Maths Optional — Study Guide

Paper 1

Dynamics & Statics

• Rectilinear motion under variable force Occasional · Rectilinear motion under variable force
• Simple harmonic motion (free, damped, forced) Core · Simple harmonic motion (free, damped, forced)
• Motion in a Plane (Resolved Components / Polar) Rare · Motion in a plane (resolved components / polar)
• Projectile motion Reliable · Projectile motion
• Constrained motion Reliable · Constrained motion; circular motion
• Work-energy theorem Occasional · Work-energy theorem; conservation of energy
• Central force motion and Kepler's laws Reliable · Central force motion; Kepler's laws
• Orbits under inverse-square central force Occasional · Orbits under inverse-square central force
• Equilibrium of a system of particles Reliable · Equilibrium of a system of particles
• Work and Potential Energy; Conservation Rare · Work and potential energy; conservation
• Friction (limiting friction) Occasional · Friction (limiting friction; equilibrium with friction)
• Common catenary Occasional · Common catenary
• Principle of virtual work Reliable · Principle of virtual work
• Stability of equilibrium (energy criterion) Reliable · Stability of equilibrium (energy criterion)
• Equilibrium of Forces in Three Dimensions Rare · Equilibrium of forces in three dimensions

ODEs

• Formulation of differential equations Occasional · Formulation of differential equations
• Variables separable Occasional · Variables separable
• Homogeneous Equations and Reduction Rare · Homogeneous equations and reduction
• Linear first-order Occasional · Linear first-order; integrating factor
• Exact equations Reliable · Exact equations; integrating factors for non-exact
• Orthogonal trajectories (cartesian and polar) Reliable · Orthogonal trajectories (cartesian and polar)
• First-order higher-degree ODEs Occasional · First-order higher-degree: solvable for p, x, y
• Clairaut's equation Reliable · Clairaut's equation; singular solution
• Linear ODE with constant coefficients Occasional · Linear ODE with constant coefficients: complementary function
• Particular integral via operator method Reliable · Particular integral via operator method
• Euler-Cauchy equation Core · Euler-Cauchy equation
• Method of variation of parameters Core · Method of variation of parameters
• Reduction of order with one solution known Occasional · Reduction of order with one solution known
• Laplace transform Occasional · Laplace transform: definition; transforms of elementary functions
• Properties of Laplace transform (linearity, shift, derivative, convolution) Occasional · Properties of Laplace transform (linearity, shift, derivative, convolution)
• Inverse Laplace transform Occasional · Inverse Laplace transform; partial fractions
• Laplace transform applied to IVP for second-order linear ODE with constant coefficients Core · Laplace transform applied to IVP for second-order linear ODE with constant coefficients
• Picard's Existence/Uniqueness Theorem; Lipschitz Condition Rare · Picard's existence/uniqueness theorem; Lipschitz condition

Vector Analysis

• Scalar and Vector Fields Rare · Scalar and vector fields
• Differentiation of a vector function of a scalar variable Occasional · Differentiation of a vector function of a scalar variable
• Gradient: definition, geometric meaning, computation Occasional · Gradient: definition, geometric meaning, computation
• Curl: definition, physical meaning, computation Occasional · Curl: definition, physical meaning, computation
• Higher order derivatives; Laplacian Occasional · Higher order derivatives; Laplacian
• Vector identities (curl of grad, div of curl, product rules) Occasional · Vector identities (curl of grad, div of curl, product rules)
• Curves in space: tangent, normal, binormal Occasional · Curves in space: tangent, normal, binormal
• Curvature and torsion Reliable · Curvature and torsion
• Serret-Frenet formulae Occasional · Serret-Frenet formulae
• Line integrals Core · Line integrals; circulation
• Gauss divergence theorem Core · Gauss divergence theorem
• Stokes' theorem Core · Stokes' theorem

