Maths Deep Dives

Every topic rebuilt 3Blue1Brown-style: foundations from zero, the derivation in pictures, worked examples for every scenario, graded exercises, and concept-trap question banks.

484 topics · 2401 pages — more land as generation runs

1.1

Arithmetic & Number Systems

21 deep-dive topics

1.1.2Place value system — units, tens, hundreds, thousands, lakhs, crores

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1.1.3Addition and subtraction — carrying, borrowing, word problems

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1.1.4Multiplication — tables (1–20), long multiplication, area model

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1.1.5Division — long division, remainder, dividend - divisor - quotient vocabulary

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1.1.6Order of operations — BODMAS - PEMDAS with nested brackets

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1.1.7Factors and multiples — all factors of a number, factor pairs

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1.1.8Prime numbers — Sieve of Eratosthenes, primality testing

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1.1.9Prime factorization — factor trees, ladder method

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1.1.10HCF (GCD) — prime factorization method, Euclidean algorithm

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1.1.11LCM — prime factorization method, relationship HCF × LCM = product

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1.1.12Fractions — proper, improper, mixed numbers

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1.1.13Equivalent fractions, simplifying fractions

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1.1.14Addition, subtraction, multiplication, division of fractions

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1.1.15Decimals — place value, reading and writing

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1.1.16Converting - fractions ↔ decimals ↔ percentages

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1.1.17Operations on decimals — all four operations

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1.1.18Percentages — finding %, % of a quantity, % increase - decrease

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1.1.19Ratio and proportion — equivalent ratios, dividing in a ratio

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1.1.20Unitary method — direct and inverse proportion

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1.1.21Profit, loss, discount, simple interest — basic applications

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1.1.22Absolute value - modulus — definition, number line interpretation

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1.2

Basic Geometry

17 deep-dive topics

1.2.1Points, lines, line segments, rays — notation and differences

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1.2.2Types of angles — acute, right, obtuse, straight, reflex, complete

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1.2.3Angle measurement — protractor use, angle relationships (complementary, supplementary)

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1.2.4Parallel and perpendicular lines — properties, transversal, alternate - co-interior angles

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1.2.5Triangles — scalene, isosceles, equilateral; acute, right, obtuse

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1.2.6Triangle properties — angle sum = 180°, exterior angle theorem

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1.2.7Quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium, kite

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1.2.8Properties of each quadrilateral — diagonals, angles, symmetry

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1.2.9Circles — centre, radius, diameter, chord, arc, sector, segment

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1.2.10Circumference and area of a circle

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1.2.11Perimeter of polygons — regular and irregular

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1.2.12Area — triangle, parallelogram, trapezium, composite shapes

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1.2.133D shapes — cube, cuboid, cylinder, cone, sphere, prism, pyramid

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1.2.14Surface area and volume of all above 3D shapes

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1.2.15Nets of 3D shapes

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1.2.16Symmetry — line symmetry, rotational symmetry, order of symmetry

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1.2.17Transformations — translation, reflection, rotation, enlargement (basic)

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1.3

Basic Data & Probability

7 deep-dive topics

1.3.1Data collection — primary vs secondary, tally charts, frequency tables

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1.3.2Bar charts, pictograms, pie charts — drawing and reading

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1.3.3Line graphs and scatter plots — basic interpretation

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1.3.4Mean, median, mode — calculation for raw and grouped data

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1.3.5Range — definition and calculation

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1.3.6Probability basics — sample space, events, P(E) = favourable - total

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1.3.7Complementary events — P(A') = 1 − P(A)

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2.1

Algebra — Introduction & Intermediate

24 deep-dive topics

2.1.1Variables, constants, coefficients — algebraic expressions

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2.1.2Like and unlike terms — simplification

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2.1.3Addition and subtraction of algebraic expressions

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2.1.4Multiplication of algebraic expressions — monomial × polynomial, polynomial × polynomial

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2.1.5Algebraic identities — (a+b)², (a−b)², (a+b)(a−b), (a+b)³, (a−b)³, (a³+b³), (a³−b³)

