TORUS KNOT3D
Subject notebook
Mathematics
“The universe is written in mathematics — learn to read it fluently.”
703 topics · 21 chapters
Topics for Mathematics
Ch. 1
Number Systems and Arithmetic
Number systems
23 topics
- Natural numbers
- Whole numbers
- Integers
- Rational numbers
- Irrational numbers
- Real numbers
- Even and odd numbers
- Prime and composite numbers
- Divisibility rules
- HCF and LCM
- Fractions and mixed numbers
- Decimals
- Recurring decimals
- Percentages
- Ratio and proportion
- Unitary method
- Exponents and powers
- Laws of exponents
- Surds and radicals
- Rationalisation of denominators
- Scientific notation
- Significant figures
- Estimation and rounding
Ch. 2
Algebra (School Level)
Algebraic expressions
13 topics
- Variables and constants
- Like and unlike terms
- Addition and subtraction of polynomials
- Multiplication of polynomials
- Division of polynomials
- Special products and algebraic identities
- Factorisation by grouping
- Factorisation of quadratics
- Factorisation of cubic polynomials
- Remainder theorem
- Factor theorem
- Rational expressions and simplification
- Partial fractions
Equations and inequalities
18 topics
- Linear equations in one variable
- Linear equations in two variables
- Simultaneous linear equations
- Quadratic equations
- Solving quadratics by factorisation
- Completing the square
- Quadratic formula
- Discriminant and nature of roots
- Sum and product of roots
- Formation of equations from roots
- Equations reducible to quadratics
- Theory of equations (Descartes' rule of signs)
- Location of roots
- Linear inequalities in one variable
- Linear inequalities in two variables
- Graphical solution of inequalities
- Modulus equations and inequalities
- Logarithmic equations
Logarithms
6 topics
Sequences, series and the binomial theorem
18 topics
- Arithmetic progression
- nth term of an AP
- Sum of an AP
- Geometric progression
- nth term of a GP
- Sum of a GP
- Sum of an infinite GP
- Harmonic progression
- Arithmetic mean
- Geometric mean
- Harmonic mean
- AM-GM-HM inequality
- Sum of squares and cubes of natural numbers
- Special series and telescoping sums
- Principle of mathematical induction
- Binomial theorem for positive integral index
- General term and middle terms
- Properties of binomial coefficients
Permutations and combinations
9 topics
Sets, relations and functions
24 topics
- Representation of sets
- Types of sets (finite, infinite, empty, singleton)
- Subsets and supersets
- Power sets
- Universal sets
- Venn diagrams
- Union and intersection of sets
- Complement of a set
- Difference and symmetric difference
- De Morgan's laws
- Cartesian product of sets
- Relations and types
- Equivalence relations and partitions
- Functions as special relations
- Domain, codomain and range
- One-one and onto functions
- Composition of functions
- Invertible functions
- Constant, identity and polynomial functions
- Rational and modulus functions
- Signum and greatest integer functions
- Exponential and logarithmic functions
- Even and odd functions
- Graphs and transformations of functions
Matrices and determinants (introductory)
16 topics
- Order and types of matrices
- Equality of matrices
- Addition and subtraction of matrices
- Scalar multiplication
- Multiplication of matrices
- Transpose of a matrix
- Symmetric and skew-symmetric matrices
- Determinants of order two and three
- Properties of determinants
- Minors and cofactors
- Adjoint of a matrix
- Inverse of a square matrix
- Solution of linear equations by matrix method
- Cramer's rule
- Consistency of linear systems
- Area of a triangle using determinants
Ch. 3
Trigonometry
Trigonometric ratios and identities
14 topics
- Trigonometric ratios in a right triangle
- Values for standard angles
- Trigonometric ratios of allied angles
- Pythagorean identities
- Reciprocal and quotient identities
- Sum and difference formulae
- Double-angle formulae
- Triple-angle formulae
- Half-angle formulae
- Product-to-sum and sum-to-product formulae
- Conditional trigonometric identities
- Graphs of trigonometric functions
- Amplitude, period and phase shift
- Maximum and minimum values of trig expressions
Trigonometric equations and inverse functions
6 topics
Ch. 4
Coordinate Geometry and Vectors
Two-dimensional geometry
42 topics
- Cartesian coordinate system
- Distance formula
- Section formula (internal and external)
- Midpoint formula
- Centroid, incentre and orthocentre
- Area of a triangle and quadrilateral
- Slope of a line
- Intercepts of a line
- Slope-point and slope-intercept forms
- Two-point and intercept forms
- Normal and symmetric forms
- General equation of a line
- Parallel, perpendicular and intersecting lines
