SINGHANIA UNIVERSITY

CURRICULUM AND SYLLABUS

Bachelor of Science in Mathematics

SINGHANIA UNIVERSITY

BACHELOR OF SCIENCE IN MATHEMATICS

SYLLABUS

F.Y.B.Sc.

Compulsory English

Term-I Term-II

Prose- 1, 2, 3, 4 Prose- 5, 6, 7, 8

Poetry- 9, 10, 11 Poetry- 12, 13, 14

Grammar- 1, 2, 3 Grammar- 4, 5

Communication Skills- 1,2,3,4,5 Communication Skills- 6,7,8,9,10

Prose1. An Astrologer’s Day - R.K Narayan

2. Our Urgent Need of Self-esteem - Nathaniel Branden

3. The Gift of Magi - O’ Henry

4. Karma - Khushwant Singh

5. Tryst with Destiny - Jawaharlal Nehru

6. Youth and the Tasks Ahead - Karan Singh

7. Prospects of Democracy in India - B. R. Ambedkar

8. The Eyes are not Here - Ruskin Bond

Poetry9. A Red, Red Rose - Robert Burns

10. Where the Mind is without Fear - Rabindranath Tagore

11. If You Call Me - Sarojini Naidu

12. Upon Westminster Bridge - William Wordsworth

13. An old Woman - Arun Kolatkar

14. Success is Counted Sweetest - Emily Dickinson

Grammar and Communication SkillsGrammar :

1. Articles2. Prepositions

FIRST YEAR

3. Verbs :

3.1 Regular and Irregular Verbs

3.2 Auxiliaries (Primary and Modal)

4. Tenses :

4.1 Present tense:

A) Simple present, B) Present progressive, C) Present perfect, D) Presentperfect progressive.

4.2 Past tense:

A) Simple past, B) Past progressive, C) Past perfect, D) Past perfect pro-gressive

4.3 Future tense :

A) Simple future, B) future progressive, C) Future perfect, D) Future perfectprogressive

5. Subject-Verb Agreement (Concord)

Communication skills:

1. Taking Leave

2. Introducing Yourself

3. Introducing People to One Another

4. Making Requests and Asking for Directions

5. Making and Accepting an Apology

6. Inviting and Accepting/Declining an Invitation

7. Making a Complaint

8. Congratulating, Expressing Sympathy and Offering Condolences

9. Making Suggestions, Offering Advice and Persuading

10. Expressing Agreement/Disagreement and Seeking Clarification

F. Y. B.Sc. Mathematics

Applied Mathematics - I

Unit No. Topic

1 Numbers, Sets, and Functions 1.1. The Quadratic Formula 1.2. Elementary Inequalities 1.3. Sets 1.4. Functions 1.5. How to Approach Problems

2 Language and Proofs 2.1. Two Theorems about Equations 2.2. Quantifiers and Logical Statements 2.3. Compound Statements 2.4. Elementary Proof Techniques, 2.5. How to Approach Problems

3 Induction 3.1. The Principal of Induction 3.2. Applications 3.3. Strong Induction 3.4. How to Approach Problems

4 Bijection and Cardinality 4.1. Representation of Natural Numbers 4.2. Bijections 4.3. Injection and surjections 4.4. Composition of Functions 4.5. Cardinality 4.6. How to Approach Problems from the fields of

5 Combinatorial Reasoning 5.1. Arrangements and Selections 5.2. Binomial Coefficients 5.3. Permutations 5.4. Functional Digraphs 5.5. How to Approach Problems

6 Divisibility 6.1. Factors and Factorization 6.2. The Euclidean Algorithm 6.3. The Dart Board Problem 6.4. Polynomials

Syllabus

F. Y. B.Sc. Mathematics

Algebra & Geometry

Unit No. Topic

1 Integers 1.1 Well Ordering Principle for N. Principle of Mathematical induction (strong form). 1.2 Divisibility in Z: Definition and elementary properties. Division Algorithm, Euclidean Algorithm (Without proof) G.C.D. and L.C.M of integers, Relatively prime integers, Definition Prime numbers ,Euclid’s lemma, Basic properties of G.C.D., G.C.D of any two integers and if it exists is unique

and can be expressed in the form ax+by, where x,yZ. 1.3 Equivalence Relations, Equivalences classes, properties of Equivalences classes, Definition of partition, every partition gives an equivalence relation and vice-versa, Definition of Congruence, Congruence as equivalence relation on , Residue classes, Partition of , Addition modulo n , Multiplication modulo n.