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2.1.6Factoring — common factor extraction, grouping, using identities

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2.1.7Linear equations in one variable — solving, transposition method

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2.1.8Word problems using linear equations

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2.1.9Linear equations in two variables — graphical and algebraic solutions

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2.1.10Simultaneous equations — substitution, elimination, cross-multiplication

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2.1.11Inequalities — linear, solving, number line representation

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2.1.12Compound inequalities — AND, OR

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2.1.13Polynomials — degree, types (monomial, binomial, trinomial)

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2.1.14Polynomial long division and synthetic division

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2.1.15Remainder theorem and factor theorem — proof and applications

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2.1.16Quadratic equations — factoring, completing the square

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2.1.17Quadratic formula — derivation by completing the square

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2.1.18Discriminant — nature of roots (real - equal - complex)

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2.1.19Vieta's formulas — sum and product of roots

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2.1.20Formation of quadratic with given roots

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2.1.21Rational expressions — simplification, operations

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2.1.22Radical (surd) expressions — simplification, rationalization

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2.1.23Equations with radicals — squaring both sides, extraneous solutions

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2.1.24Absolute value equations and inequalities

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2.2

Functions

11 deep-dive topics

2.2.1Concept of a function — input, output, mapping

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2.2.2Domain, codomain, range

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2.2.3Function notation — f(x), g(x)

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2.2.4Vertical line test for functions

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2.2.5Types — constant, linear, quadratic, polynomial, rational, radical, piecewise

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2.2.6Graphs of functions — plotting, reading key features

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2.2.7Transformations — vertical - horizontal shifts, reflections, stretches - compressions

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2.2.8Composition of functions — f(g(x)), g(f(x))

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2.2.9Inverse functions — finding f⁻¹(x), horizontal line test

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2.2.10Even and odd functions — graphical and algebraic tests

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2.2.11Increasing and decreasing functions — intuitive definition

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2.3

Coordinate Geometry

14 deep-dive topics

2.3.1Cartesian plane — axes, quadrants, ordered pairs

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2.3.2Distance formula — derivation using Pythagoras

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2.3.3Midpoint formula

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2.3.4Section formula — internal and external division

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2.3.5Slope (gradient) — definition, formula, interpretation

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2.3.6Equations of a line — slope-intercept, point-slope, two-point, standard (ax+by+c=0)

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2.3.7Intercepts — x-intercept, y-intercept

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2.3.8Parallel lines — equal slopes

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2.3.9Perpendicular lines — product of slopes = −1

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2.3.10Distance from a point to a line

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2.3.11Area of triangle using coordinate formula

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2.3.12Collinearity of three points

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2.3.13Circle equation — standard form (x−h)² + (y−k)² = r²

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2.3.14General form of circle — converting, finding centre and radius

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2.4

Trigonometry — Foundation

7 deep-dive topics

2.4.1Pythagorean theorem — proof (by similar triangles, rearrangement), converse

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2.4.2Trigonometric ratios in right triangle — sin, cos, tan, cosec, sec, cot

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2.4.3SOH-CAH-TOA mnemonic

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2.4.4Trig ratios of standard angles — 0°, 30°, 45°, 60°, 90° (derive, don't memorize blindly)

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2.4.5Complementary angle relationships — sin(90−θ) = cos θ etc.

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2.4.6Reciprocal identities — cosec, sec, cot in terms of sin, cos, tan

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2.4.7Applications — heights and distances problems

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2.5

Number Theory (Intermediate)

11 deep-dive topics

2.5.1Divisibility rules — 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 (with proofs where possible)

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2.5.2Integers — operations, number line, absolute value

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2.5.3Rational numbers — definition, decimal expansion (terminating - repeating)

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2.5.4Irrational numbers — √2, π, e — proof that √2 is irrational

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2.5.5Real number system — ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ

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2.5.6Modular arithmetic — definition, addition, multiplication, congruence

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2.5.7Euclidean algorithm — GCD computation

