# «B. A. (Hons.) / B. Sc. (Hons.) Mathematics Syllabus for Semester System First Semester Code Course Periods/ Week Credits BHM-101 Calculus 4 4 BHM-102 ...»

1. Program in C for finding greater among three numbers using conditional operators.

2. Program in C to find sum of square of natural numbers.

3. Program in C to generate prime numbers.

4. Program in C to calculate simple interest using loops.

5. Program in C to find square using function.

6. Program in C to find factorial of an integer using recursive function.

7. Program in C to generate Fibonacci numbers.

8. Program in C to check the number is Armstrong.

9. Program in C to check the number palindrome.

10. Program in C to find sum of series (

11. Program in C to find sum of series.

12. Program in C to swap the values using call by value and call by reference.

13. Program in C to calculate area and perimeter of a circle using call by value and call by reference.

14. Program in C to find the percentage of marks using pointer.

15. Program in C to find sum of two matrices.

16. Program in C to find multiplication of two matrices.

17. Program in C to the root of non-linear equation using Bisection method.

18. Program in C to the root of non-linear equation using False position method.

19. Program in C to interpolate values using Lagrange’s method.

20. Program in C to for solving system of linear equations using Gauss-Seidal method.

21. Program in C to evaluate the integral using Trapezoidal rule.

22. Program in C to evaluate the integral using Simpson’s rule.

BHM-501: Analysis – I Unit I Bounded and unbounded sets, Infimum and supremum of a set and their properties, Order completeness property of R, Archimedian property of R, Density of rational and irrational numbers in R, Dedekind form of completeness property, Equivalence between order completeness property of R and Dedekind property. Order completeness in, Neighbourhood, Open set, Interior of a set, Limit point of a set, Closed set, Countable and uncountable sets, Derived set, closure of a set, Bolzano- Weierstrass theorem for sets.

Unit II Sequence of real numbers, Bounded sequence, limit points of a sequence, limit interior and limit superior convergent and non-convergent sequences, Cauchy’s sequence, Cauchy’s general principle of convergence, Algebra of sequences, Theorems on limits of sequences, Subsequences, Monotone sequences, Monotone convergence Theorem.

Unit III Infinite series and its convergence, Test for convergence of positive term series, Comparison test, Ratio test, Cauchy’s root test, Raabe’s test, Logarithmic test, Integral test, Alternating series, Leibnitz test, Absolute and conditional convergence.

Unit IV Continuous and discontinuous functions, Types of discontinuities, Theorems on continuity, Uniform continuity, Relation between continuity and uniform continuity, Derivative of a function, Relation between continuity and differentiability, Increasing and decreasing functions, Darboux theorem, Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Taylor’s theorem with Cauchy’s and Lagrange’s form of remainders.

**Books Recommended:**

• R. G. Bartle and D.R. Sherbert, Introduction to Real Analysis ( 3rd Edition), John Wiley and Sons ( Asia) Pvt Ltd., Singapore, 2002.

• S.C. Malik and Savita Arora: Mathematical Analysis, New Age International (P) Ltd.

Publishers, 1996.

• K. A. Ross, Elementary Analysis: The Theory of Calculus, Under graduate Texts in Mathematics, Springer ( SIE), Indian reprint, 2004.

• Sudhir R Ghorpade and Balmohan V. Limaye, A course in Calculus and Real Analysis, Undergraduate Text in Math., Springer (SIE). Indian reprint, 2004.

• T.M. Apostol: Mathematical Analysis, Addison-Wesley Series in Mathematics, 1974.

BHM-502: Differential Equations – II Unit I Series solutions of differential equations- power series method, Frobenius method, Bessel and Legendre differential equation and their series solution.

Unit II Partial differential equations of the first order, Lagrange’s solutions, Solution of some standard type of equations, Charpit’s general method of solution.

Unit III Partials differential equations of second and higher orders, Classification of linear partial differential equations of second order, Homogeneous and non-homogeneous equation with constant coefficients, Partial differential equations reducible to equations with constant coefficients, Monge,s method.

Unit IV Calculus of variations, Variational problems with fixed boundaries, Euler’s equation for functionals containing first order derivatives and one independent variable, Extremals, Functional dependent on higher order derivatives, Functional dependent on more than one independent variable, Variational problems in parametric form, Invariance of Euler’s equation under coordinates transformation.

.