2 Polynomials 2.1 Definition of polynomial, Degree of polynomial, Algebra of polynomials, Division algorithm (without proof). G.C.D of two polynomials (without proof). 2.2 Remainder Theorem, Factor Theorem. 2.3 Relation between the roots and the coefficients of a polynomial, Examples.

3 Matrices and System of linear equations 3.1 Matrices, Echelon and Reduced echelon form of a matrix, Reduction of matrix to its echelon form, Definition of rank of a matrix by using echelon form. 3.2 System of linear equations, Matrix form of system of linear equations, Homogeneous and non-homogeneous system of linear equations, Gauss Elimination and Gauss Jordan Method. 3.3 Consistency of a system of linear equations, condition of consistency (without proof). 3.4 Eigen values, Eigen vectors, characteristic equation of a matrix of order up to 3×3 3.5 Statement of Cayley Hamilton theorem and its use to find the inverse of a matrix.

Syllabus

4 Analytical Geometry of two dimensions:

4.1 Change of axes, Translation and rotation.

4.2 Conic Section: General equation of second degree in x

and y. Centre of conic, Nature of conic, Reduction to

standard form.

5 Planes in 3-dimension:

Revision: Equations of the first degree in x, y, z, Transformation to the normal form, determination of plane under given conditions, Equations of the plane through three given points.

5.1 Systems of planes, two sides of a plane.

5.2 Length of the perpendicular from a point to a plane, bisectors of angles between two planes.

5.3 Joint equation of two planes, Angle between planes.

6 Lines in 3-dimension:

Revision: Equations of a line, equations of a straight line in terms of its direction cosines and the co-ordinates of a point on it, equations of a line through two points, Symmetrical and unsymmetrical forms of the equations of a line. transformation of the equations of a line to the symmetrical form. Angle between a line and a plane.

6.1 The condition that a given line may lie in a given plane, the condition that two given lines are coplanar.

6.2 Number of arbitrary constants in the equations of a straight line, sets of conditions which determine a line.

6.3 The shortest distance between two lines, the length and equations of the line of shortest distance between two straight lines, length of perpendicular from a given point to a given line.

7 Sphere

7.1 Definition and equation of the sphere in various forms.

7.2 Plane section of a sphere, intersection of two spheres.

7.3 Equation of a circle, sphere through a given circle, intersection of a sphere and a line.

7.4 Equation of a tangent plane

8 Cones and Cylinders:

8.1 Definition of cone and cylinder.

8.2 Equation of cone and cylinder with vertex at origin and

(,,).

8.3 The right circular cone, equation of a right circular cone.

8.4 The right circular cylinder, equation of a right circular cylinder.

F. Y. B. Sc.

Calculus and Differential Equations

Unit No. Topic

1 The Real Numbers:

1.1 Algebraic properties of R,

1.2 Order properties of R, Iintervals in R, neighborhoods and deleted

neighborhoods of a real number, bounded subsets of R. 1.3 The Completeness Property of R, denseness of Q in R.

2 Limit and Continuity :

2.1 - definition of limit of a function, Basic properties of limits.

2.2 Continuity of function at a point, Types of discontinuity.

2.3 Continuous functions on intervals.

2.4 Properties of continuous functions on closed and bounded interval. (i) Boundedness. (ii) Attains its bounds. (iii) Intermediate value theorem

3 Differentiation :

3.1 Definition of derivative of a real valued function at a point, notion of

differentiability, geometric interpretation of a derivative of a real

valued function at a point.

3.2 Differentiability of a function over an interval.

3.3 Statement of rules of differentiability, chain rule of finding derivative

of composite of differentiable functions (without proof), derivative of

an inverse function.