Analytic Geometry

• Straight lines in 3D Reliable · Straight lines in 3D: parametric/symmetric/vector forms
• Shortest distance between two skew lines Occasional · Shortest distance between two skew lines
• Plane Reliable · Plane: equations, distance, angle
• Second-degree equations in three variables Occasional · Second-degree equations in three variables: general form
• Reduction of Second-Degree Equation to Canonical Form Rare · Reduction of second-degree equation to canonical form
• Sphere Core · Sphere: equation, tangent plane, intersection
• Cone Core · Cone: equation; right circular cone
• Cylinder Occasional · Cylinder: equation; right circular cylinder
• Paraboloid (elliptic and hyperbolic) Reliable · Paraboloid (elliptic and hyperbolic)
• Ellipsoid Occasional · Ellipsoid
• Hyperboloid of one sheet Occasional · Hyperboloid of one sheet; ruled surface

Linear Algebra

• Linear dependence and independence Occasional · Linear dependence and independence
• Subspaces Reliable · Subspaces: definition, intersection, sum, direct sum
• Bases and dimension; coordinates in a basis Reliable · Bases and dimension; coordinates with respect to a basis
• Linear transformations Occasional · Linear transformations: definition, kernel, image
• Rank and nullity; rank-nullity theorem Reliable · Rank and nullity; rank-nullity theorem
• Matrix of a linear transformation Core · Matrix of a linear transformation; change of basis
• Algebra of matrices Occasional · Algebra of matrices: addition, multiplication, transpose
• Row and Column Reduction; Echelon Form Rare · Row and column reduction; Echelon form
• Congruence and similarity of matrices Occasional · Congruence and similarity of matrices
• Rank of a matrix Reliable · Rank of a matrix
• Inverse of a matrix (adjoint and row reduction) Occasional · Inverse of a matrix (adjoint and row reduction)
• Solution of system of linear equations Reliable · Solution of system of linear equations; Gauss elimination
• Eigenvalues and eigenvectors Core · Eigenvalues and eigenvectors: computation
• Cayley-Hamilton theorem Occasional · Cayley-Hamilton theorem
• Diagonalization via Eigenvectors Rare · Diagonalization via eigenvectors
• Symmetric and Skew-Symmetric Matrices Rare · Symmetric and skew-symmetric matrices
• Hermitian and skew-Hermitian matrices Occasional · Hermitian and skew-Hermitian matrices
• Orthogonal and unitary matrices Occasional · Orthogonal and unitary matrices
• Quotient Vector Space V/W Rare · Quotient vector space V/W: cosets; dimension formula dim(V/W)=dim V - dim W

Calculus

• Continuity of real functions Occasional · Continuity of real functions
• Differentiability Occasional · Differentiability; rules of differentiation
• Mean-value theorems (Rolle, Lagrange, Cauchy) Occasional · Mean-value theorems (Rolle, Lagrange, Cauchy)
• Taylor's theorem with remainders Occasional · Taylor's theorem with remainders
• Indeterminate forms Occasional · Indeterminate forms; L'Hopital's rule
• Maxima and minima of single-variable functions Reliable · Maxima and minima of single-variable functions
• Asymptotes Rare · Asymptotes
• Curve tracing (cartesian and polar) Occasional · Curve tracing (cartesian and polar)
• Functions of two/three variables: limits, continuity Occasional · Functions of two/three variables: limits, continuity
• Partial derivatives Reliable · Partial derivatives; equality of mixed partials
• Maxima and Minima of Multi-Variable Functions (Unconstrained) Rare · Maxima and minima of multi-variable functions (unconstrained)
• Lagrange's method of multipliers (constrained extrema) Core · Lagrange's method of multipliers (constrained extrema)
• Jacobian Occasional · Jacobian; change of variables
• Riemann Definite Integral; Integrability Rare · Riemann definite integral; integrability
• Indefinite integrals Reliable · Indefinite integrals: substitution, parts, partial fractions
• Improper integrals (unbounded interval/integrand) Occasional · Improper integrals (unbounded interval/integrand)
• Double integrals Core · Double integrals; change of order/variables
• Triple Integrals; Cylindrical and Spherical Coordinates Rare · Triple integrals; cylindrical, spherical coordinates
• Areas, surface areas, volumes via integration Occasional · Areas, surface areas, volumes via integration

