Detailed Syllabus Semester 1: First Year First Semester ... Hons.pdf · evolution, computer generations ... Isomorphism and homomorphism. ... Line integrals of linear differential

MIDNAPORE COLLEGE (AUTONOMOUS)

MIDNAPORE-721101, WEST BENGAL

SYLLABUS FOR B.SC MATHEMATICS (HONOURS)

Structure of Syllabus (10 % marks are allotted for internal assessment of each paper)

Semester 1: First Year First Semester 100

Paper MTMH-101:i)Geometry 40

ii) Vector Algebra 10

Paper MTMH-102:i) Algebra I 25

ii) Statics 25

Semester 2: First Year Second Semester 150

Paper MTMH-201:i) Mathematical Analysis I 25

ii) Differential Equation I 25

Paper MTMH-202:i) computer programming 25

ii) Computer practical 25

Paper MTMH-203:i) Algebra II 50

Semester 3: Second Year First Semester 150

Paper MTMH-301: Mathematical Analysis II 50

Paper MTMH-302: i) Classical Mechanics I 25

ii)Vector Calculus 25

Paper MTMH-303:i)Probability 20

ii) Statistics 30

Semester 4: Second Year Second Semester 150

Paper MTMH-401: Numerical Analysis 50

Paper MTMH-402:i)Numerical Practical 20

ii)Optimization 30

Paper MTMH-403: i) Tensor calculus 20

ii) Differential Equations II 30

Semester 5: Third Year First Semester 150

Paper MTMH-501: Algebra III 50

Paper MTMH-502: Classical Mechanics II and Hydrostatics 50

Paper MTMH-503: Mathematical Analysis III 50

Semester 6: Third Year Second Semester 100

Paper MTMH-601: i) Mathematical Modeling 20

ii) Mathematical Analysis IV 30

Paper MTMH-602: i) Algebra IV 25

ii) Project Work 25

Detailed Syllabus

Semester 1: First Year First Semester Marks: 100

SYLLABUS FOR B.SC MATHEMATICS HONOURS(Midnapore College)

Page 2 of 16

PAPER MTMH-101

i)Geometry. Marks : 40

Analytical geometry of two dimensions: Transformation of rectangular axes. General equation of second

degree and its reduction to normal form. Pole and polar, conjugate diameter, Polar equation of a conic.

Analytical geometry of three dimensions: Direction cosines. Straight line. Plane; Sphere : General

Equation. Circle, Sphere through the intersection of two spheres. Radical Plane, Tangent, Normal; Cone :

Right circular cone. General homogeneous second degree equation. Section of cone by a plane as a conic

and as a pair of lines. Condition for three perpendicular generators. Reciprocal cone; Cylinder : Generators

parallel to either of the axes, general form of equation. Right-circular cylinder; Ellipsoid, Hyperboloid,

Paraboloid :Canonical equations only; Tangent planes, Normal, Enveloping cone; Surface of Revolution

(about axes of reference only). Ruled surface. Generating lines of hyperboloid of one sheet and hyperbolic

paraboloid; Transformation of rectangular axes by translation, rotation and their combinations; Knowledge

of Cylindrical, Polar and Spherical polar co-ordinates, their relations (no deductions required). Central

conicoids, paraboloids, plane sections of conicoids. Generating lines. Reduction of second degree equations

to normal form; classification of quadrics.

ii) Vector Algebra. Marks : 10

Operations with vectors. Scalar and vector product of three vectors. Product of four vectors. Reciprocal

vectors.

PAPER MTMH-102

i)Algebra I. Marks : 25

Elementary Matrix Theory and Linear Algebra: Basic matrix operations, row reduction, row-equivalence,

Rank of a matrix, Determinants, Systems of linear equations, Gauss elimination and consistency, Systems

(homogeneous and non-homogeneous) of linear equations as matrix equations and the invariance of its

solution set under row-equivalence.

Trigonometry: De-Moivre's theorem and applications. Direct and inverse, circular and hyperbolic,

functions. Logarithm of a complex quantity. Expansion of trigonometric functions.

Theory of Equations: Polynomials in one variable and the division algorithm. Relations between the roots

and the coefficients. Transformation of equations. Descartes rule of signs. Solution of cubic and biquadratic

(quartic) equations (Ferrais method,Eulers method).

.Graph Theory: Graphs and planar graphs, Paths and circuits, Hamiltonian paths, Shortest paths, Trees,

Spanning trees(Definations and Examples only.).

ii) Statics. Marks : 25

Statics in plane: Equilibrium of a particle, Equilibrium of a system of particles, Work and potential energy,

Mass centers and centers of gravity, Friction, Thin beams, Flexible cables, Frames.

Statics in space: General force systems, Equilibrium of a system of particles, Reduction of force systems,

Equilibrium of a rigid body, Displacements of a rigid body, Generalized coordinates and constraints, Work

and potential energy.


Page 3 of 16

Semester 2: First Year Second Semester Marks : 150

PAPER MTMH-201

i)Mathematical Analysis I. Marks : 25

Set Theory: Rings and Algebras of Sets; Relations and Functions; Basic Algebra, counting and Arithmetic;

Infinite Direct Products, Axiom of Choice and Cardinal Numbers.