- Angle between two lines
- Distance of a point from a line
- Distance between parallel lines
- Equation of angle bisectors
- Family of lines and concurrency
- Transformation of axes (translation)
- Locus and its equation
- Circle: standard and general equations
- Centre-radius form of a circle
- Tangent to a circle
- Normal to a circle
- Chord of contact
- Pair of tangents from a point
- Radical axis
- Coaxial circles
- Parabola: standard forms
- Focus, directrix and latus rectum of a parabola
- Tangent and normal to a parabola
- Ellipse: standard equations
- Eccentricity of an ellipse
- Foci, vertices and latus rectum of an ellipse
- Tangent and normal to an ellipse
- Hyperbola: standard equations
- Asymptotes of a hyperbola
- Rectangular hyperbola
- Tangent and normal to a hyperbola
- General second-degree equation and conic classification
- Pair of straight lines
- Homogeneous equations
Three-dimensional geometry and vector algebra
21 topics
- Scalars and vectors
- Types of vectors
- Addition and subtraction of vectors
- Scalar multiplication of vectors
- Section formula in vector form
- Dot (scalar) product
- Cross (vector) product
- Scalar triple product
- Vector triple product
- Projection of a vector
- Direction ratios and direction cosines
- Equation of a straight line in space
- Angle between two lines in space
- Shortest distance between skew lines
- Equation of a plane
- Intercept and normal forms of a plane
- Angle between two planes
- Distance of a point from a plane
- Line-plane intersection
- Sphere: equation and properties
- Tangent plane to a sphere
Ch. 5
Calculus
Limits and continuity
11 topics
Differentiation
23 topics
- Derivative from first principles
- Rules of differentiation (sum, product, quotient)
- Chain rule
- Derivatives of trigonometric functions
- Derivatives of inverse trigonometric functions
- Derivatives of exponential and logarithmic functions
- Logarithmic differentiation
- Differentiation of parametric functions
- Implicit differentiation
- Differentiation of a function w.r.t. another function
- Second and higher order derivatives
- Rolle's theorem
- Lagrange's mean value theorem
- Cauchy's mean value theorem
- Increasing and decreasing functions
- Tangents and normals
- Rate of change of quantities
- Approximations using differentials
- Maxima and minima
- First and second derivative tests
- Optimisation and word problems
- Curve sketching and concavity
- Points of inflection
Indefinite integration
8 topics
Definite integration
8 topics
Applications of integration
6 topics
Differential equations (introductory)
8 topics
Multivariable and vector calculus
18 topics
- Functions of several variables
- Partial derivatives
- Higher order partial derivatives
- Total differential
- Chain rule for several variables
- Jacobians
- Maxima and minima of two variables
- Double integrals
- Triple integrals
- Change of variables in multiple integrals
- Gradient of a scalar field
- Divergence and curl of a vector field
- Line integrals
- Surface integrals
- Volume integrals
- Green's theorem
- Stokes' theorem
- Divergence (Gauss) theorem
Ch. 6
Probability and Statistics
Descriptive statistics
15 topics
- Collection and classification of data
- Frequency distributions
- Cumulative frequency
- Histograms and frequency polygons
- Ogives (cumulative frequency curves)
- Mean (arithmetic, geometric, harmonic)
- Median and quartiles
- Mode
- Percentiles and deciles
- Range and interquartile range
- Mean deviation
- Variance and standard deviation
- Coefficient of variation
- Skewness and kurtosis
- Moments of a distribution
Probability theory
25 topics
- Random experiments and sample space
- Types of events
- Classical definition of probability
- Axiomatic approach to probability
- Addition theorem of probability
- Multiplication theorem of probability
- Conditional probability
- Independent events
- Total probability theorem
- Bayes' theorem
- Random variables (discrete and continuous)
- Probability mass and density functions
- Cumulative distribution functions
- Expected value and variance
- Moment generating functions (intro)
- Bernoulli trials
- Binomial distribution
- Poisson distribution
- Geometric and negative binomial distributions
- Uniform distribution
- Exponential distribution
- Normal distribution and its properties
- Standard normal variate
- Law of large numbers
- Central limit theorem
Sampling and inference
13 topics
Correlation, regression and applied statistics
10 topics
Ch. 7
Discrete Mathematics
Mathematical logic
11 topics
- Propositions and truth values
- Logical connectives (AND, OR, NOT, implies, iff)
- Truth tables
- Tautologies and contradictions
- Logical equivalence
- Laws of logic
- Predicates and quantifiers