Paper 2

Algebra

• Groups: definition, axioms, examples Occasional · Groups: definition, axioms, examples
• Subgroups; Subgroup Criterion Rare · Subgroups; subgroup criterion
• Cyclic groups Core · Cyclic groups: structure, generators, order of element
• Cosets and Lagrange's theorem Occasional · Cosets and Lagrange's theorem
• Normal subgroups; quotient groups Occasional · Normal subgroups; quotient groups
• Group homomorphisms: kernel, image Occasional · Group homomorphisms: kernel, image
• Isomorphism theorems (First, Second, Third) Occasional · Isomorphism theorems (First, Second, Third)
• Permutation Groups (S_n): Cycle Decomposition, Sign, A_n Rare · Permutation groups (S_n): cycle decomposition, sign, A_n
• Cayley's Theorem Rare · Cayley's theorem
• Rings: Definition, Axioms, Examples Rare · Rings: definition, axioms, examples
• Subrings and ideals Occasional · Subrings and ideals
• Ring homomorphisms; quotient rings Occasional · Ring homomorphisms; quotient rings
• Integral domains; characteristic Occasional · Integral domains; characteristic
• Principal Ideal Domains (PID) Rare · Principal ideal domains (PID)
• Euclidean domains Reliable · Euclidean domains
• Fields and finite fields Reliable · Fields: definition; finite fields F_p; characteristic
• Field Extensions; Tower Law; Algebraic Closure Rare · Field/algebraic extensions; tower law; algebraic closure

Complex Analysis

• Analytic Functions: Complex Differentiability Rare · Analytic functions: complex differentiability
• Cauchy-Riemann equations (necessary and sufficient) Reliable · Cauchy-Riemann equations (necessary and sufficient)
• Harmonic functions and harmonic conjugate Reliable · Harmonic functions; harmonic conjugate
• Cauchy's theorem (Cauchy-Goursat) Occasional · Cauchy's theorem (Cauchy-Goursat)
• Power Series of Analytic Functions; Radius of Convergence Rare · Power series of analytic functions; radius of convergence
• Taylor's Series for Analytic Functions Rare · Taylor's series for analytic functions
• Singularities: removable, pole, essential Occasional · Singularities: removable, pole, essential
• Laurent's series in an annulus Reliable · Laurent's series in an annulus
• Residues: computation at poles of various orders Occasional · Residues: computation at poles of various orders
• Cauchy's residue theorem Reliable · Cauchy's residue theorem
• Contour integration of real integrals using residues Core · Contour integration of real integrals using residues

Linear Programming

• LPP: standard form; basic, basic feasible, optimal solutions Occasional · LPP: standard form; basic, basic feasible, optimal solutions
• Graphical method Reliable · Graphical method (two-variable LPPs)
• Simplex method (basic) Core · Simplex method (basic)
• Big-M / two-phase method (artificial variables) Reliable · Big-M / two-phase method (artificial variables)
• Duality Occasional · Duality: dual formulation, weak/strong duality, complementary slackness
• Transportation problem Reliable · Transportation problem: BFS (NWCR/LCM/VAM), MODI optimality
• Assignment problem (Hungarian method) Reliable · Assignment problem (Hungarian method)

Numerical Analysis

• Bisection Method (Convergence, Error) Rare · Bisection method (convergence, error)
• Regula Falsi (False Position) Rare · Regula Falsi (false position)
• Newton-Raphson method (convergence, geometric meaning) Reliable · Newton-Raphson method (convergence, geometric meaning)
• Gaussian Elimination Rare · Gaussian elimination
• Gauss-Jordan method Occasional · Gauss-Jordan method
• Gauss-Seidel iteration Reliable · Gauss-Seidel iteration; convergence criteria
• Newton's forward difference interpolation Occasional · Newton's forward difference interpolation
• Newton's Backward Difference Interpolation Rare · Newton's backward difference interpolation
• Lagrange's interpolation Reliable · Lagrange's interpolation
• Trapezoidal rule (composite; error) Occasional · Trapezoidal rule (composite; error)
• Simpson's 1/3 and 3/8 rules Reliable · Simpson's 1/3 and 3/8 rules
• Gaussian quadrature Occasional · Gaussian quadrature
• Euler's method (and modified Euler) Occasional · Euler's method (and modified Euler)
• Runge-Kutta methods (RK2/RK4) Occasional · Runge-Kutta methods (RK2/RK4)
• Number systems Reliable · Number systems: binary, octal, hex; conversions
• Algebra of Binary Numbers Rare · Algebra of binary numbers
• Logic gates and truth tables Occasional · Logic gates and truth tables
• Boolean algebra Core · Boolean algebra; normal forms (DNF, CNF)
• Representation of Integers, Signed Integers, and Reals (incl. Double Precision) Rare · Representation of integers, signed integers, reals (incl. double precision)
• Algorithms and flowcharts for numerical analysis problems Occasional · Algorithms and flowcharts for numerical analysis problems