Real numbers: Algebraic Axioms, Order Axioms(bounds, bounded sets, and their properties, sup and inf of

sets), Completeness Axiom (or the least upper bound property and equivalent conditions including the

nested interval property), Countability of Z and Q, Uncountability of R. Bolzano-Weierstrass theorem.

Sequences and series of Real Numbers: Sequences. Bounded sequences, monotone sequences and their

convergence, limsup and liminf and convergence criterion using them, sub sequences, Cauchy sequences

and their convergence criterion. Infinite series and their convergence. Geometric series. The comparison

test. Series of non-negative terms. The condensation test. Integral test. Ratio and root tests. Absolute and

conditional convergence. Alternating series and Leibnitz's theorem.

Limits of Functions: Bounded and Monotone functions; Limits of functions; Properties of limits. One

sided limits and limits involving Infinity; Landau’s little ―oh‖ and big ―Oh‖.

ii)Differential Equations I. Marks : 25

Elementary Methods in Ordinary Differential Equations: First order exact equations and integrating

factors. First order higher degree equations solvable for x, y, p. Clairaut's form and singular solutions.

Orthogonal trajectories.

Linear differential equations of second order with constant coefficient and variable coefficient.

Transformation of the equation by changing the dependent variable and the independent variable. Method

of variation of parameters.

Ordinary simultaneous differential equations. Series solution at an ordinary point: Power Series solution of

ordinary differential equations. Simple problems only.

PAPER MTMH-202

i)Computer Programming. Marks : 25

Fundamentals of Computer Science and Computer Programming: Computer fundamentals : Historical

evolution, computer generations, functional description, operating system, hardware & software. Positional

number system : binary, octal, decimal, hexadecimal system. Binary arithmetic. Storing of data in a

computer : BIT, BYTE, Word. Coding of data –ASCII, EBCDIC, etc. Algorithm and Flow Chart : Important

features, Ideas about the complexities of algorithm. Application in simple problems. Programming

languages : General concepts, Machine language, Assembly Language, High Level Languages, Compiler

and Interpreter. Object and Source Program. Ideas about some major HLL.


Page 4 of 16

Introduction to ANSI C: Character set in ANSI C. Key words : if, while, do, for, int, char, float etc. Data

type : character, integer, floating point, etc. Variables, Operators : =,= =, !!<,>, etc. (arithmetic, assignment,

relational, logical, increment, etc.) .

Expressions: e.g. (a = = b ) !! (b = = c), Statements : e.g. if (a>b) small = a; else small = b. Standard

input/output. Use of while, if…. Else, for, do…while, switch, continue, etc. Arrays, strings functions.

Running simple C Programs. Header file.

Boolean Algebra: Huntington Postulates for Boolean Algebra. Algebra of sets and Switching Algebra as

examples of Boolean Algebra. Statement of principle of duality. Disjunctive normal and Conjunctive

normal forms of Boolean Expressions. Design of simple switching circuits. Truth functional logic and

prepositional connectives. Switching circuits. Boolean algebras. Duality. Boolean functions. Normal forms.

Karnaugh maps.

ii)Computer practical. Marks : 25

1. General programs: (i) Area of circle, triangle, (ii) Summation of finite and convergent infinite series, (iii)

Maximum and minimum among three number and numbers, (iv) Roots of quadratic equation, (v) G.C.D.

and L.C.M.,between two integers, (vi) Testing of prime numbers, (vii) Split a ~number into digits, (vii)

Computation of nPr and nCr (viii) Searching and sorting (bubble sort only) etc., String functions, Random

number generation.

2. Problems on matrices: (i) Addition and subtraction, (ii) Product, (iii) Trace and (iv) Transpose.

PAPER MTMH-203

Algebra II. Marks: 50

Groups: Definition of a group, with examples and simple properties. Groups of transformations.

Subgroups. Generation of groups and cyclic groups. Various subgroups of GL2(R). Coset decomposition.

Lagrange's theorem and its consequences. Fermat's and Euler's theorems. Permutation groups. Even and

odd permutations. The alternating groups An. Isomorphism and homomorphism. Normal subgroups.

Quotient groups. First homomorphism theorem. Cayley's theorem.

Marks: 25

Rings and Fields: Commutative rings, integral domains, and their elementary properties. Ordered

integral domain: The integers and the well-ordering property of positive elements. Finite induction.

Divisibility, the division algorithm, primes, GCDs, and the Euclidean algorithm. The fundamental theorem

of arithmetic. Congruence modulo n and residue classes. The rings Z„ and their properties. Units in Zn, and

Zp

for prime p. Subrings and ideals. Characteristic of a ring. Fields.

Rings homeomorphisms. Ideals and quotient rings. Prime and maximal ideals. The quotient field of an

integral domain. Euclidean rings. Polynomial rings. Polynomials over Q and Eisenstein's criterion.

