# Mathematics Masters Program

Announcement (12-June-2014)

The examination topics for the Entrance Examination to the Master's Programme are as follows:

Calculus (40)

Foundations of Mathematics (15)

Linear Algebra (15)

Mathematical Analysis (15)

Ordinary Differential Equations (15)

Note: The number in brackets next to each subject indicates the relative weight that subject in the Examination.

More detailed information on the subtopics of each topic is given below:

1. Calculus
1. Differentiation
2. Curve Sketching
3. Integration
4. Double and Triple Integrals
5. Polar and Cylindrical Coordinate Systems
6. Infinite Series
7. Series Tests for Convergence or Divergence
8. Taylor Series
9. Partial Derivatives
10. Three-Dimensional Coordinate Systems
11. Vector Algebra

2. Foundations of Mathematics
1. Mathematical Logic: General Statements, Open Names, Inference, Quantifiers
2. Mathematical Proofs: Mathematical Induction, Direct Proof, Proof by Contradiction
3. Set theory: Operations on sets, DeMorgan’s Laws and Power Sets
4. Relations: Composition of Relations, Types of Relations and Invertible Relations, Partitions and Equivalence Relations
5. Functions: Composition of Functions, One-to-One and Onto Functions and Invertible Functions, Injective, Surjective and Bijective Functions
6. Denumberable and Nondenumberable Sets
7. Cardinal Numbers

3. Linear Algebra
1. Systems of Linear Equations
2. Row Reduction and Echelon Forms
3. Solutions to Linear Systems of Equations
4. Linear Independence and Bases
5. Change of Basis
6. Matrix Operations
7. Invertible Matrices
8. LU Factorisation
9. Dimension and Rank
10. Linear Transformations
11. Eigenvalues and Eigenvectors
12. The Characteristic Equation
13. Orthogonality
14. The Gram-Schmidt Process

4. Mathematical Analysis
1. The Real Number System
2. Sequence of Real Numbers
3. Metric Spaces and Convergent Sequences in Metric Spaces
4. Continuous Functions in Metric Spaces
5. Compactness and Connectedness in Metric Spaces
6. The Riemann Integral

5. Ordinary Differential Equations
1. Solving First and Second ODEs and their Applications
2. Solving ODEs using the Series Method
3. Systems of ODE's
4. Solving Differential Equations using Laplace Transforms