B.A./B.Sc. First Year Mathematics Syllabus

Calculus I

This course introduces students to the fundamental concepts of differential and integral calculus, including limits, continuity, differentiation, and basic integration techniques.

Key Topics:

• Limits and Continuity: Definitions, properties, and applications
• Differentiation: Rules, chain rule, implicit differentiation, related rates
• Applications of Derivatives: Optimization, curve sketching, mean value theorem
• Integration: Antiderivatives, definite integrals, fundamental theorem of calculus
• Integration Techniques: Substitution, integration by parts, partial fractions
• Applications of Integration: Area between curves, volumes of solids of revolution

Linear Algebra I

This course covers the basic concepts and techniques of linear algebra, including systems of linear equations, matrices, determinants, and vector spaces.

Key Topics:

• Systems of Linear Equations: Matrix representation, Gaussian elimination
• Matrices: Operations, inverses, elementary matrices
• Determinants: Properties, cofactor expansion, Cramer's rule
• Vector Spaces: Subspaces, linear independence, bases, dimension
• Linear Transformations: Matrix representations, kernel and range
• Eigenvalues and Eigenvectors: Diagonalization, applications

Analytical Geometry and Trigonometry

This course focuses on geometric concepts and their algebraic representations, along with advanced trigonometric functions and identities.

Key Topics:

• Coordinate Geometry: Lines, circles, conics, transformations
• Three-Dimensional Geometry: Planes, lines in space, quadric surfaces
• Trigonometric Functions: Graphs, inverse trigonometric functions
• Trigonometric Identities: Pythagorean, sum and difference, double-angle formulas
• Solving Trigonometric Equations: Methods and applications
• Vector Geometry: Dot product, cross product, applications

Calculus II

This course continues the study of calculus, covering advanced integration techniques, sequences, series, and introduces functions of several variables.

Key Topics:

• Advanced Integration Techniques: Trigonometric substitution, improper integrals
• Applications of Integration: Arc length, surface area, work, center of mass
• Sequences: Convergence, monotonic sequences, bounded sequences
• Series: Convergence tests, power series, Taylor and Maclaurin series
• Functions of Several Variables: Limits, continuity, partial derivatives
• Multiple Integrals: Double and triple integrals, change of variables

Discrete Mathematics

This course introduces students to mathematical structures that are fundamentally discrete (as opposed to continuous), with applications in computer science.

Key Topics:

• Logic: Propositional logic, predicate logic, logical equivalences
• Proofs: Direct, contrapositive, contradiction, and induction proofs
• Set Theory: Operations, relations, equivalence relations, partial orders
• Functions: One-to-one, onto, inverse functions, composition
• Number Theory: Divisibility, prime numbers, modular arithmetic
• Combinatorics: Permutations, combinations, pigeonhole principle
• Recurrence Relations: Linear recurrence relations, solving methods

Probability and Statistics I

This course provides an introduction to probability theory and basic statistical concepts, forming the foundation for data analysis.

Key Topics:

• Descriptive Statistics: Measures of central tendency, dispersion, visualization
• Probability: Sample spaces, events, probability axioms
• Conditional Probability: Bayes' theorem, independence
• Random Variables: Discrete and continuous distributions
• Expectation: Mean, variance, moment generating functions
• Common Distributions: Binomial, Poisson, geometric, normal, exponential
• Joint Distributions: Marginal and conditional distributions

Calculus

• Thomas' Calculus by George B. Thomas Jr. and Maurice D. Weir
• Calculus: Early Transcendentals by James Stewart
• Calculus by Michael Spivak

Linear Algebra

• Introduction to Linear Algebra by Gilbert Strang
• Linear Algebra and Its Applications by David C. Lay
• Linear Algebra Done Right by Sheldon Axler

Analytical Geometry

• Coordinate Geometry by Loney
• Analytic Geometry by George F. Simmons

Discrete Mathematics

• Discrete Mathematics and Its Applications by Kenneth H. Rosen
• Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik

Probability and Statistics

• Introduction to Probability by Joseph K. Blitzstein and Jessica Hwang
• Probability and Statistics by Morris H. DeGroot
• Mathematical Statistics with Applications by Dennis Wackerly

Students in the first year mathematics program are typically assessed through a combination of:

• Written Examinations: Mid-term and end-of-semester exams testing theoretical understanding and problem-solving skills.
• Assignments: Regular problem sets to reinforce concepts and provide feedback.
• Quizzes: Short assessments to check understanding of specific topics.
• Project Work: Small projects exploring applications of mathematical concepts.
• Class Participation: Engagement in discussions and problem-solving sessions.

• Practice consistently: Mathematics requires regular practice to develop problem-solving skills.
• Focus on understanding proofs: Understanding why mathematical statements are true is as important as knowing how to apply them.
• Form study groups: Collaborating with peers can enhance understanding and provide new perspectives.
• Seek help early: If you're struggling with a concept, ask instructors or teaching assistants for clarification.
• Connect concepts: Try to understand how different mathematical topics relate to each other.
• Use technology wisely: Utilize mathematical software and online resources as supplements, not replacements for understanding fundamentals.