Polynomial rings over arbitrary commutative rings. UFDs. If A is a UFD, then so is A ,x2,..., x

n]

Marks: 25

Semester 3: Second Year First Semester Marks: 150


Page 5 of 16

PAPER MTMH-301

Mathematical Analysis II Marks: 50

Topology of R and Continuity: Open sets and Closed sets, Compact and connected subsets of R; The

Cantor set, Continuous functions, One-sided continuity; Discontinuity and Monotonicity; Exterme value

and Intermediate value theorems; Uniform continuity; Approximation by Step, Piecewise linear and

Polynomial functions. Marks: 15

The Derivative: Differentiability; Geometric Meaning of the Derivative; Derivatives of elementary

functions; Higher-order Derivatives, The Basic Theorems of Differential Calculus, Fermat’s Lemma and

Roll’es Theorem , The theorems of Lagrange and Cauchy, Taylor's Formula, Problems. Differential

Calculus Used to Study Functions, Conditions for a Function to be Monotonic, Conditions for an Interior

Extremum of a Function. Conditions for a Function to be Convex ,L'Hopital's Rule,

Applications of differential calculus: Asymptotes, Envelopes, Radius of curvatures, Curve tracing.

Marks: 15 The Riemann Integral: Tagged partitions and Riemann sums; Some classes of Riemann Integrable

functions; Sets of measure zero and Lebesgue’s integrability criterion; Properties of the Riemann integral.

Fundamental theorem of Calculus. Some Applications of Integration, Additive Interval Functions and the

Integral, Arc Length, The Area of a Curvilinear Trapezoid, Volume of a Solid of Revolution , Work and

Energy.

Improper Riemann Integrals: Convergence of an Improper Integral, Improper Integrals with More than one

Singularity. Beta and Gamma functions. Marks: 20

PAPER MTMH-302

i) Classical Mechanics I. Marks: 25

1. Mechanics of a Single Particle: Velocity and acceleration of a particle in (i) plane polar coordinates -

radial and cross-radial Components (ii) spherical polar and (iii) cylindrical polar co-ordinate system; Time

and path integral of force; work and energy; Conservative force and concept of potential; Dissipative forces;

Conservation of linear and angular momentum.

2. Mechanics of a System of Particles: Linear momentum, angular momentum and energy - centre of mass

decomposition; Equations of motion, conservation of linear and angular momentum.

3. Rotational Motion: Moment of inertia, radius of gyration; Energy and angular momentum of rotating

systems of Particles; Parallel and perpendicular axes theorems of moment of inertia; Calculation of moment

of inertia for simple symmetric systems; Ellipsoid of inertia and inertia tensor; Setting up of principal axes

in simple symmetric cases. Rotating frames of reference -Coriolis and centrifugal forces, simple examples.

Force free motion of rigid bodies - free spherical top and free symmetric top.

ii)Vector Calculus. Marks: 25

Scalar-valued functions over the plane and the space. Vector function of a scalar variable: Curves

and Paths. Vector fields.

Vector differentiation: Directional derivatives, the tangent plane, total differential, gradient,

divergence, and curl.

Vector integration: Path, line, surface, and volume integrals. Line integrals of linear differential

forms, integration of total differentials, conservative fields, conditions for line integrals to depend

only on the endpoints, the fundamental theorem on exact differentials. Serret-Frenet Formulas.

Theorems of Green, Gauss, Stokes, and problems based on these.

.


Page 6 of 16

PAPER MTMH-303

i)Probability. Marks: 20

Random variables: Concept, cumulative distribution function, discrete and continuous random variables,

expectation, mean, variance, moment generating function.

Discrete random variables: Bernoulli random variable, Binomial random variable, geometric random

variable, Poisson random variable.

Continuous random variables: Uniform random variable, exponential random variable, Gamma random

variable, Normal random variable.

Conditional probability and conditional expectations, Bayes theorem, independence, computing

expectation by conditioning; some applications — a list model, a random graph, Polya's urn model.

Bivariate random variables: Joint distribution, joint and conditional distributions, the correlation

coefficient. Bivariate normal distribution.

Functions of random variables: Sum of random variables, the laws of large numbers, central limit

theorem, approximation of distributions.

ii)Statistics. Marks: 30

Random sample. Concept of sampling and various types of sampling. Sample and population. Collection,

tabulation and graphical representation. Grouping of data. Sample characteristic and their computation.

Sampling distribution of a statistic. Estimates of a population characteristic or parameter. Unbiased and

consistent estimates. Sample characteristics as estimates of the corresponding population characteristics.

Sampling distributions of the sample mean and variance. Exact sampling distributions for the normal

populations.

Bivariate samples. Scatter diagram. Sample correlation co-efficient. Least square regression lines and

parabolas. Estimation of parameters. Method of maximum likelihood. Applications to binomial, Poisson and

normal population. Confidence intervals. Interval estimation for parameters of normal population.

Statistical hypothesis. Simple and composite hypothesis. Best critical region of a test. Neyman-Pearson

theorem (Statement only) and its application to normal population. Likelihood ratio testing and its

application to normal population. Simple applications of hypothesis testing.