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2.5.8Extended Euclidean algorithm

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2.5.9Bézout's identity

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2.5.10Chinese Remainder Theorem (intro)

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2.5.11Fermat's little theorem (statement)

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2.6

Matrices & Determinants — Introduction

12 deep-dive topics

2.6.1Matrix definition — rows, columns, order, elements

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2.6.2Types of matrices — row, column, square, diagonal, identity, zero, symmetric, skew-symmetric

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2.6.3Matrix operations — addition, subtraction (conditions)

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2.6.4Scalar multiplication

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2.6.5Matrix multiplication — conditions, process, non-commutativity

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2.6.6Transpose — definition, properties

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2.6.7Determinant of 2×2 matrix

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2.6.8Determinant of 3×3 matrix — cofactor expansion

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2.6.9Properties of determinants

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2.6.10Inverse of 2×2 matrix

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2.6.11Solving 2×2 systems using Cramer's rule

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2.6.12Solving systems using matrix inversion

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2.7

Statistics & Probability — Intermediate

13 deep-dive topics

2.7.1Measures of central tendency — mean (grouped - ungrouped), median (grouped), mode (grouped)

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2.7.2Cumulative frequency — ogive, median from graph

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2.7.3Measures of dispersion — variance, standard deviation

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2.7.4Box-and-whisker plots — quartiles, IQR

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2.7.5Probability — classical, empirical, axiomatic (Kolmogorov axioms)

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2.7.6Mutually exclusive events — addition rule

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2.7.7Independent events — multiplication rule

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2.7.8Conditional probability — P(A - B) = P(A∩B) - P(A)

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2.7.9Bayes' theorem — derivation and applications

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2.7.10Permutations — nPr, arrangements with restrictions

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2.7.11Combinations — nCr, Pascal's triangle

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2.7.12Binomial theorem — expansion, general term

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2.7.13Binomial distribution — PMF, mean, variance

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3.1

Advanced Trigonometry

22 deep-dive topics

3.1.1Unit circle definition of trig functions — all 6 trig functions for any angle

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3.1.2Radian measure — definition, conversion formula degrees ↔ radians

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3.1.3Arc length and sector area using radians

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3.1.4Trig functions for angles beyond 90° — ASTC rule (All, Sin, Tan, Cos)

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3.1.5Reference angles

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3.1.6Graphs of sin x, cos x, tan x — key features, period, amplitude

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3.1.7Graphs of cosec x, sec x, cot x

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3.1.8Transformations of trig graphs — A·sin(Bx + C) + D (amplitude, period, phase, vertical shift)

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3.1.9Pythagorean identities — sin² + cos² = 1, derivations of the other two

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3.1.10Reciprocal identities, quotient identities

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3.1.11Co-function identities

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3.1.12Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs

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3.1.13Double angle formulas — sin 2A, cos 2A (three forms), tan 2A

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3.1.14Half angle formulas — derivations from double angle

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3.1.15Product-to-sum formulas

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3.1.16Sum-to-product formulas

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3.1.17Inverse trig functions — arcsin, arccos, arctan — domain, range, graphs

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3.1.18Solving trig equations — general solutions, solutions in given range

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3.1.19Law of sines — proof and applications (ambiguous case)

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3.1.20Law of cosines — proof and applications

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3.1.21Area of triangle = ½ab·sin C

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3.1.22Heron's formula (derivation using trig)

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3.2

Exponentials & Logarithms

13 deep-dive topics

3.2.1Exponential functions aˣ — graphs, properties, asymptote

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3.2.2Laws of exponents — review with real exponents

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3.2.3The number e — definition as limit of (1+1 - n)ⁿ, natural growth context

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3.2.4Natural exponential function eˣ — graph, derivative preview

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3.2.5Exponential growth and decay models — half-life, doubling time

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3.2.6Logarithm — definition as inverse of exponential

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3.2.7Common log (log₁₀) and natural log (ln x)

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3.2.8Laws of logarithms — product, quotient, power rules — proofs