3.4. Mean Value Theorems: Rolle•fs Theorem, Lagrange•fs Mean

Value Theorem, Cauchy•fs Mean Value Theorem

3.5 Indeterminate forms. L-Hospitals rule.

3.6 Higher order derivatives, examples, Leibnitz Theorem and its

applications 3.7 Taylor•fs and Maclaurin•fs Theorem with Lagrange•fs form of

remainder (without proof), Examples with assuming convergence of series.

4 Integration :

4.1 Integration of rational function by using partial fraction.

4.2 Integration of some irrational functions:

i) where n is a positive integer, ii)

Syllabus

iii)

4.3 Reduction formula

i) ii) , n is a positive integer

iii) iv) v) 5 Differential Equations of first order and first degree:

5.1 Introduction to function of two, three variables, homogenous

functions, Partial derivatives.

5.2 Differential equations, General solution of Differential equations.

5.3 Methods of finding solution of Differential equations of first order

and first degree, Variable separable form, Homogenous Differential

equations, Differential equations reducible to homogeneous form.

Exact Differential equations.

Differential equations reducible to exact Differential equations, Integrating factors, Linear Differential equations. Bernoulli’s Differential equations.

6 Application of Differential Equations :

6.1 Orthogonal trajectories.

6.2 Kirchhofff’s law of electrical circuit (RC & LR Circuit)

7 Methods of finding general solution of Differential Equations of

first order and higher degree:

7.1 Equations solvable for p.

7.2 Equations solvable for x.

7.3 Equations solvable for y.

7.4 Equation in Clairaut’s form.

F. Y. B. Sc.

Discrete Mathematics

Unit No.

1 Logic and Proofs: 1.1 Propositional logic. 1.2 Propositional equivalences. 1.3 Predicates and quantifiers. 1.4 Nested quantifiers. 1.5 Rules of inference. 1.6 Introduction to proofs.

2 Counting: 2.1 The basics of counting. 2.2 Permutation and combinations. 2.3 Generalized permutation and combinations.

3 Advanced Counting Technique: 3.1 Inclusion-Exclusion (without proof).

4 The Laplace Transform: 4.1 Definition, Laplace Transform of some elementary functions. 4.2 Some important properties of Laplace Transform. 4.3 Laplace Transform of derivatives, Laplace Transform of

Integrals. 4.4 Methods of finding Laplace Transform, Evaluation of

Integrals. 4.5 The Gamma function, Unit step function and Dirac delta

function.

5 The Inverse Laplace Transform: 5.1 Definition, Some inverse Laplace Transform. 5.2 Some important properties of Inverse Laplace Transform. 5.3 Inverse Laplace Transform of derivative, Inverse Laplace

Transform of integrals. 5.4 Convolution Theorem, Evaluation of Integrals.

6 Applications of Laplace Transform: 6.1 Solution of Ordinary Differential Equations with constant

coefficients.

7 Fourier Series : 7.1 Definition and examples of Fourier Series.

Syllabus

Topic

F. Y. B. Sc. Mathematics

Linear Algebra

Unit

No. Topic

1 Vector Spaces

Definition, examples, linear dependence, basis and dimension,

vector subspace, Necessary and sufficient condition for

subspace, vector space as a direct sum of subspaces.

2 Inner Product Spaces

Inner product, norm as length of a vector, distance between two

vectors, orthonormal basis, orthonormal projection,Gram Schmidt

processs of ortogonalization, null space, range space, rank,

nullity, Sylvester Inequality.

3 Linear Transformations

Definition, examples, properties of linear transformations, equality

of linear transformations, kernel and rank of linear

transformations, composite transformations, Inverse of a linear

transformation, Matrix of a linear transformation, change of basis,

similar matrices

Syllabus

SyllabusF.Y.B.Sc. Mathematics

Introduction to Mathematics

Unit

No.

Topic

NumbersNatural, whole, integers, rational, irrational, real numbersOperations on numbers: addition, subtraction, multiplication

and divisionUse of brackets

Indices, squares, square roots, cube, cube roots.

Unitary method, variation- direct and inverse

Ratio and proportionB) Algebra

Basics of Algebra

Use of letters in place of numbers.