Time series: Definition. Why to analyze Time series data? Components. Measurement of Trend – (i)

Moving average method. (ii) curve fittings (linear and quadric curve). (Ideas of other curves e.g.,

exponential curve etc.). Ideas about the measurement of other components.

Index Number: Meaning of index number. Construction of Price Index Number. Consumer Price Index

number. Calculation of purchasing power of rupee.


Page 7 of 16

Semester 4: Second Year Second Semester Marks: 150

PAPER MTMH-401

Numerical Analysis Marks: 50

What is Numerical Analysis?

Errors in Numerical computation : Gross error, Round off error, Truncation error. Approximate numbers.

Significant figures. Absolute, relative and percentage error.

Operators : , ∇ , E, μ, δ (Definitions and simple relations among them).solution of differences equation.

Marks: 5 Interpolation : Problems of interpolation, Weierstrass’ approximation theorem

only statement). Polynomial interpolation. Equispaced arguments. Difference table. Deduction of Newton’s

forward and backward interpolation formulae.

Statements of Stirling’s and Bessel’s interpolation formulae. error terms.

General interpolation formulae : Deduction of Lagrange’s interpolation formula. Divided difference.

Newton’s General Interpolation formula (only statement). Inverse interpolation.

Interpolation formulae using the values of both f(x) and its derivative f(x) : Idea of Hermite interpolation

formula (only the basic concepts). Numerical Differentiation based on Newton’s forward & backward and

Lagrange’s formulae. Marks: 15

Numerical Integration : Integration of Newton’s interpolation formula.

Newton-Cote’s formula. Basic Trapezoidal and Simpson’s ⅓rd. formulae. Their composite forms. Weddle’s

rule (only statement). Statement of the error terms associated with these formulae. Degree of precision (only

definition).

Solution of non-linear equations (algebraic and transcendental): Solution of a single equation by (i)

Graphical method, (ii) Method of Bisection, (iii) Regular falsi method, (iv) Fixed point iteration method, (v)

Newton-Raphson method. Marks: 15

Geometrical interpretation of these method and Convergence of fixed point iteration and Newton-Raphson

method. Numerical solution of a system of linear equations : Gauss elimination method. Iterative method – Gauss-

Seidal method. Matrix inversion by Gauss elimination method (only problems – up to 3×3 order).

Eigen value Problems : Power method for numerically extreme Eigen values. Jacobi’s method. Numerical

solution or Ordinary Differential Equation : Basic ideas, nature of the problem. Picard, Euler and Runge-

Kutta (4th

order) methods (emphasis on the problems only).

Marks: 15

PAPER MTMH-402

i) Numerical and Statistical Practical (Lab) using C programming. Marks: 20

a) Numerical Methods Lab: Programming in C from the following set of problems:

• Bisection method.

• Regula Falsi method.

• Fixed-point method.

• Newton-Raphson method.

• Lagrange interpolation.

• Newton's forward and backward interpolation.

• Hermite interpolation.

• Gauss Quadrature.

• Gauss elimination method.


Page 8 of 16

• LU decomposition.

• Jacobi's method (eigenvalue).

• Power method (eigenvalue).

• Euler's method.

• Fitting a polynomial function.

b)Statistical Methods Lab: Programming in C from the following set of problems:

• Mean.

• Median.

• Mode.

• Moments

• Correlation coefficients.

• Skewness and Kurtosis.

ii)Linear Programming Problems and Games Marks: 30

Basic solutions and Basic Feasible Solution (BFS) with reference to L.P.P. Matrix formulation of L.P.P.

Degenerate and Non-degenerate B.F.S.

Some basic properties of convex sets, convex functions, and concave functions. Theory and application of

the simplex method of solution of a linear programming problem, Charne's M-technique. The two phase

method, principle of duality in linear programming problem, fundamental duality theorem, simple problems,

the transportation and assignment problems, Game theory.

PAPER MTMH-403

i)Differential Equations II. Marks: 25

Partial differential equation (PDE) : Introduction. Formation of P.D.E., Solution of PDE by Lagrange’s

method of solution and by Charpit’s method.,Jacobi’s Method.

Laplace transforms: Introduction to infinite integrals. Linearity of Laplace trans-forms. Existence theorem

for Laplace transforms. Laplace transforms of derivatives and integrals. Shifting theorems. Differentiation

and integration of transforms. Convolution theorem. Solution of integral equations and systems of

differential equations using Laplace transforms.

ii)Tensor Analysis. Marks: 25

Tensor Algebra: Vector spaces; The dual spaces; Tensor product of vector spaces; Transformation

formulae; Contraction; Special tensors; Inner product; Associated tensors; Exterior algebra

Tensor calculus: Differentiable Manifolds; Tangent vectors; Affine tensors and tensorial forms;

connections; covariant differentiation; connections over submanifolds; Absolute derivation of tensorial

forms.

Riemannian geometry: Riemannian manifolds; Metric; The fundamental theorem of local Riemannian

geometry; Differential parameters; Curvature tensors; Geodesics ; Geodesic curvature; Geometrical

interpretation of curvature tensor.