- Rules of inference
- Methods of proof (direct, contrapositive, contradiction)
- Proof by mathematical induction (advanced)
- Strong induction
Counting and combinatorics (advanced)
7 topics
Graph theory
15 topics
- Graphs, vertices and edges
- Types of graphs
- Degree of a vertex and handshaking lemma
- Walks, paths and circuits
- Connected graphs
- Eulerian trails and circuits
- Hamiltonian paths and cycles
- Trees and their properties
- Spanning trees and minimum spanning trees
- Planar graphs and Euler's formula
- Graph colouring
- Bipartite graphs and matching
- Directed graphs
- Shortest path algorithms (Dijkstra, Floyd)
- Network flows (intro)
Ch. 8
Number Theory
General
20 topics
- Divisibility and the division algorithm
- Prime numbers and composites
- Fundamental theorem of arithmetic
- Sieve of Eratosthenes
- Euclid's algorithm for GCD
- Linear Diophantine equations
- Modular arithmetic
- Congruence classes
- Linear congruences
- Fermat's little theorem
- Euler's totient function
- Euler's theorem
- Wilson's theorem
- Chinese remainder theorem
- Quadratic residues and Legendre symbol
- Continued fractions
- Arithmetic and multiplicative functions
- Perfect and amicable numbers
- Distribution of primes (intro)
- Cryptography basics (RSA, Diffie-Hellman)
Ch. 9
Euclidean Geometry and Mensuration
Plane geometry
17 topics
- Points, lines, rays and angles
- Parallel lines and transversals
- Triangles: types and properties
- Congruence of triangles (SSS, SAS, ASA, RHS)
- Similarity of triangles
- Pythagoras theorem and converse
- Basic proportionality theorem (Thales)
- Angle bisector theorem
- Quadrilaterals: types and properties
- Parallelograms, rectangles, rhombi, squares
- Trapeziums and kites
- Polygons: interior and exterior angles
- Circles: chords, arcs and sectors
- Tangent-secant theorems
- Cyclic quadrilaterals
- Geometric constructions with compass and ruler
- Loci in plane geometry
Mensuration
17 topics
- Perimeter of triangles and quadrilaterals
- Area of triangles (base-height, Heron's formula)
- Area of rectangles, squares, parallelograms
- Area of trapeziums and rhombi
- Circumference and area of circles
- Area of sectors and segments
- Surface area of cubes and cuboids
- Surface area of cylinders
- Surface area of cones
- Surface area of spheres and hemispheres
- Volume of cubes and cuboids
- Volume of cylinders
- Volume of cones
- Volume of spheres and hemispheres
- Volume of prisms and pyramids
- Frustum of a cone
- Combinations of solids
Ch. 10
Linear Programming and Optimisation (Introductory)
Ch. 11
Real Analysis
General
29 topics
- The real number system and field axioms
- Order axioms and completeness
- Supremum and infimum
- Archimedean property
- Sequences of real numbers
- Convergence and divergence of sequences
- Monotone sequences
- Subsequences and Bolzano-Weierstrass theorem
- Cauchy sequences and completeness
- Series of real numbers
- Tests of convergence (comparison, ratio, root, integral)
- Absolute and conditional convergence
- Alternating series and Leibniz test
- Power series and radius of convergence
- Limits of functions (rigorous epsilon-delta)
- Continuity and uniform continuity
- Differentiability and the mean value theorems
- Taylor's theorem with remainder
- Riemann integration
- Riemann sums and integrability criteria
- Fundamental theorem of calculus (rigorous)
- Improper integrals and convergence tests
- Sequences and series of functions
- Pointwise and uniform convergence
- Weierstrass M-test
- Metric spaces
- Open, closed and compact sets in metric spaces
- Completeness and Baire category (intro)
- Contraction mapping theorem
Ch. 12
Complex Analysis
General
27 topics
- Algebra of complex numbers (advanced)
- Argand plane and polar form
- De Moivre's theorem
- Roots of unity
- Stereographic projection and the Riemann sphere
- Functions of a complex variable
- Limits and continuity in the complex plane
- Analytic (holomorphic) functions
- Cauchy-Riemann equations
- Harmonic functions and conjugates
- Elementary complex functions (exp, log, trig)
- Branches of multi-valued functions
- Complex line integrals
- Cauchy's integral theorem
- Cauchy's integral formula
- Liouville's theorem
- Fundamental theorem of algebra (proof)
- Taylor series of analytic functions
- Laurent series
- Isolated singularities (removable, poles, essential)
- Residues and the residue theorem
- Evaluation of real integrals by contour integration
- Argument principle and Rouche's theorem
- Maximum modulus principle
- Conformal mappings
- Mobius transformations
- Schwarz lemma (intro)
Ch. 13
Abstract Algebra
General
25 topics
- Binary operations and algebraic structures
- Semigroups and monoids (intro)
- Groups: definition and examples