PDEs

• Family of surfaces Reliable · Family of surfaces; PDE formulation
• Quasilinear first-order PDEs (Lagrange's method) Reliable · Quasilinear first-order PDEs (Lagrange's method)
• Cauchy's method of characteristics Occasional · Cauchy's method of characteristics
• Classification and reduction to canonical form Core · Classification and reduction to canonical form
• Second-order linear PDEs with constant coefficients (CF, PI) Core · Second-order linear PDEs with constant coefficients (CF, PI)
• Wave equation Reliable · Wave equation: D'Alembert and separation of variables
• Heat equation Occasional · Heat equation: separation of variables; Fourier solutions
• Laplace equation: Dirichlet/Neumann, separation of variables Occasional · Laplace equation: Dirichlet/Neumann, separation of variables
• Charpit's method Occasional · Charpit's method; complete integrals of nonlinear first-order PDEs

Real Analysis

• Real number system as ordered field with LUB property Occasional · Real number system as ordered field with LUB property
• Sequences Occasional · Sequences: limit, bounded, monotone convergence
• Cauchy sequences; completeness of R Occasional · Cauchy sequences; completeness of R
• Series of real terms: convergence, standard tests Occasional · Series of real terms: convergence, standard tests
• Absolute and conditional convergence Occasional · Absolute and conditional convergence
• Rearrangement of Series; Riemann's Theorem Rare · Rearrangement of series; Riemann's theorem (statement)
• Continuity of Functions on R; Epsilon-Delta Rare · Continuity of functions on R; epsilon-delta
• Uniform continuity Occasional · Uniform continuity
• Properties of Continuous Functions on Compact Sets Rare · Properties of continuous functions on compact sets
• Riemann integral Core · Riemann integral: Darboux sums, integrability
• Improper integrals (analysis perspective) Occasional · Improper integrals (analysis perspective)
• Fundamental theorems of integral calculus Occasional · Fundamental theorems of integral calculus
• Pointwise vs. Uniform Convergence of Sequences of Functions Rare · Pointwise vs. uniform convergence of sequences of functions
• Uniform Convergence: Term-by-Term Differentiation Rare · Uniform convergence: term-by-term differentiation
• Uniform convergence of series Occasional · Uniform convergence of series; Weierstrass M-test
• Partial derivatives; equality of mixed partials (Schwarz) Occasional · Partial derivatives; equality of mixed partials (Schwarz)
• Maxima and minima of multi-variable functions (analytic criteria) Occasional · Maxima and minima of multi-variable functions (analytic criteria)
• Maxima and minima of single-variable functions Occasional · Maxima and minima of single-variable functions

Mechanics & Fluid Dynamics

• D'Alembert's Principle Rare · D'Alembert's principle
• Lagrange's equations Core · Lagrange's equations
• Hamilton's equations Core · Hamilton's equations
• Moment of inertia Reliable · Moment of inertia; perpendicular/parallel axis theorems
• Motion of rigid bodies in two dimensions Occasional · Motion of rigid bodies in two dimensions
• Equation of Continuity Rare · Equation of continuity
• Euler's equation of motion for inviscid flow Occasional · Euler's equation of motion for inviscid flow
• Potential flow Core · Potential flow; velocity potential
• Two-Dimensional and Axisymmetric Flow Rare · Two-dimensional and axisymmetric flow
• Sources, sinks, doublets Core · Sources, sinks, doublets
• Vortex motion; circulation Occasional · Vortex motion; circulation
• Navier-Stokes equation for a viscous fluid Occasional · Navier-Stokes equation for a viscous fluid