Semester 5: Third Year First Semester Marks: 150


Page 9 of 16

PAPER MTMH-501

Algebra III. Marks: 50

Vector spaces and linear transformations: Vector spaces over a field, subspaces. Sum and direct sum of

sub-spaces. Linear span. Linear dependence and independence. Basis. Finite dimen-sional spaces. Existence

theorem for bases in the finite dimensional case. Invariance of the number of vectors in a basis, dimension.

Existence of complementary sub-space of any subspace of a finite-dimensional vector space. Dimensions of

sums of subspaces. Quotient space and its dimension. Marks: 15

Linear transformations and matrix representation. Non-singular transformations. Inverse of a Matrix. Rank-

nullity theorem. Equivalence of row and column ranks. Elementary matrices and elementary operations.

Equivalence and canonical form. Diagonalization, Determinants. Eigen-values, eigenvectors, and the

characteristic equation of a matrix. Cayley-Hamilton theorem and its use in finding the inverse of a matrix.

Marks: 15

Inner product spaces: Cauchy-Schwartz inequality. Orthogonal vectors and or-thogonal complements.

Orthonormal sets and bases. Bessel's inequality. Gram-Schmidt orthogonalization method. Hermitian, Self-

Adjoint, Unitary, and Orthogo-nal transformation for complex and real spaces. Bilinear and Quadratic

forms. The Spectral Theorem. The structure of orthogonal transformations in real Euclidean spaces.

Marks: 20

PAPER MTMH- 503

Mathematical Analysis III. Marks: 50

The Space and its Subsets: The Set and the Distance in it; Open and Closed Sets in ; Compact

Sets in , Limits and Continuity of Functions of Several Variables ,The Limit of a Function, Continuity

of a Function of Several Variables.

Differential Calculus in Several Variables: The Linear Structure on , as a Vector Space; Linear

Transformations L : ; The Norm in ; The Euclidean Structure on ; Derivatives; Higher

derivatives; Smoothness Classes; Implicit and Inverse Functions; The Rank Theorem; Lagrange Multipliers.

Integral Calculus in Several Variables: Multiple Integrals; Differential Forms ; The General Stokes' Formula.

Marks: 30

Complex Variables: Confonnal transformations of the plane and Cauchy Riemann equations.

Continuity and differentiability of complex functions. Analytic functions. Harmonic functions.

Elementary functions. Mapping by elementary functions. Mobius transformations. Fixed points.

Cross ratio. Inverse points and critical mappings. Conformal maps. Gregory's series.

Marks: 20


Page 10 of 16

PAPER MTMH-502

i)Classical mechanics II. Marks: 30

1. Central force problem: Motion under central force; Nature of orbits in an attractive inverse square field; Kepler's

laws of planetary motion. Rutherford scattering as an example of repulsive potential.

2. Mechanics of Ideal Fluids: Streamlines and flow lines; Equation of continuity; Euler's equation of motion;

Streamline motion - Bernoulli's equation and its applications. Definition of Newtonian and non-Newtonian fluids.

3. Lagrangian and Hamiltonian formulation of Classical Mechanics:

Generalised coordinates, constraints and degrees of freedom; D’Alembart’s principle; Lagrange’s equation for

conservative systems (from D'Alembert's principle; variational principle not required) and its application to simple

cases; Generalised momentum; Idea of cyclic coordinates, its relation with conservation principles.

Waves and Vibrations:

1. Simple Harmonic Motion: Differential equation and its solution.

2. Superposition of Simple Harmonic Motion: Analytical treatment, Lissajous figures, natural, damped and forced

vibration, resonance, sharpness of resonance.

3. Waves: Plane progressive wave in 1-d and 3-d. Plane wave and spherical wave solutions. Dispersion: phase velocity

and group velocity.

4. Differential Equation of Wave Motion: Plane progressive wave - energy and intensity. Bel, decibel and phon.

Superposition of waves, beats. Velocity of longitudinal wave in solid and in gas, velocity of transverse wave in string,

Doppler Effect.

ii)Hydrostatics Marks: 20

1. Definition of Fluid, Perfect Fluid, pressure. To prove that the Pressure at a point in a fluid in equilibrium is the

same in every direction. Transmissibility of liquid pressure. Pressure of heavy fluids. To prove

(a) In a fluid at rest under gravity the pressure at all points in the same at all points in the same horizontal

plane.

(b) In a homogeneous fluid at rest under gravity the difference between the pressures at two points is

proportional to the difference of their depths.

(c) In a fluid at rest under gravity horizontal planes are surfaces of equal density.

(d) When two fluids of different densities at rest under gravity do not mix, their surface of separation is a

horizontal plane. Pressure in heavy homogeneous liquid. Thrust of heavy homogeneous liquid of plane

surfaces.

2. Definition of centre of pressure. Formula for the depth of the centre of pressure of a plane area. Position of the

centre of pressure. Centre of pressure of a triangular area whose angular points are at different depths. Centre

of pressure of circular area. Position of the centre of pressure referred to co-ordinate axes through the centroid

of the area. Centre of pressure of an elliptical area when its major axis is vertical or along the line of greatest

slope. Effect of additional depth on centre of pressure.