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3.2.9Change of base formula — proof

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3.2.10Solving exponential equations using logarithms

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3.2.11Solving logarithmic equations

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3.2.12Graphs of logarithmic functions

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3.2.13Logarithmic scale — decibels, Richter, pH

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3.3

Sequences & Series

14 deep-dive topics

3.3.1Arithmetic progression (AP) — nth term, sum of n terms — derivations

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3.3.2Geometric progression (GP) — nth term, sum of n terms — derivations

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3.3.3Sum of infinite GP — when it converges, proof

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3.3.4Harmonic progression — definition, HM

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3.3.5AM-GM-HM inequalities — proofs

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3.3.6Arithmetic-geometric progression — finding sum

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3.3.7Sigma (Σ) notation — evaluating, telescoping sums

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3.3.8Formulae — Σ1, Σn, Σn², Σn³ — proofs

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3.3.9Mathematical induction — principle, steps, problems

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3.3.10Strong induction

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3.3.11Binomial theorem — statement, proof by induction

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3.3.12Pascal's triangle — combinatorial interpretation

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3.3.13General term of binomial expansion — finding specific terms

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3.3.14Binomial theorem for rational indices (approximate values)

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3.4

Conic Sections

12 deep-dive topics

3.4.1Definition via focus, directrix, eccentricity e

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3.4.2Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

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3.4.3Reflective property of parabola (application in telescopes, antennas)

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3.4.4Ellipse — standard forms, semi-major - minor axes, foci, eccentricity, latus rectum

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3.4.5Sum of focal radii = 2a property

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3.4.6Kepler's connection — orbits are ellipses (motivation)

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3.4.7Hyperbola — standard forms, asymptotes, foci, eccentricity

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3.4.8Difference of focal radii = 2a property

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3.4.9Rectangular hyperbola xy = c²

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3.4.10Circle as degenerate conic (e = 0)

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3.4.11General second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0 — discriminant classification

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3.4.12Parametric forms of all conics

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3.5

Complex Numbers

13 deep-dive topics

3.5.1Imaginary unit i = √(−1), i² = −1, powers of i cycle

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3.5.2Complex number a+bi — real part, imaginary part

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3.5.3Argand plane — geometric representation

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3.5.4Modulus - z - and argument arg(z)

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3.5.5Polar form — r(cos θ + i sin θ) = r·cis θ

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3.5.6Euler's formula — e^(iθ) = cos θ + i sin θ (proof via Taylor series)

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3.5.7Exponential form z = re^(iθ)

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3.5.8Algebraic operations — add, subtract, multiply, divide (rectangular and polar)

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3.5.9Complex conjugate — properties, applications in division

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3.5.10De Moivre's theorem — statement, proof, applications

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3.5.11nth roots of complex numbers — finding all n roots

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3.5.12Roots of unity — cube roots, nth roots, geometric interpretation

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3.5.13Applications — solving polynomial equations with complex roots

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3.6

3D Geometry

12 deep-dive topics

3.6.1Coordinate system in 3D — x, y, z axes, octants

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3.6.2Distance formula in 3D

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3.6.3Section formula in 3D

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3.6.4Direction cosines and direction ratios

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3.6.5Relation between direction cosines - l² + m² + n² = 1

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3.6.6Equation of a line in 3D — vector, symmetric, parametric forms

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3.6.7Angle between two lines

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3.6.8Equation of a plane — normal form, intercept form, general form

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3.6.9Angle between two planes

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3.6.10Angle between line and plane

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3.6.11Distance from a point to a plane

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3.6.12Skew lines — shortest distance

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4.1

Calculus I — Limits & Derivatives

33 deep-dive topics

4.1.1Intuitive concept of a limit — table of values, graphical

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4.1.2Limit laws — sum, product, quotient, constant multiple

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4.1.3One-sided limits — left-hand, right-hand

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4.1.4Infinite limits and limits at infinity — vertical - horizontal asymptotes

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4.1.5Squeeze theorem (sandwich theorem)