Algebraic expressions, addition, subtraction, multiplication and

division of algebraic expressions.

Polynomials: factors and multiples.

Identities.

Equations: equations with one variable, linear equations in two

variables, quadratic equations.

Basics of Geometry

Basic concepts

Unit - IA) Arithmetic

Unit - IIA) Geometry

Angles, pairs of angles, triangles and quadrilaterals-types and

properties.

Triangles- congruence and similarity.

Circle – basic concepts, circumference, area, theorems.

Quadrilaterals – properties of different quadrilaterals, theorems.

Geometric constructionB) Applied Mathematics

Mathematics in day to day life

Profit and loss

Percentages

Simple and compound interest

Discount and commission

Statistics – measures of central tendency and variability,

Graphs.

Mensuration – Area and volumes of different geometrical

figures.

Co- ordinate geometry.

S. Y. B.Sc. Mathematics

Applied Mathematics-II

Unit

No. Topic

1 Modular Arithmetic :

1.1 Relations

1.2. Congruences

1.3. Applications

1.4. Fermat’s Little Theorem

1.5. Congruence and Groups

2 Two Principles of Counting:

2.1. The Pigeonhole Principle

2.2. The Inclusion

3 Graph Theory :

3.1. The Königsberg Bridge Problem

3.2. Isomorphism of Graphs

3.3. Connection and Trees

3.4. Bipartite graphs

3.5. Coloring Problems

3.6. Planar Graphs

4 Recurrence Relations:

4.1. General Properties

4.2. First-Order Recurrences

4.3. Second-Order Recurrences

4.4. General Linear Recurrences

4.5. Other Classical Recurrences

4.6. Generating Functions

Syllabus

SECOND YEAR

S. Y. B.Sc. Mathematics

Real Analysis-I

Unit

No.

1 Sets and functions:

Operations on sets, Functions, Real-valued functions,

Equivalence countability, Real numbers, Cantor set, Least

upper bounds.

2 Sequences of Real Numbers:

Definition of sequence and subsequence, Limit of a sequence,

Convergent sequences, Monotone sequences, Divergent

sequences, Limit superior, Limit inferior, Cauchy sequences.

3 Series of Real numbers:

Convergent and divergent series, series with non-negative

terms, alternating series, Conditional and Absolute

convergence, Rearrangement of series, Tests of absolute

convergence, series whose terms form a non-increasing

sequence, The class l2.

Syllabus

Topic

S. Y. B.Sc. Mathematics

Computational Geometry

Unit

No.

1 Two-dimensional Transformations Representation of Points in Two-dimensional Plane

Transformations and Matrices

Transformation of Points

Transformation of Straight Lines

Midpoint Transformation

Transformation of Parallel Lines

Transformation of Intersecting Lines

Transformation: Rotations, Reflections , Scaling, Shearing

Concatenation of Transformations or Combined

Transformations

Transformation of a Unite Square

Solid Body Transformations

Transformation and Homogenous Co-ordinates

Rotation about an Arbitrary Point

Reflection Through an Arbitrary Line

Projection- A Geometric Interpretation of Homogeneous Co-

ordinates

Overall Scaling

Points of Infinity

2 Three-dimensional Transformations

Three Dimensional- Scaling, Shearing, Rotation, Reflection,

Translation

Multiple Transformations (Concatenated Translation)

Rotation about an axis Parallel to Co-ordinate Axes, an

Arbitrary Axis in Space

Syllabus

Topic

Reflection through Co-ordinate Planes, Planes Parallel to

Co-ordinate Planes, Arbitrary Planes

Affine and Perspective Transformations

Orthographic Projections

Axonometric Projections

Oblique Projections

Single Point Perspective Transformations

Vanishing Points

3 Plane Curves Curve Representation

Non-parametric Curves

Parametric Curves

Parametric Representation of a Circle and Generation of

Circle

Parametric Representation of an Ellipse and Generation of

Ellipse

Parametric Representation of a Parabola and Generation of

Parabolic Segment Generation of n Points on Hyperbola

4 Space Curves Bezier Curves

S. Y. B.Sc. Mathematics

Ordinary Differential Equations

Unit

No.