3. Equilibrium of fluids in given fields of force. Definition of field of force, line of force. Pressure derivative in

terms of force. Surface of equi-pressure. To find the necessary and sufficient condition of equilibrium of a

fluid under the action of a force whose components are X,Y, Z along the co-ordinate axes.

To prove

(a) That surfaces of equal pressure are the surfaces intersecting orthogonally the lines of force.

(b) When the force system is conservative, the surfaces of equal pressure are equi-potential surfaces and are

also surface of equal density.

To find the differential equation of equal pressure and density.

4. Rotating fluids. To determine the pressure at any point and the surface of equal pressure when a mass of

homogeneous liquid contained in a vessel, revolves uniformly about a vertical axis.

5. The stability of the equilibrium of floating bodies. Definition, stability of equilibrium of a floating body,

metacentre, plane of floatation, surface of buoyancy. General proposition about small rotational displacement.

To derive the condition for stability.

6. Pressure of gases. The atmosphere. Relation between pressure, density and temperature. Pressure in an

isothermal atmosphere. Atmosphere in convective equilibrium.


Page 11 of 16

PAPER MTMH- 503

Mathematical Analysis III. Marks: 50

The Space and its Subsets: The Set and the Distance in it; Open and Closed Sets in ; Compact

Sets in , Limits and Continuity of Functions of Several Variables ,The Limit of a Function, Continuity

of a Function of Several Variables.

Differential Calculus in Several Variables: The Linear Structure on , as a Vector Space; Linear

Transformations L : ; The Norm in ; The Euclidean Structure on ; Derivatives; Higher

derivatives; Smoothness Classes; Implicit and Inverse Functions; The Rank Theorem; Lagrange Multipliers.

Integral Calculus in Several Variables: Multiple Integrals; Differential Forms ; The General Stokes' Formula.

Marks: 30

Complex Variables: Confonnal transformations of the plane and Cauchy Riemann equations.

Continuity and differentiability of complex functions. Analytic functions. Harmonic functions.

Elementary functions. Mapping by elementary functions. Mobius transformations. Fixed points.

Cross ratio. Inverse points and critical mappings. Conformal maps. Gregory's series.

Marks: 20


Page 12 of 16

Semester 6: Third Year Second Semester Marks: 100

PAPER MTMH-601

i)Mathematical Modeling. Marks: 20 Introduction, basic steps of Mathematical Modeling, its needs, types of models, limitations. Elementary

ideas of dynamical systems, autonomous dynamical systems in the plane-linear theory. Equilibrium point,

node, saddle point, focus, centre and limit-cycle ideas with simple illustrations and figures. Linearization of

non-linear plane autonomous systems. Mathematical Modeling in the biological environment.

Mathematical modeling of epidemics. Basic concepts. Simple epidemic model, formulation, solution,

interpretation, and limitations. General epidemic model, formulation, solution, interpretation, and

limitations.

ii)Mathematical Analysis IV. Marks: 30

Metric spaces: Definition and examples, neighbourhoods, limit points, interior, and boundary points. Open

and closed sets. Closure, interior, and boundary of a set. Subspaces. Cauchy sequences and complete

spaces, e.g., C[a, b] is a complete metric space. Cantor's intersection theorem and the contraction mapping

principle. Dense and nowhere dense subsets. Baire Category Theorem. Compactness: Sequential

compactness and Heine-Borel property, totally bounded spaces, finite intersection property, continuous

functions on compact sets. Marks: 15

Sequence and series of functions: Pointwise convergence. Uniform convergence, and its relation to

continuity, integration, and differentiation. Weierstrass M-test. Power series, radius of convergence.

Analytic functions and examples. Fourier series: Periodic functions and Trigonometric poly-nomials.

Definition of Fourier coefficients and series. Riemann Lebesgue lemma. Bessel's inequality. Parseval's

identity. Dirichlet's conditions for convergence of Fourier series. Examples of Fourier expansions and

summation results for series.

Marks: 15

PAPER MTMH-602

Algebra IV. Marks: 25

Advanced Group Theory: Group automorphisms, inner automorphisms. Automorphism groups and their

computations. Center of a group Conjugacy relation. Normalizer. Counting principle and the class equation

of a finite group. Cauchys theorem, Sylows theorems. Free abelian groups. Structure theorem of finitely

generated abelian groups.

ii) Project Work. Marks: 25

REFERENCES

1. Abraham and Becker, Electricity and Magnetism.

2. Ahlfors, Complex Analysis.

3. Apostol, T., Mathematical Analysis.

4. Apostol, T., Calculus, Volumes I and II.

5. Artin, Algebra.

6. Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley and Sons, 1978.

7. Bailey, N., The Mathematical Theory of Infectious Diseases, Haftier Press, New York, 1975.


Page 13 of 16

8. Balaguruswamy, E., Programming in ANSI C, 2nd ed., Tata McGraw Hill.

9. Balaguruswamy, E., Numerical Methods, Tata McGraw Hill, 2000.

10. Beightler, C, Phillips, D., and Wilde, D., Foundations of Optimization, Prentice Hall, En-glewood

Clifls, New Jersey, 1979.