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4.1.6Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e

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4.1.7Continuity — definition, types of discontinuity (removable, jump, infinite)

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4.1.8Intermediate Value Theorem

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4.1.9Epsilon-delta definition of a limit — formal proofs

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4.1.10Derivative from first principles — difference quotient definition

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4.1.11Interpretation — instantaneous rate of change, slope of tangent

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4.1.12Power rule — proof for integer, rational exponents

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4.1.13Sum, difference, constant multiple rules

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4.1.14Product rule — proof

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4.1.15Quotient rule — proof

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4.1.16Chain rule — proof, composite function derivatives

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4.1.17Derivatives of sin x, cos x — proofs from first principles

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4.1.18Derivatives of all six trig functions

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4.1.19Derivatives of eˣ and aˣ — proofs

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4.1.20Derivatives of ln x and logₐ(x)

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4.1.21Derivatives of inverse trig functions — all six

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4.1.22Implicit differentiation — technique, applications

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4.1.23Parametric differentiation — dy - dx, d²y - dx²

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4.1.24Higher-order derivatives — notation, physical meaning

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4.1.25Related rates — setting up and solving

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4.1.26L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms

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4.1.27Mean Value Theorem — proof, Rolle's theorem

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4.1.28Applications — increasing - decreasing, local extrema (first derivative test)

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4.1.29Second derivative test — concavity, inflection points

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4.1.30Curve sketching — systematic approach

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4.1.31Optimization — constrained, unconstrained, real-world problems

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4.1.32Linear approximation and differentials

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4.1.33Newton-Raphson method for root finding

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4.2

Calculus II — Integration

18 deep-dive topics

4.2.1Antiderivative — definition, family of solutions (+C)

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4.2.2Basic integration rules — power, trig, exponential, log

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4.2.3Riemann sums — left, right, midpoint; formal definition of definite integral

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4.2.4Fundamental Theorem of Calculus — Part 1 and Part 2 — full proofs

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4.2.5Net change theorem

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4.2.6U-substitution — technique, change of limits for definite integrals

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4.2.7Integration by parts — derivation from product rule, LIATE mnemonic

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4.2.8Trigonometric integrals — sinᵐ·cosⁿ cases, tan and sec cases

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4.2.9Trigonometric substitution — x = a sin θ, a tan θ, a sec θ cases

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4.2.10Partial fractions — linear, repeated, irreducible quadratic factors

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4.2.11Improper integrals — Type I (infinite limits), Type II (discontinuous integrand)

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4.2.12Convergence tests for improper integrals — comparison

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4.2.13Area between curves — horizontal and vertical slices

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4.2.14Volume of revolution — disk method, washer method

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4.2.15Volume of revolution — shell method

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4.2.16Arc length formula — derivation

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4.2.17Surface area of revolution

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4.2.18Average value of a function

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4.3

Calculus III — Sequences & Series

19 deep-dive topics

4.3.1Sequences — convergence, divergence, boundedness, monotonicity

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4.3.2Squeeze theorem for sequences

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4.3.3Series — partial sums, convergence definition

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4.3.4Geometric series — convergence condition, proof

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4.3.5Telescoping series

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4.3.6Divergence test (necessary but not sufficient)

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4.3.7Integral test — proof, p-series

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4.3.8Direct comparison test

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4.3.9Limit comparison test

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4.3.10Alternating series test — Leibniz test, proof

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4.3.11Absolute vs conditional convergence

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4.3.12Ratio test — proof, limitations

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4.3.13Root test

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4.3.14Power series — centre, radius of convergence, interval of convergence

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4.3.15Term-by-term differentiation and integration of power series

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4.3.16Taylor series — derivation from power series

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4.3.17Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all

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4.3.18Taylor's remainder theorem — error estimation

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4.3.19Applications — approximation, evaluating limits

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4.4

Multivariable Calculus

34 deep-dive topics

4.4.1Functions of several variables — graphs, level curves, level surfaces

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4.4.2Limits and continuity in 2D — path-dependence issue