1 Linear Differential Equations with constant coefficients:

The auxiliary equations. Distinct roots, repeated roots,

Complex roots, particular solution. The operator 1/f (D) and

its evaluation for the functions eaxu & xu and the operator

1/(D2+a2) acting on sin ax and cost ax with proofs.

2 Non-Homogeneous Differential Equations:

Method of undetermined coefficients, Method of variation of

parameters, Method of reduction of order, The use of a

known solution to find another.

3 Power series solutions:

Introduction and review of power series, Linear equations

and power series, Convergence of power series,Ordinary

points and regular singular points.

4 System of First-Order Equations:

Introductory remarks, linear systems, homogeneous linear

systems with constant Coefficients, Distinct roots, repeated

roots, Complex roots.

Syllabus

Topic

S. Y. B.Sc. Mathematics

Number Theory

Unit

No.

1 Divisibility :

Divisibility in integers, Division Algorithm, GCD, LCM,

Fundamental theorem of Arithmetic, Infinitude of primes, Mersene

Numbers and Fermat Numbers.

2 Congruences :

Properties of Congruences, Residue classes, complete and

reduced residue system, their properties, Fermat•fs theorem.

Euler’s theorem, Wilson’s theorem, x2 º -1 (mod p) has a solution

if and only if p = 2 or p º 1 (mod 4), where p is a prime. Linear

Congruences of degree 1, Chinese remainder theorem.

3 Greatest integer function:

Arithmetic functions Euler’s function, the number of divisors d(n),

sum of divisors s(n), w(n) and W(n). Multiplicative functions, Mo

bius function, Mo bius inversion formula.

4 Quadratic Reciprocity:

Quadratic residues, Legendre• fs symbol. Its properties, Law of

quadratic reciprocity.

5 Diophantine Equations :

Diophantine Equations ax + by = c and Pythagorean triplets.

Syllabus

Topic

S. Y. B.Sc. Mathematics

Multivariable Calculus- I Unit

No.

1 Limit and Continuity of Multivariable functions:

1.1. Functions of several variables, graphs and level curves of

function of two variables.

1.2. Limit and Continuity in higher dimensions.

2 Partial Derivatives:

2.1. Definition and examples.

2.2. Second order partial derivative, the mixed derivative

theorem.

2.3. Partial derivatives of higher order

3 Differentiability:

3.1. Differentiability, the increment theorem for functions of two

variables (without proof).

3.2. Chain rules for composite function.

3.3. Directional derivatives, gradient vectors.

3.4. Tangent planes, normal lines and differentials.

4 Extreme Values:

4.1. Extreme values, First derivative test and Second derivative

test for local extreme values.

4.2. Lagrange’s multipliers method for finding extreme values of

constraint function (One Constraint)

4.3. Taylors Formula for two variables.

5 Multiple Integrals:

5.1. Double Integral over rectangles, Fubini’s theorem for

calculating double integrals (Without proof).

5.2. Double integrals in polar form.

5.3. Triple integrals in rectangular coordinates.

5.4. Triple integral in cylindrical and spherical coordinates.

5.5. Substitution in multiple integrals, Application to area and

volumes.

Syllabus

Topic

T. Y. B. Sc. Mathematics

Complex Analysis

Unit No. Topic

1 Complex Numbers Sums and products, Basic algebraic properties, Further properties, Vectors and Moduli, Complex Conjugates, Exponential Form, Products and powers in exponential form, Arguments of products and quotients, Roots of complex numbers, Examples, Regions in the complex plane.

2 Analytic functions Functions of Complex Variables, Limits, Theorems on limits, Limits involving the point at infinity, Continuity, Derivatives, Differentiation formulas, Cauchy-Riemann Equations, Sufficient Conditions for differentiability, Polar coordinates, Analytic functions, Harmonic functions.

3 Elementary Functions The Exponential functions, The Logarithmic function, Branches and derivatives of logarithms, Some identities involving logarithms, Complex exponents, Trigonometric functions, Hyperbolic functions.

