Dhaka University - First Year Mathematics Syllabus Notes

Dhaka University Affiliated Colleges - First Year Mathematics Syllabus

Overview

  • This document outlines the syllabus for the first year of the Four-Year B.S. Honours Course in Mathematics for Dhaka University Affiliated Colleges, effective from the 2017-2018 session.

Course Structure and Marks Distribution

  • First Year Courses:
    • MAT 101: Fundamentals of Mathematics (100 Marks, 3 Credits)
    • MAT 102: Calculus I (100 Marks, 3 Credits)
    • MAT 103: Linear Algebra I (100 Marks, 3 Credits)
    • MAT 104: Analytic and Vector Geometry (100 Marks, 3 Credits)
    • MAT 150: Math Lab I (100 Marks, 2 Credits)
    • COM 100: History of the Emergence of Independent Bangladesh (100 Marks, 4 Credits)
    • Minor Subjects (Choose Two):
      • Physics (6 Credits)
      • Chemistry (6 Credits)
      • Statistics (6 Credits)
      • Economics (6 Credits)

Detailed Syllabus

MAT 101: Fundamentals of Mathematics
  • Marks: 100
  • Credits: 3
  • Hours: 45
  • Elements of Logic:
    • Mathematical statements and logical connectives.
    • Conditional and bi-conditional statements.
    • Truth tables and tautologies.
    • Quantifiers.
    • Logical implication and equivalence.
    • Deductive reasoning.
  • Set Theory:
    • Sets and subsets, set operations.
    • Cartesian product of two sets.
    • Operations on family of sets.
    • De Morgan’s laws.
  • Relations and Functions:
    • Relations, Order relation, Equivalence relations.
    • Functions, Images and inverse images of sets.
    • Injective, surjective, and bijective functions.
    • Inverse functions.
  • Real Number System:
    • Field and order properties.
    • Natural numbers, Integers, and rational numbers.
    • Absolute value and their properties.
    • Basic inequalities (inequalities of means, powers, Cauchy, Chebyshev, Weierstrass).
  • Complex Number System:
    • Field of Complex numbers.
    • De Moivre's theorem and its applications.
  • Theory of Equations:
    • Relations between roots and coefficients.
    • Symmetric functions of roots.
    • Sum of the powers of roots.
    • Synthetic division.
    • Des Cartes rule of signs.
    • Multiplicity of roots.
    • Transformation of equation.
  • Elementary Number Theory:
    • Divisibility.
    • Fundamental theorem of arithmetic.
    • Congruences (basic properties only).
  • Summation of Series:
    • Summation of algebraic and trigonometric series.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 3 hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • S. Lipschutz, Set Theory, Schaum’s Outline Series.
    • S. Barnard & J. M. Child, Higher Algebra.
    • W.L. Ferrar, Algebra.
    • P.R. Halmos, Naive Set Theory.
    • H. S. Hall and S. R. Knight, Higher Algebra.
MAT 102: Calculus I
  • Marks: 100
  • Credits: 3
  • Hours: 45
  • Functions & Their Graphs:
    • Polynomial and rational functions.
    • Logarithmic and exponential functions.
    • Trigonometric functions & their inverses.
    • Hyperbolic functions & their inverses.
    • Combinations of such functions.
  • Limit and Continuity:
    • Definitions and basic theorems on limit and continuity.
    • Limit at infinity & infinite limits, Computation of limits.
    • Indeterminate forms (L’Hospital’s rule).
  • Differentiation:
    • Tangent lines and rates of change.
    • Definition of derivative.
    • One-sided derivatives.
    • Rules of differentiation (proofs and applications).
    • Successive differentiation.
    • Leibnitz's theorem (proof and application).
    • Related rates.
    • Linear approximations and differentials.
  • Applications of Differentiation:
    • Rolle’s theorem.
    • Mean value theorem.
    • Maximum and minimum values of functions and related problems.
    • Concavity and points of inflection.
    • Optimization problems.
  • Integration:
    • Antiderivatives and indefinite integrals.
    • Techniques of integration.
    • Definite integration using antiderivatives.
    • Fundamental theorems of calculus (proofs and applications).
    • Basic properties of integration. Integration by reduction.
  • Applications of Integration:
    • Arc lengths.
    • Plane areas.
    • Surfaces of revolution.
    • Volumes of solids of revolution.
    • Volumes by cylindrical shells.
    • Volumes by cross sections.
  • Graphing in Polar Coordinates:
    • Tangents to polar curves.
    • Arc length in polar coordinates.
    • Areas in polar coordinates.
  • Improper Integrals:
    • Tests of convergence and their applications.
    • Gamma and Beta functions.
  • Approximation and Series:
    • Taylor polynomials and series.