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4.4.3Partial derivatives — notation, calculation, geometric meaning

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4.4.4Clairaut's theorem — mixed partials are equal (under conditions)

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4.4.5Tangent planes and linear approximations to surfaces

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4.4.6Differentiability in multiple variables

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4.4.7Chain rule for multivariable functions — all cases

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4.4.8Directional derivative — definition, formula

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4.4.9Gradient vector ∇f — definition, properties

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4.4.10Gradient as direction of steepest ascent

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4.4.11Gradient perpendicular to level curves - surfaces

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4.4.12Critical points — finding, classifying

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4.4.13Second derivative test — Hessian determinant

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4.4.14Absolute extrema on closed bounded regions

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4.4.15Lagrange multipliers — one and two constraints

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4.4.16Double integrals over rectangles — Fubini's theorem

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4.4.17Double integrals over general regions — Type I and II

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4.4.18Changing order of integration

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4.4.19Double integrals in polar coordinates — Jacobian r

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4.4.20Triple integrals in Cartesian, cylindrical, spherical coordinates

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4.4.21Change of variables — general Jacobian

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4.4.22Applications — mass, centre of mass, moments of inertia

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4.4.23Vector fields — definition, visualization

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4.4.24Divergence — definition, physical meaning (flux density)

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4.4.25Curl — definition, physical meaning (rotation)

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4.4.26Conservative vector fields — potential functions

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4.4.27Line integrals — scalar and vector, work done

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4.4.28Fundamental theorem for line integrals

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4.4.29Green's theorem — proof sketch, both forms

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4.4.30Parametric surfaces — tangent planes, surface area

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4.4.31Surface integrals — scalar and vector (flux)

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4.4.32Stokes' theorem — statement, curl-circulation connection

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4.4.33Divergence theorem (Gauss's theorem) — statement, flux-divergence connection

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4.4.34Unification — all three theorems as generalized Stokes

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4.5

Linear Algebra (Full)

44 deep-dive topics

4.5.1Vectors in ℝⁿ — operations, geometric interpretation

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4.5.2Dot product — formula, cosine formula, Cauchy-Schwarz inequality proof

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4.5.3Cross product — formula, geometric meaning (area), right-hand rule

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4.5.4Projection of vectors

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4.5.5Lines and planes in 3D — vector equations

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4.5.6Matrices — review, operations, types

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4.5.7Matrix multiplication — definition, associativity, non-commutativity

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4.5.8Systems of linear equations — matrix form Ax = b

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4.5.9Gaussian elimination — forward elimination, back substitution

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4.5.10Row echelon form and reduced row echelon form

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4.5.11Pivot positions, free variables

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4.5.12Rank of a matrix — definition, row rank = column rank theorem

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4.5.13Null space (kernel) and column space (image) — basis, dimension

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4.5.14Rank-nullity theorem — proof

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4.5.15Linear independence — formal definition, testing

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4.5.16Span — definition

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4.5.17Basis — definition, uniqueness of representation

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4.5.18Dimension — basis cardinality

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4.5.19Coordinate vectors — change of basis

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4.5.20Change of basis matrix

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4.5.21Determinants — cofactor expansion along any row - column

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4.5.22Properties — row operations, multiplicativity

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4.5.23Geometric interpretation — signed volume

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4.5.24Cramer's rule

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4.5.25Invertible matrix theorem — 12+ equivalent conditions

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4.5.26LU decomposition — algorithm, applications

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4.5.27Linear transformations — definition, kernel, image

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4.5.28Matrix representation of linear transformations

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4.5.29Eigenvalues and eigenvectors — characteristic polynomial

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4.5.30Finding eigenspaces

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4.5.31Diagonalization — conditions, procedure

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4.5.32Complex eigenvalues — rotation-scaling interpretation

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4.5.33Inner product spaces — dot product generalization

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4.5.34Orthogonal sets and orthonormal basis

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4.5.35Gram-Schmidt orthogonalization — algorithm