4 Integrals Derivatives of functions, Definite integrals of functions, Contours, Contour integral, Examples, Upper bounds for Moduli of contour integrals, Anti-derivatives, Examples, Cauchy-Groursat’s Theorem (without proof), Simply and multiply Collected domains. Cauchy integral formula, Derivatives of analytic functions. Liouville’s Theorem and Fundamental Theorem of Algebra.

5 Series Convergence of sequences and series, Taylor’s series, Laurent series (without proof), examples.

6 Residues and Poles Isolated singular points, Residues, Cauchy residue theorem, residue at infinity, types of isolated singular points, residues at poles, zeros of analytic functions, zeros and poles.

Syllabus

THIRD YEAR

T. Y. B.Sc. Mathematics

Real Analysis - II

Unit

No.

1 Riemann Integral:

Sets of measure zero, Definition and existence of Riemann integral,

properties of Riemann integral, Fundamental theorem of integral

calculus, mean value theorems of integral calculus.

2 Improper Integrals:

Definition of improper integral of first kind, comparison test, absolute

and conditional convergence, integral test for convergence of series,

definition of improper integral of second kind, Cauchy principal value.

3 Sequences and series of functions:

Point wise and uniform convergence of sequences of functions,

consequences of uniform convergence, convergence and uniform

convergence of series of functions, integration and differentiation of

series of functions.

Syllabus

Topic

T. Y. B.Sc. Mathematics

Ring Theory

Unit

No.

1 Rings and Fields:

Rings and Fields, Integral Domains, The Fields of Quotients of an

Integral Domain, Rings of Polynomials, Factorization of Polynomials

over a Field.

2 Ideals and Factor Rings:

Homomorphisms and Factor Rings, Prime and Maximal Ideals.

3 Factorization:

Unique Factorization Domains, Euclidean Domains, Gaussian Integers

and Multiplicative Norms

Syllabus

Topic

T. Y. B.Sc. Mathematics

Partial Differential Equation

Unit

No.

1 Rings and Fields:

Rings and Fields, Integral Domains, The Fields of Quotients of an

Integral Domain, Rings of Polynomials, Factorization of Polynomials

over a Field.

2 Ideals and Factor Rings:

Homomorphisms and Factor Rings, Prime and Maximal Ideals.

3 Factorization:

Unique Factorization Domains, Euclidean Domains, Gaussian Integers

and Multiplicative Norms

Syllabus

Topic

T. Y. B. Sc. Mathematics

Graph Theory

Unit

No. Topic

1 An Introduction to Graphs

The definition of a Graph, Graphs and Models, More

Definitions, Vertex Degree, Sub graphs, Paths and Cycles, The

Matrix Representation of Graphs, Fusion.

2 Trees and Connectivity

Definition and Simple Properties, Bridges, Spanning Trees,

Connector Problems, Shortest Path Problems, Cut Vertices and

Connectivity.

3 Euler Tours and Hamiltonian Cycles

Euler Tours, The Chinese Postman Problem, Hamiltonian

Graphs, The Travelling Salesman Problem.

4 Directed Graphs

Definitions, In degree and Out degree, Tournament, Traffic

Flow.

Syllabus

T. Y. B.Sc. Mathematics

Multivariable Calculus - II

Unit

No.

1 Vector Valued Function :

1.1 Vector valued function.

1.2 Limit and Continuity of vector function.

1.3 Derivative of vector function and motion.

1.4 Differentiations rules.

1.5 Constant vector function and its necessary and

sufficient condition.

1.6 Integration of vector function of one scalar

variable.

1.7 Arc length and unit tangent vector T. Curvature and the

unit normal vector N.

2 Line Integrals :

2.1 Definition and evaluation of line integral.

2.2 Properties of line integrals.

2.3 Vector fields, work, circulation and flux across

smooth curves.

2.4 Path independence, Potential functions,

Conservative fields.

2.5 Green’s theorem in plane, evaluating integrals

using Green’s theorem.

Syllabus

Topic

3 Surface and Volume Integrals :

3.1 Surface area and surface integrals.

3.2 Surface integral for parameterized surfaces.

3.3 Stokes theorem (without proof).

3.4 The Gauss divergence theorem (proof for special

regions).