11. Bell, R- J. T., Elementary Treatise on Coordinate Geometry.

12. Beltrami, E., Mathematics for Dynamic Modeling, Academic Press, Orlando, Florida, 1987.

13. Besant, W. H., Ramsey, A. S., A Treatise on Hydromechanics (Part I).

14. Bhat, B. R., Modern Probability Theory.

15. Mathematical Modeling in the Biological Environment, MTE-14, Indira Gandhi National Open

University, New Delhi, 1998.

16. Birkhoff G., and S. Mac Lane, A Survey of Modern Algebra, 4th ed.

17. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul, Basic Abstract Algebra, 2nd ed.

18. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul, First Course in Linear Algebra.

19. Bolza, Oskar, Lectures on the Calculus of Variations, Dover.

20. Boyce and Diprima, Elementary Differential Equations and Boundary Value Problems.

21. Braun, M., Differential Equations and their Applications, Springer, New York, 1978.

22. Chaki, M. C, A Textbook of Analytical Geometry, Calcutta Publishers.

23. Chakraborty, J. G., and Ghosh, Advanced Higher Algebra.

24. Chakraborty, J. G., and Ghosh, Linear Programming.

25. Chakraborty, J. G., and Ghosh, P. R., Advanced Analytical Dynamics.

26. Chakraborty, J. G., and Ghosh, Advanced Analytical Geometry.

27. Chatterjee, B. C, Linear Algebra

28. Chorltan, F., Textbook of Dynamics.

29. Cormen, T. H., Leiserson, C. E., Rivest, R. L., and Stein, C, Introduction to Algorithms, Second

Edition.

30. Corwin and Szczarba, Multivariate Calculus.

31. Coulson, C. A., Electricity.

32. Courant, R., and F. John, Introduction to Calculus and Analysis, Volume I.

33. Courant, R_, and F. John, Introduction to Calculus and Analysis, Volume II.

34. Das and Mukherjee, Higher Trigonometry.

35. Datta, K. B., Matrix and Linear Algebra.

36. Davis, M., and Weyuker, E. J., Computability, Complexity, and Languages.

37. Burton, David M., Elementary Number Theory, 1989.

38. Deitel, H. N., and Deitel, P. J., C How to Program, Prentice Hall.

39. do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, Inc., 1976.

40. Elsgolts, L., Differential Equations and the Calculus of Variations, Mir Publishers, Moeow.

41. Ferraro, V. C. A., Electromagnetic Theory.

42. Fraleigh, A First Course in Abstract Algebra.

43. Friedberg, Insel, and Spence, Linear Algebra.

44. Froberg, C. E., Introduction to Numerical Analysis, 2nd ed., Addison Wesley, 1979.

45. Gass, S. I., Linear Programming.

46. Ghosh, M. C, Statics.

47. Goldberg, Methods of Real Analysis.

48. Gottfried, Byron S., Theory and Problems of Programming with C, Tata McGraw Hill, 1998.

49. Gupta, A. S., Calculus of Variations, Prentice Hall of India, New Delhi.

50. Hadley, G., Linear Programming.

51. Halmos, Finite Dimensional Vector Spaces.

52. Halmos, Naive Set Theory.

53. Hancock, Les, and Krieger, Morris, The C primer, McGraw Hill, 1998.

54. Hardy, G. H., and Wright, E. M., An Introduction to the Theory of Numbers, 5th Edition,

Clarendon Press(Oxford), 1979.


Page 14 of 16

55. Herstein, I. N., Topics in Algebra.

56. Hildebrand, Advanced Calculus for Applications.

57. Hirschfelder, and Hirschfelder, Introduction to Discrete Mathematics.

58. Hoffman and Kunze, Linear Algebra. 59. Hopcroft, and Ullman, Introduction to Automata Theory, Languages, and Computation.

60. Iyenger, and Bali, A Textbook of Engineering Mathematics.

61. Jacobeon, Basic Algebra, Volumes I and II.

62. Jain, M. K., Iyenger, S. R. K., and Jain, R. K., Numerical Methods for Scientific and Engineering

Computation, New Age International (P) Ltd., 1999.

63. Jackson, J. D., Classical Electrodynamics.

64. Anderson, J. L., Principles of Relativity Physics.

65. Kanetkar, Y., Let ui C, 3rd ed., BPB pub.

66. Kapur, J. N., Insight into Mathematical Modeling, Indian National Science Academy, New Delhi,

1983. 67. Kar, J. M.

t Hydrostatics.

68. Kernighan, B., and Pike, R., The Unix Programming Environment.

69. Knuth, D., The Art of Computer Programming, Volumes I, II, and III.

70. Kernighan, B. W., and Ritchie, D. M., The C Programming Language, 2nd ed., Prentice Hall,

1989. 71. Kolmogorov, A. N., and Fomin, S. V., Elements of the Theory of Functions and Functional Analysis,

Vot 1, Metric and normed spaces, Graylock, Rochester; Dover, New York.

72. Kolmogorov, A. N., and Fbmin, S. V., Introductory Real Analysis, Dover, New York. 73. Kreyszig, Advanced Engineering Mathematics.

