CSS Syllabus — Pure Mathematics

Section A — 40 Marks

I. Modern Algebra

1. Groups

• Groups and subgroups

• Lagrange’s Theorem

• Cyclic groups

• Normal subgroups

• Quotient groups

• Fundamental Theorem of Homomorphism

• Isomorphism Theorems of Groups

• Inner automorphisms

• Conjugate elements

• Conjugate subgroups

• Commutator subgroups

2. Rings and Fields

• Rings

• Subrings

• Integral domains

• Quotient fields

• Isomorphism theorems

• Field extensions

• Finite fields

3. Vector Spaces and Linear Algebra

• Vector spaces

• Linear independence

• Bases

• Dimension of a finitely generated space

• Linear transformations

• Matrices and their algebra

• Reduction of matrices to echelon form

• Rank and nullity of a linear transformation

• Solution of systems of homogeneous and non-homogeneous linear equations

• Properties of determinants

Section B — 40 Marks

II. Calculus & Analytic Geometry

1. Calculus

• Real numbers

• Limits

• Continuity

• Differentiability

• Indefinite integration

• Mean Value Theorems

• Taylor’s Theorem

• Indeterminate forms

• Asymptotes

• Curve tracing

• Definite integrals

• Functions of several variables

• Partial derivatives

• Maxima and minima

• Jacobians

• Double and triple integration (techniques only)

• Applications of Beta and Gamma functions

• Areas and volumes

• Riemann–Stieltjes integral

• Improper integrals and their conditions of existence

• Implicit Function Theorem

2. Analytic Geometry

• Conic sections in Cartesian coordinates

• Plane polar coordinates and their use in representing straight lines and conic sections

• Cartesian and spherical polar coordinates in three dimensions

• The plane

• The sphere

• The ellipsoid

• The paraboloid

• The hyperboloid in Cartesian and spherical polar coordinates

Section C — 20 Marks

III. Complex Variables

• Functions of a complex variable

• De Moivre’s Theorem and its applications

• Analytic functions

• Cauchy’s Theorem

• Cauchy’s Integral Formula

• Taylor’s and Laurent’s series

• Singularities

• Cauchy Residue Theorem

• Contour integration

• Fourier series

• Fourier transforms

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