    • Convergence of series.
    • Taylor's series.
    • Taylor's theorem and remainders.
    • Differentiation and integration of series.
    • Validity of Taylor expansions and computations with series.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 3 hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • H. Anton, I. C. Bivens and S. Davis, Calculus: Early Transcendentals, Wiley.
    • E.W. Swokowski, Calculus with Analytic Geometry, Brooks/Cole.
    • G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry, Addison Wesley.
    • J. Stewart, Single Variable Calculus: Early Transcendentals, Cengage Learning.
    • G. Strang, Calculus, Wellesley-Cambridge.
    • R. Larson, R. P. Hostetler, F. H. Edwards and D. E. Heyd, Calculus with Analytic Geometry, Houghton Mifflin College Div.
MAT 103: Linear Algebra I
  • Marks: 100
  • Credits: 3
  • Hours: 45
  • Matrices and Determinants:
    • Notion of matrix, types of matrices.
    • Algebra of matrices.
    • Determinant function, properties of determinants.
    • Minors, Cofactors, expansion and evaluation of determinants.
    • Elementary row and column operations and row-reduced echelon matrices.
    • Invertible matrices.
    • Different types of matrices.
    • Rank of matrices.
  • Vectors in RnR^n and CnC^n:
    • Review of geometric vectors in R2R^2 and R3R^3 spaces.
    • Vectors in RnR^n and CnC^n.
    • Inner product.
    • Norm and distance in RnR^n and CnC^n.
  • System of Linear Equations:
    • System of linear equations (homogeneous and non-homogeneous) and their solutions.
    • Application of matrices and determinants for solving system of linear equations.
    • Applications of system of equations in real life problems.
  • Vector Spaces:
    • Notion of groups and fields.
    • Vector spaces, subspaces.
    • Linear combination of vectors.
    • Linear dependence of vectors.
    • Basis and dimension of vector spaces.
    • Row and column space of a matrix.
    • Rank of matrices.
    • Solution spaces of systems of linear equations.
  • Linear Transformation:
    • Linear transformations.
    • Kernel and image of a linear transformation and their properties.
    • Matrix representation of linear transformations.
    • Change of bases.
  • Eigenvalues and Eigenvectors:
    • Eigenvalues and Eigenvectors.
    • Diagonalization.
    • Cayley-Hamilton theorem and its application.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 3 hours): 70 Marks (Answer any five out of eight questions).
  • Books Recommended:
    • Howard Anton & Chris Rorres – Elementary Linear Algebra with Application.
    • Seymour Lipschutz (Schaum's Outline Series) - Linear Algebra.
    • Md. Abdur Rahman - Linear Algebra.
MAT 104: Analytic and Vector Geometry
  • Marks: 100
  • Credits: 3
  • Hours: 45
  • Two-dimensional Geometry:
    • Transformation of coordinates.
    • Pair of straight lines (homogeneous second-degree equations).
    • General second-degree equations representing pair of straight lines.
    • Angle between pair of straight lines.
    • Bisectors of angle between pair of straight lines.
    • General equations of the second degree (reduction to standard forms, identifications, properties, and tracing of conics).
  • Three-dimensional Geometry:
    • Coordinates, Distance.
    • Direction cosines and direction ratios.
    • Planes (equation of a plane, angle between two planes, distance of a point from a plane).
    • Straight lines (equations of lines, relationship between planes and lines, shortest distance).
    • Spheres, Conicoids (basic properties).
  • Vector Geometry:
    • Vectors in plane and space.
    • Algebra of vectors.
    • Rectangular Components.
    • Scalar and Vector products.
    • Triple scalar product.
    • Applications of vectors to geometry (vector equations of straight lines and planes, areas, and volumes).
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 3 hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • A.F.M. Abdur Rahman & P.K. Bhattacharjee, Analytic Geometry and Vector Analysis.
    • Khosh Mohammad, Analytic Geometry and Vector Analysis.
    • J. A. Hummel, Vector Geometry.
    • H. Anton, I. C. Bivens and S. Davis, Calculus: Early Transcendentals, Wiley.
    • E.W. Swokowski, Calculus with Analytic Geometry, Brooks/Cole; Alternate.
MAT 150: Math Lab I
  • Marks: 100
  • Credits: 2
  • Hours: 45
  • Problem-solving in concurrent courses (e.g., Algebra, Calculus, Linear Algebra, and Geometry) using MATHEMATICA/MATLAB.
  • Lab Assignments: At least 5 lab assignments.
  • Evaluation:
    • Internal Assessment (Laboratory works): 20 marks
    • Final Examination (Lab, 3 hours): 80 marks