74. Kumaresan, S., Linear Algebra, A Geometric Approach.

75. Kuroah, A., Higher Algebra.

76. Lewis, Matrix Theory.

T7. Lewis, and Papadimitriou, Elements of the Theory of Computation.

78. Schrage, Linus, Optimising modeling with LINDO.

79. Lipechutz, S., Set Theory and Related Topics, Schaum'a Outline Series.

80. Liu, C. L., Elements of Discrete Mathematics.

81. Loney, S. L., Dynamics. 82. Loney, S. L., Elements of Coordinate Geometry.

83. Loney, S. L., An Elementary Treatise on the Dynamics of 1'article and of Rigid Bodies.

84. Loney, S. L., Statics.

85. Mac Lane, S., and G. Birkhoff, Algebra.

86. Maity, S. R., and Maity, A., Dynamics of a Particle. 87. Maity, K. C., and R. K. Ghosh, Differential Calculus: An Introduction to Analysis, Parts I and II.

88. Malik, Mathematical Analysis.

89. Marsden, J., and Tromba, Vector Calculus.

90. MathCAD Premium, User's Manual.

91. MATLAB - High Performance Numeric Computation and Visualization Software: User's Guide.

92. Maulik, T. N., Linear Programming: Theory and Applications.

93. May, R., Theoretical Ecology: Principles and Applications, Saunders, Philadelphia, 1976.

94. Meerschaert, Mark M., Mathematical Modeling, Academic Press, New York, 1993.

95. Meyer, W., Concepts of Mathematical Modeling, McGraw Hill, New York, 1994.

96. Miller, F. H., Partial Differential Equations, John Wiley and Sons.

97. Millman, R. S. and Parker, G. D., Elements of Differential Geometry, Prentice Hall, 1977. 98. Moller, C., The Theory of Relativity.


Page 15 of 16

99. Mullish, Henry, and Herbert, L., Spirit of C: An Introduction to Modern Programming, Jaico

publishers.

100. Murray, D., Introductory Course in Differential Equations.

101. Niven, I., ZucJcerman, H. S., and Montgomery, H. L, An Introduction to the Theory of Numbers,

5th Edition, John Wiley and Sons., Inc. 102. O'Neil, Barret, Elementary Differential Geometry, 1966.

103. Parzen, E., Modern Probability Theory and Its Applications.

104. Schwartz, R., Learning Perl. 105. Bergmann, P. G., Introduction to the Theory of Relativity.

106. Maliur Rahman, Md., Hydrostatics.

107. Maliur Rahman, Md., Statics.

108. Ramsey, A. S., Dynamics (Part I). 109. Rayward-Smith, V. J., A First Course in Formal Language Theory.

110. Press, W. H., Teukolsky, S. A., Vellerling, W. T., and Flannery, B. P., Numerical Recipes in C, 2nd

ed., Cambridge University Press. 111. Resnick, R., Introduction to Special Relativity.

112. Roman, Introduction to Discrete Mathematics.

113. Ross, S., Differential Equations.

114. Rot man, Theory of Groups.

115. Rudin, Principles of Mathematical Analysis.

116. Sastry, S. S., Introductory Methods of Numerical Analysis, 3rd ed., Prentice Hall India, 1998.

117. Scarborough, James B., Numerical Mathematical Analysis, Oxford and IBH publishing co.

118. Shanti Narayan, A Course in Mathematical Analysis.

119. Shanti Narayan, Analytical Geometry of Three Dimensions.

120. Sharma, B. D., Hydrostatics.

121. Simmons, Introduction to Topology and Modern Analysis.

122. Simmons, Differential Equations.

123. Sneddon, I. N., Elements of Partial Differential Equations, McGraw Hill.

124. Stevens, W. R., Advanced Programming in the Unix Environment.

125. Stevens, W. R., TCIP/IP Illustrated, Volume I.

126. Stevens, W. R., Unix Network Programming, Volume I.

127. Stoll, Set Theory and Logic.

128. Str&ng, Linear Algebra.

129. Synge, J. L., Relativity: The Special Theory.

130. Ugarov, V. A., Special Theory of Relativity.

131. Bach, M., The Design of the Unix Opearting System.

132. Verma, R. S., A Textbook on Statics.

133. Dixon, W. G., Special Relativity: The Foundation of Macroscopic Physics.

134. Widder, Advanced Calculus.

135. Willmore, T. J., An Introduction to Differential and Riemannian Geometry, Oxford University

Press, 1965.

136. Wolfram, S., Mathematics, Cambridge.

137. Rindler, W., Essential Relativity.

138. Xevier, C, C Language and Numerical Methods, New Age International.

139.Mathematical analysis Vol I &II ,Vladimir Zorich,Springer.

140.Real Mathematical Analysis-Charles Chapman Pagh, Springer.

141.Calculus,M.Spivak,CUP


Page 16 of 16

142.Depth In C; Sribastab Sribastab

Documents

Detailed Syllabus Semester 1: First Year First Semester ... Hons.pdf · evolution, computer generations ... Isomorphism and homomorphism. ... Line integrals of linear differential