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4.5.36QR decomposition

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4.5.37Orthogonal matrices — properties, det = ±1

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4.5.38Symmetric matrices — spectral theorem (real eigenvalues, orthogonal eigenvectors)

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4.5.39Quadratic forms — positive definite, negative definite, indefinite

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4.5.40Singular Value Decomposition (SVD) — full derivation

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4.5.41Least squares — normal equations, QR approach

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4.5.42Pseudoinverse

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4.5.43Abstract vector spaces — axioms, examples beyond ℝⁿ

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4.5.44Subspaces — four fundamental subspaces of a matrix

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4.6

Ordinary Differential Equations

33 deep-dive topics

4.6.1Classification — order, degree, linear vs nonlinear, autonomous vs non-autonomous

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4.6.2Direction fields and Euler's method — visual - numerical intuition first

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4.6.3Separable ODEs — technique, implicit solutions

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4.6.4First-order linear ODEs — integrating factor method (derivation)

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4.6.5Bernoulli equations — substitution

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4.6.6Exact equations — exactness condition, finding potential function

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4.6.7Integrating factors for non-exact equations

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4.6.8Existence and uniqueness theorem — Picard-Lindelöf (statement)

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4.6.9Second-order linear ODEs — superposition principle, general theory

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4.6.10Homogeneous with constant coefficients — characteristic equation

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4.6.11Case 1 - two distinct real roots

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4.6.12Case 2 - repeated real root — reduction of order

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4.6.13Case 3 - complex conjugate roots — Euler's formula connection

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4.6.14Non-homogeneous — method of undetermined coefficients (annihilator method)

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4.6.15Non-homogeneous — variation of parameters

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4.6.16Cauchy-Euler (Equidimensional) equation

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4.6.17Power series solutions — ordinary points

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4.6.18Frobenius method — regular singular points

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4.6.19Bessel's equation and Bessel functions (intro, physical relevance)

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4.6.20Legendre's equation and Legendre polynomials (intro)

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4.6.21Systems of first-order linear ODEs — matrix method

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4.6.22Phase plane analysis — trajectories, critical points

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4.6.23Stability of equilibria — stable, unstable, saddle, spiral, centre

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4.6.24Linearization of nonlinear systems

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4.6.25Laplace transform — definition, region of convergence

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4.6.26Transforms of standard functions — proofs

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4.6.27Properties — linearity, first - second shift theorems, scaling

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4.6.28Laplace of derivatives — key property for solving ODEs

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4.6.29Inverse Laplace transform — partial fractions, tables

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4.6.30Solving ODEs with Laplace (including discontinuous forcing)

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4.6.31Heaviside step function and Dirac delta function

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4.6.32Convolution theorem — proof, applications

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4.6.33Impulse response and transfer function (GNC connection)

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4.7

Partial Differential Equations

11 deep-dive topics

4.7.1Classification — elliptic, parabolic, hyperbolic (discriminant test)

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4.7.2Initial value problems (IVP) vs boundary value problems (BVP)

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4.7.3Fourier series — motivation from periodic functions

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4.7.4Dirichlet conditions for convergence

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4.7.5Full Fourier series — coefficients derivation

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4.7.6Half-range sine and cosine series

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4.7.7Parseval's theorem

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4.7.8Heat equation (parabolic) 1D — derivation from Fourier's law

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4.7.9Solving heat equation — separation of variables

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4.7.10Wave equation (hyperbolic) 1D — derivation

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4.7.11Solving wave equation — D'Alembert's solution

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4.8

Numerical Methods

17 deep-dive topics

4.8.9Secant method

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4.8.10Polynomial interpolation — Lagrange form, Newton's divided differences

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4.8.11Error in polynomial interpolation

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4.8.12Cubic spline interpolation — natural, clamped

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4.8.13Numerical differentiation — forward, backward, central differences

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4.8.18Solving linear systems — Gaussian elimination with partial pivoting

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4.8.19LU decomposition (numerical)