**Books Recommended:**

• Dennis G. Zill, A first course in differential equations,

• Tyn Mint-U and Lokenath Debnath, Linear Partial Differential Equations

• D.A. Murray: Introductory Course on Differential Equations, Orient Longman (India), 1967.

• A.S. Gupta: Calculus of variations with applications, Prentice Hall of India, 1997.

• I.N. Sneddon: Elements of Partial Differential Equations, McGraw Hill Book Company, 1988.

BHM-503: Metric Spaces Unit I Definition and examples of metric spaces, open spheres and closed spheres, Neighbourhood of a point, Open sets, Interior points, Limit points, Closed sets and closure of a set, Boundary points, diameter of a set, Subspace of a metric space.

Unit II Convergent and Cauchy sequences, Complete metric space, Dense subsets and separable spaces, Nowhere dense sets, Continuous functions and their characterizations, Isometry and homeomorphism.

Unit III Compact spaces, Sequential compactness and Bolzano-Weierstrass property, Finite Intersection property, Continuous functions and compact sets.

Unit IV Disconnected and connected sets, Components, Continuous functions and connected sets.

**Books Recommended:**

1. G.F. Simmons: Introduction to Topology and Modern Analysis, McGraw Hill, 1963.

2. E.T. Copson, Metric spaces, Cambridge University Press, 1968.

3. P.K. Jain and Khalil Ahmad: Metric spaces, Second Edition, Narosa Publishing House, New Delhi, 2003.

4. B. K. Tyagi, first course in metric spaces, Cambridge University Press, 2010.

BHM-504: Linear Algebra Unit I Definition, examples and basic properties of a vector space. Subspaces. Linear independence. Linear combinations and span. Basis and dimension. Sum and intersection of subspaces. Direct sum of subspaces.

Unit II Definition and examples of linear transformations. Properties of linear transformations.

Rank and kernel. The rank and nullity of a matrix. Rank-Nullity Theorem and its consequence. The matrix representation of a linear transformation. Change of basis.

Isomorphism.

Unit III Scalar product in and Inner product spaces. Orthogonality in inner product spaces. Normed linear spaces. Inner product on complex vector spaces. Orthogonal complements. Orthogonal sets and the Gram-Schmidt process. Unitary matrices.

Unit IV Eigenvalues and eigen vectors. Characteristic equation and polynomial. Eigenvectors and eigenvalues of linear transformations and matrices. The Caley-Hamilton Theorem.

Similar matrices and diagonalization. Eigenvalues and eigenvectors of symmetric and Hermitian matrices. Orthogonal diagonalization. Quadratic forms and conic sections.

**Books recommended:**

• David C. Lay: Linear algebra and its applications (3rd Edition), Pearson Education asia, Indian Reprint, 2007.

• Geory Nakos and David Joyner: Linear algebra with Applications, Brooks/ Cole Publishing Company, International Thomson Publishing, Asia, Singapore, 1998.

• Stephen H. Friedberg, Arnold J. Insel and L.E.Space- Linear Algebra, 4th dition, PHI Pht Ltd., New Delhi, 2004.

• I. V. Krishnamurty, V.P. Mainra, J.L. Arora- An introduction to Linear Algebra, East West Press, New Delhi, 2002.

BHM-505: Probability and Statistics

**Unit I:**

Sample space and events, algebra of events, axiomatic approaches, conditional probability, basic laws of total probability and compound probability, Byes’ theorem, Independence.

**Unit- II:**

Discrete and continuous random variables, mathematical expectation, variance, moment about a point, central moment, moment generating function, Binomial, Poisson, Normal and Rectangular distributions.

**Unit III:**

Two-dimensional random variables, joint distribution functions, marginal distributions, covariance, linear regression and correlation, rank correlation, least square method of fitting regression lines.

**Unit IV:**

Sampling, random sampling, large sample tests of means and proportion. t-student, χ 2 (chi square) and F distributions (without derivation) and testing of hypothesis based on them.

Reference Books

1. Irwin Miller and Marylees Miller, John E. Freund's Mathematical Statistics with Applications, Pearson Education.

2. Robert V. Hogg, Allen Craig Deceased and Joseph W. McKean, Introduction to Mathematical Statistics, Pearson Education

3. Sheldon M. Ross, Introduction to probability and statistics for engineers and scientists, Elsevier Academic Press.

4. J.N. Kapur and H.C. Saxena, Mathematical Statistics, S. Chand.

BHM-601: Analysis – II Unit I Definition, existence and properties of Riemann integral of a bounded function, Darboux theorem, Condition of integrability, Rieman integrability for continuous functions, bounded functions, monotonic function and functions with finite or infinite number of discontinuities (without proof). The integral as the limit of the sums, Properties of Riemann integral, Fundamental theorem of calculus, First Mean value theorems, Change of variables, Second mean value theorem, Generalised mean value Theorems.