Mathematics Minor Courses

  • For Honours Students of Departments Other Than Mathematics
First Year
  • MAM 101: Fundamentals of Mathematics (100 Marks, 2 Credits)
  • MAM 102: Calculus I (100 Marks, 2 Credits)
  • MAM 103: Analytic and Vector Geometry (100 Marks, 2 Credits)
  • MAM 104: Linear Algebra (100 Marks, 2 Credits)
Detailed Syllabi for Minor Courses
MAM 101: Fundamentals of Mathematics
  • Marks: 100
  • Credits: 2
  • Hours: 30
  • Content:
    1. Sets and subsets, Set operations, Family of Sets, De Morgan’s laws. Relations and functions: Cartesian product of sets. Relations. Equivalence relations. Functions. Images and inverse images of sets. Injective, surjective, and bijective functions. Inverse functions.
    2. The Real number system: Field and order properties. Natural numbers, integers, and rational numbers. Absolute value. Basic inequalities. (including inequalities involving means, powers; inequalities of Cauchy, Chebyshev, Weierstrass).
    3. The Complex number system: Geometrical representation Polar form. De Moivre’s theorem and its applications. Elementary number theory: Divisibility. Fundamental theorem of arithmetic. Congruences (basic properties only).
    4. Summation of finite series: Arithmetic-geometric series. Method of difference. Successive differences.
    5. Theory of equations: Synthetic division. Number of roots of polynomial equations. Relations between roots and coefficients. Multiplicity of roots. Symmetric functions of roots. Transformation of equations.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 2 ½ hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • S. Lipschutz, Set Theory, Schaum’s Outline Series.
    • S. Barnard & J. M. Child, Higher Algebra.
    • W.L. Ferrar, Algebra.
    • P.R. Halmos, Naive Set Theory.
MAM 102: Calculus I
  • Marks: 100
  • Credits: 2
  • Hours: 30
  • A. Differential Calculus
    1. Functions and their graphs (polynomial and rational functions, logarithmic and exponential functions, trigonometric functions and their inverses, hyperbolic functions and their inverses, combination of such functions). Limits of Functions: definition. Basic limit theorems (without proofs).
    2. Limit at infinity and infinite limits. Continuous functions. Properties Continuous functions on closed and boundary intervals (no proofs required).
    3. Differentiation: Tangent lines and rates of change. Definition of derivative. One-sided derivatives. Rules of differentiation (with applications). Linear approximations and differentials. Successive differentiation. Leibnitz theorem. Rolle’s theorem: Lagrange’s mean value theorems. Extrema of functions, problems involving maxima and minima.
  • B. Integral Calculus
    1. Integrals: Antiderivatives and indefinite integrals. Techniques of integration. Definite integration using antiderivatives.
    2. Definite integral as a limit of a sum. The fundamental theorem of calculus. Integration by reduction.
    3. Application of integration: Plane areas. Solids of revolution. Volumes by cylindrical shells. Volumes by cross-sections. Arc length and surface of revolution.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 2 ½ hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • H. Anton et al, Calculus with Analytic Geometry.
    • E.W. Swokowski, Calculus with Analytic Geometry.
    • L. Bers & P. Karal, Calculus.
    • S. Lang, A First Course in Calculus.
MAM 103: Analytic and Vector Geometry
  • Marks: 100
  • Credits: 2
  • Hours: 30
  • Two-dimensional geometry
    1. Coordinates in two dimension. Transformations of coordinates.
    2. Reduction of second-degree equations to standard forms. Pairs of straight lines. Identifications of conics. Equations of conics in polar coordinates.
  • Three-dimensional geometry
    1. Coordinates in three dimensions. Direction cosines, and direction ratios.
    2. Planes, straight lines, and conicoids (basic definitions and properties only)
  • Vector geometry
    1. Vectors in plane and space. Algebra of vectors. Scalar and vector products. Triple scalar products. Applications to Geometry.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 2 ½ hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • A.F.M. Abdur Rahman & P.K. Bhattacharjee, Analytic Geometry and Vector Analysis.
    • Khosh Mohammad, Analytic Geometry and Vector Analysis.
    • J. A. Hummel, Vector Geometry.
MAM 104: Linear Algebra
  • Marks: 100
  • Credits: 2
  • Hours: 30
  • Content:
    1. Matrices and Determinants: Notion of matrix. Types of matrices. Matrix operations, laws of matrix Algebra. Determinant function. Properties of determinants. Minors, Cofactors, expansion and evaluation of determinants. Elementary row and column operations and row-reduced echelon matrices. Invertible matrices. Block matrices.
    2. System of Linear Equations: Linear equations. System of linear equations (homogeneous and non-homogeneous) and their solutions. Application of matrices and determinants for solving system of linear equations.
    3. Vector Spaces: Vectors in RnR^n and CnC^n: Review of geometric vectors in R2R^2 and R3R^3 space. Vectors in RnR^n and CnC^n. Inner product. Norm and distance in RnR^n and CnC^n. Abstract vector space over R and C. Subspace. Sum and direct sum of subspaces. Linear independence of vectors; basis and dimension of vector spaces. Row and column space of a matrix; rank of matrices. Solution spaces of systems of linear equation.
    4. Linear transformations. Kernel and image of a linear transformation and their properties. Matrix representation of linear transformations. Change of bases.
    5. Eigenvalues and eigenvectors. Diagonalization. Cayley Hamiton theorem. Applications.
  • Evaluation:
    • In-course Assessment: 30 Marks
    • Final Examination (Theory, 2 ½ hours): 70 Marks (Answer any five out of eight questions).
  • References:
    • H. Anton, and C.Rorres, Linear Algebra with Applications, 7th Edition
    • S. Lipshutz, Linear Algebra, Schaum’s Outline Series.
    • W. Greub, Linear Algebra.