- Subgroups and cosets
- Cyclic groups
- Lagrange's theorem
- Normal subgroups and quotient groups
- Group homomorphisms and isomorphisms
- First isomorphism theorem
- Permutation groups and symmetric groups
- Cayley's theorem
- Group actions (intro)
- Sylow theorems
- Direct products of groups
- Rings: definition and examples
- Subrings and ideals
- Quotient rings
- Ring homomorphisms
- Integral domains and fields
- Polynomial rings
- Division algorithm for polynomials
- Irreducibility criteria (Eisenstein)
- Fields and field extensions (intro)
- Finite fields (intro)
- Galois theory (intro)
Ch. 14
Linear Algebra (Advanced)
General
23 topics
- Vector spaces over a field
- Subspaces and quotient spaces
- Linear independence and dependence
- Basis and dimension
- Coordinates and change of basis
- Linear transformations
- Kernel and image (rank-nullity theorem)
- Matrix representation of linear maps
- Eigenvalues and eigenvectors
- Characteristic and minimal polynomials
- Diagonalisation of matrices
- Cayley-Hamilton theorem
- Inner product spaces
- Norms and orthogonality
- Gram-Schmidt orthogonalisation
- Orthogonal projections
- Spectral theorem for symmetric matrices
- Quadratic forms and definiteness
- Bilinear forms (intro)
- Jordan canonical form
- Dual spaces (intro)
- Tensor products (intro)
- Applications: least squares, Markov chains
Ch. 15
Differential Equations (Advanced)
General
24 topics
- Review of first-order methods
- Existence and uniqueness theorems
- Higher-order linear ODEs with constant coefficients
- Wronskian and linear independence of solutions
- Method of undetermined coefficients
- Variation of parameters
- Cauchy-Euler equations
- Power series solutions
- Frobenius method
- Legendre and Bessel equations (intro)
- Laplace transforms and inverse transforms
- Solving ODEs with Laplace transforms
- Systems of first-order linear ODEs
- Phase plane analysis (intro)
- Sturm-Liouville theory
- Classification of second-order PDEs
- First-order partial differential equations
- Charpit's method (intro)
- Heat equation and separation of variables
- Wave equation (d'Alembert's solution)
- Laplace's equation and harmonic functions
- Boundary and initial value problems
- Fourier series and convergence
- Fourier transforms (intro)
Ch. 16
Differential Geometry and Topology
General
21 topics
- Parametrised curves in plane and space
- Arc length and reparametrisation
- Curvature and torsion
- Frenet-Serret formulae
- Regular surfaces
- First fundamental form
- Second fundamental form
- Normal and Gaussian curvature
- Mean curvature
- Geodesics
- Gauss-Bonnet theorem (intro)
- Topological spaces and open sets
- Basis and subbasis
- Continuous maps and homeomorphisms
- Subspace and quotient topology
- Connectedness and path-connectedness
- Compactness
- Separation axioms (T0-T4)
- Metric topology (review)
- Fundamental group (intro)
- Manifolds (intro)
Ch. 17
Functional Analysis (Introductory)
Ch. 18
Numerical Methods
General
22 topics
- Floating-point arithmetic and error analysis
- Truncation and round-off errors
- Bisection method
- Fixed-point iteration
- Newton-Raphson method
- Secant and regula falsi methods
- Systems of nonlinear equations (intro)
- Gaussian elimination
- LU decomposition
- Matrix inversion by elimination
- Iterative methods: Jacobi and Gauss-Seidel
- Lagrange interpolation
- Newton's divided differences
- Spline interpolation (intro)
- Numerical differentiation
- Trapezoidal rule
- Simpson's one-third and three-eighth rules
- Gaussian quadrature (intro)
- Euler's method for ODEs
- Runge-Kutta methods
- Least squares curve fitting
- Numerical solution of PDEs (intro)
Ch. 19
Operations Research
General
16 topics
- Linear programming: simplex method
- Big-M and two-phase methods
- Duality theory
- Dual simplex method
- Sensitivity analysis
- Transportation problem
- Assignment problem
- Hungarian method
- Game theory: pure and mixed strategies
- Queuing theory basics (M/M/1)
- Inventory models (EOQ)
- Replacement theory (intro)
- Network analysis: PERT and CPM
- Dynamic programming
- Integer programming (intro)
- Nonlinear programming basics (KKT conditions, intro)
Ch. 20
Financial and Actuarial Mathematics
General
17 topics
- Time value of money
- Simple and compound interest (applied)
- Nominal and effective rates
- Annuities: immediate, due and perpetuities
- Amortisation schedules
- Sinking funds
- Bonds: price and yield
- Term structure of interest rates (intro)
- Portfolio theory basics
- Options and derivatives (intro)
- Binomial option pricing
- Black-Scholes model (intro)
- Value at risk (intro)
- Mortality tables
- Life annuities and assurances
- Premium calculation
- Policy reserves (intro)
Ch. 21