Unit II Definition of improper integrals, Convergence of improper integrals, Test for convergence of improper integrals Comparison test, Cauchy’s test for convergence, Absolute convergence, Abel’s Test, Dirichlet’s Test, Beta and Gamma functions and their properties and relations.

Unit III Definition of pointwise and uniform convergence of sequences and series of functions, Cauchy’s criterion for uniform convergence, Weierstrass M-test, Uniform convergence and continuity, Uniform convergence and differentiation, Uniform convergence and integration..

Unit IV Fourier Series, Fourier Series for even and odd functions, Fourier Series on intervals other than [ ]. Power series, Radius of convergence, uniform and absolute convergence, Abel’s Theorem (without proof), exponential and logarithmic functions.

**Books Recommended:**

• R. G. Bartle and D.R. Sherbert, Introduction to Real Analysis ( 3rd Edition), John Wiley and Sons ( Asia) Pvt Ltd., Singapore, 2002.

• S.C. Malik and Savita Arora: Mathematical Analysis, New Age International (P) Ltd. Publishers, 1996.

• K. A. Ross, Elementary Analysis: The Theory of Calculus, Under graduate Texts in Mathematics, Springer ( SIE), Indian reprint, 2004.

• Sudhir R Ghorpade and Balmohan V. Limaye, a course in Calculus and Real Analysis, Undergraduate Text in Math., Springer (SIE). Indian reprint, 2004.

• T.M. Apostol: Mathematical Analysis, Addison-Wesley Series in Mathematics, 1974.

BHM-602: Geometry of Curves and Surfaces Unit I Curves in Representation of curves, Unit and arbitrary speed curves, Tangent, Principal normal and binormal, Curvature and torsion, Frenet formula.

Unit II Behaviour of a curve near one of its points, The curvature and torsion of a curve as the intersection of two surfaces, Contact between curves and surface, Osculating circle, and osculating sphere, involutes and evolutes, Helics.

Unit III Introduction of Differential forms, covariant derivative, Frame field and connection forms, Surfaces in examples, tangent plane and surface normal, Orientability, The first fundamental form and its properties.

Unit IV Direction coefficients on a surface, orthogonal trajectories, Double family of curves, Second fundamental form, normal curvature, principal curvature, Gaussian and mean curvature.

**Books recommended:**

• Barrett O’Neill, Elementary Differential Geometry, Academic Press.

• W. Klingenberg, A course in differential geometry, Spriger-Verlag.

• T. Willmore, An introduction to Differential Geometry, Clarendon Press, Oxford C.

• E. Weatherburn, Differential Geometry of three dimensions, University press, Cambridge.

BHM-603: Complex Analysis Unit I Complex numbers as ordered pairs, Geometrical representation of complex numbers, stereographic projection, Limit and continuity, Complex derivative, Derivative and Analyticity, analytic functions, Cauchy-Riemann equations, Harmonic equations.

Unit II Elementary functions, Exponential functions, Trigonometric functions, Hyperbolic functions, Logarithmic functions, Analyticity of Log functions, Inverse trigonometric and Hyperbolic functions. Mapping by elementary functions, Mobius Transformations, Fixed points, Cross ratio, Inverse points and critical mappings, Conformal mappings.

Unit III Integration of complex-valued functions, Contours, Contour integrals, Anti derivatives, Cauchy Theorem (without proof), Simply and multiply connected regions, Cauchy integral formula (without proof), Line integration, Complex line integration, Contour integration and Green’s theorem (without proof), Path Independence, Indefinite Integrals, Fundamental theorem of calculus in the complex plane.

Unit IV Convergence of sequences and series, Taylor series, Laurent series, Maclaurin series.

**Books Recommended:**

• R. V. Churchill and J. W. Brown: Complex Variables and Applications, McGraw Hill Publishing Company, 1990.

• E. Hille: Analytic Functions Theory, Vol. 2, Ginn and Co. 2nd Ed. New York, 1973.