
  7. 4.7.7Parseval's theorem
  8. 4.7.8Heat equation (parabolic) 1D — derivation from Fourier's law
  9. 4.7.9Solving heat equation — separation of variables
  10. 4.7.10Wave equation (hyperbolic) 1D — derivation
  11. 4.7.11Solving wave equation — D'Alembert's solution
  12. 4.7.12Solving wave equation — separation of variables
  13. 4.7.13Laplace's equation (elliptic) — physical meaning (steady-state)
  14. 4.7.14Laplace on rectangle — separation of variables
  15. 4.7.15Laplace on disk — polar coordinates, Bessel functions connection
  16. 4.7.16Neumann and Dirichlet boundary conditions
  17. 4.7.17Sturm-Liouville theory — eigenvalue problems, orthogonality of eigenfunctions
  18. 4.7.18Fourier transform — definition, properties
  19. 4.7.19Solving PDEs with Fourier transforms (heat equation on infinite domain)
  20. 4.7.20Convolution with Fourier transform
  21. 4.7.21Intro to finite difference methods for PDEs
4.8

Numerical Methods

29 topics
  1. 4.8.1Sources of error — truncation error, round-off error
  2. 4.8.2IEEE 754 floating-point standard — significant bits, special values
  3. 4.8.3Machine epsilon — what it means in practice
  4. 4.8.4Condition number — absolute, relative; ill-conditioned problems
  5. 4.8.5Numerical stability vs instability — catastrophic cancellation
  6. 4.8.6Root finding — bisection method (convergence analysis)
  7. 4.8.7Fixed-point iteration — convergence conditions
  8. 4.8.8Newton-Raphson method — derivation, quadratic convergence
  9. 4.8.9Secant method
  10. 4.8.10Polynomial interpolation — Lagrange form, Newton's divided differences
  11. 4.8.11Error in polynomial interpolation
  12. 4.8.12Cubic spline interpolation — natural, clamped
  13. 4.8.13Numerical differentiation — forward, backward, central differences
  14. 4.8.14Error analysis of finite differences
  15. 4.8.15Numerical integration — trapezoidal rule (composite), error
  16. 4.8.16Simpson's 1 - 3 rule, 3 - 8 rule (composite) — derivation
  17. 4.8.17Gaussian quadrature — Gauss-Legendre
  18. 4.8.18Solving linear systems — Gaussian elimination with partial pivoting
  19. 4.8.19LU decomposition (numerical)
  20. 4.8.20Iterative methods — Jacobi, Gauss-Seidel, convergence
  21. 4.8.21Eigenvalue computation — power method, inverse iteration
  22. 4.8.22ODE solvers — Euler's method (derivation, global error)
  23. 4.8.23Modified Euler (Heun's method)
  24. 4.8.24Runge-Kutta 4th order (RK4) — derivation
  25. 4.8.25Adaptive step-size — RK45, error control
  26. 4.8.26Stiff equations — implicit methods, backward Euler
  27. 4.8.27Systems of ODEs — RK4 for systems
  28. 4.8.28Boundary value problems — shooting method, finite difference
  29. 4.8.29Solving nonlinear systems — Newton's method in n dimensions
4.9

Probability Theory & Statistics

25 topics
  1. 4.9.1Probability space — sample space Ω, sigma-algebra F, measure P — Kolmogorov axioms
  2. 4.9.2Inclusion-exclusion principle
  3. 4.9.3Discrete random variables — PMF, CDF
  4. 4.9.4Expected value, variance, standard deviation — properties
  5. 4.9.5Moment generating function (MGF) — definition, use
  6. 4.9.6Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial
  7. 4.9.7Continuous random variables — PDF, CDF, percentiles
  8. 4.9.8Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta
  9. 4.9.9Chi-squared, t, F distributions — definition, degrees of freedom
  10. 4.9.10Joint distributions — joint PMF - PDF, marginal, conditional
  11. 4.9.11Independence of random variables — formal definition
  12. 4.9.12Covariance and correlation
  13. 4.9.13Conditional expectation
  14. 4.9.14Transformations of random variables — change-of-variable technique
  15. 4.9.15Central Limit Theorem — statement, proof sketch, significance
  16. 4.9.16Law of Large Numbers — weak and strong
  17. 4.9.17Statistical estimation — MLE, method of moments
  18. 4.9.18Properties of estimators — unbiasedness, consistency, efficiency
  19. 4.9.19Confidence intervals — derivation for mean, proportion
  20. 4.9.20Hypothesis testing — null - alternative, test statistic, p-value, errors (Type I & II)
  21. 4.9.21z-test, t-test, chi-squared goodness of fit, F-test
  22. 4.9.22Linear regression — least squares, inference on coefficients
  23. 4.9.23Multiple regression
  24. 4.9.24Bayesian statistics — prior, likelihood, posterior (intro)
  25. 4.9.25Monte Carlo simulation — law of large numbers basis
4.10

Advanced Topics (Elite Level)

27 topics
  1. 4.10.1Complex analysis — analytic functions, Cauchy-Riemann equations
  2. 4.10.2Complex integration — contour integrals
  3. 4.10.3Cauchy's integral theorem and formula
  4. 4.10.4Laurent series — principal part, annulus of convergence
  5. 4.10.5Residues and poles
  6. 4.10.6Residue theorem — computing real integrals
  7. 4.10.7Tensor analysis — scalars, vectors, rank-2 tensors
  8. 4.10.8Covariant and contravariant components
  9. 4.10.9Einstein summation convention
  10. 4.10.10Metric tensor — raising - lowering indices
  11. 4.10.11Christoffel symbols — intro
  12. 4.10.12Calculus of variations — functionals, functional derivative
  13. 4.10.13Euler-Lagrange equation — derivation
  14. 4.10.14Brachistochrone problem
  15. 4.10.15Hamilton's principle — least action
  16. 4.10.16Isoperimetric problems — constraints (Lagrange multipliers in variational sense)
  17. 4.10.17Convex optimization — convex sets, convex functions
  18. 4.10.18First-order optimality conditions — gradient = 0
  19. 4.10.19KKT conditions for constrained optimization
  20. 4.10.20Gradient descent and variants — convergence analysis
  21. 4.10.21Linear programming — simplex method (intro)
  22. 4.10.22Real analysis — rigorous epsilon-delta, metric spaces
  23. 4.10.23Uniform continuity — difference from pointwise
  24. 4.10.24Uniform convergence of function sequences
  25. 4.10.25Measure theory — Lebesgue measure (intro)
  26. 4.10.26Fourier analysis — DFT, FFT algorithm (Cooley-Tukey)
  27. 4.10.27Stochastic processes — Markov chains, steady-state, random walks