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4.8.20Iterative methods — Jacobi, Gauss-Seidel, convergence

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4.8.21Eigenvalue computation — power method, inverse iteration

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4.8.22ODE solvers — Euler's method (derivation, global error)

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4.8.23Modified Euler (Heun's method)

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4.8.24Runge-Kutta 4th order (RK4) — derivation

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4.8.25Adaptive step-size — RK45, error control

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4.8.26Stiff equations — implicit methods, backward Euler

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4.8.27Systems of ODEs — RK4 for systems

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4.8.28Boundary value problems — shooting method, finite difference

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4.8.29Solving nonlinear systems — Newton's method in n dimensions

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4.9

Probability Theory & Statistics

25 deep-dive topics

4.9.1Probability space — sample space Ω, sigma-algebra F, measure P — Kolmogorov axioms

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4.9.2Inclusion-exclusion principle

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4.9.3Discrete random variables — PMF, CDF

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4.9.4Expected value, variance, standard deviation — properties

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4.9.5Moment generating function (MGF) — definition, use

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4.9.6Common discrete distributions — Bernoulli, Binomial, Poisson, Geometric, Negative Binomial

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4.9.7Continuous random variables — PDF, CDF, percentiles

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4.9.8Common continuous distributions — Uniform, Normal, Exponential, Gamma, Beta

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4.9.9Chi-squared, t, F distributions — definition, degrees of freedom

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4.9.10Joint distributions — joint PMF - PDF, marginal, conditional

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4.9.11Independence of random variables — formal definition

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4.9.12Covariance and correlation

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4.9.13Conditional expectation

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4.9.14Transformations of random variables — change-of-variable technique

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4.9.15Central Limit Theorem — statement, proof sketch, significance

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4.9.16Law of Large Numbers — weak and strong

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4.9.17Statistical estimation — MLE, method of moments

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4.9.18Properties of estimators — unbiasedness, consistency, efficiency

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4.9.19Confidence intervals — derivation for mean, proportion

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4.9.20Hypothesis testing — null - alternative, test statistic, p-value, errors (Type I & II)

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4.9.21z-test, t-test, chi-squared goodness of fit, F-test

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4.9.22Linear regression — least squares, inference on coefficients

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4.9.23Multiple regression

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4.9.24Bayesian statistics — prior, likelihood, posterior (intro)

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4.9.25Monte Carlo simulation — law of large numbers basis

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4.10

Advanced Topics (Elite Level)

27 deep-dive topics

4.10.1Complex analysis — analytic functions, Cauchy-Riemann equations

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4.10.2Complex integration — contour integrals

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4.10.3Cauchy's integral theorem and formula

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4.10.4Laurent series — principal part, annulus of convergence

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4.10.5Residues and poles

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4.10.6Residue theorem — computing real integrals

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4.10.7Tensor analysis — scalars, vectors, rank-2 tensors

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4.10.8Covariant and contravariant components

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4.10.9Einstein summation convention

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4.10.10Metric tensor — raising - lowering indices

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4.10.11Christoffel symbols — intro

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4.10.12Calculus of variations — functionals, functional derivative

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4.10.13Euler-Lagrange equation — derivation

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4.10.14Brachistochrone problem

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4.10.15Hamilton's principle — least action

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4.10.16Isoperimetric problems — constraints (Lagrange multipliers in variational sense)

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4.10.17Convex optimization — convex sets, convex functions

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4.10.18First-order optimality conditions — gradient = 0

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4.10.19KKT conditions for constrained optimization

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4.10.20Gradient descent and variants — convergence analysis

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4.10.21Linear programming — simplex method (intro)

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4.10.22Real analysis — rigorous epsilon-delta, metric spaces

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4.10.23Uniform continuity — difference from pointwise

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4.10.24Uniform convergence of function sequences

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4.10.25Measure theory — Lebesgue measure (intro)

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4.10.26Fourier analysis — DFT, FFT algorithm (Cooley-Tukey)

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4.10.27Stochastic processes — Markov chains, steady-state, random walks

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