### 2.4 Mathematics Core Courses

#### MTH101: Symmetry

[Cr:3, Lc:2, Tt:1, Lb:0]

Course Outline

• Basic operations, row reduction, determinant and trace, Cramer’s rule.
• Basic notions of groups, permutation groups.
• Group actions, prime-power groups.
• Finite Abelian groups.
• Matrix groups, SO(3) and rotations.
• Groups of symmetries of real plane and Platonic solids.
• Vector spaces and linear transformations, eigen values and eigen vectors, diagonalization.

• M. A. Armstrong, Groups and Symmetry, Springer (1988).
• M. Artin, Algebra, Prentice-Hall of India, New Delhi (1994).
• I. S. Luthar and I. B. S. Passi, Algebra Vol. I, Narosa Publishing House, New Delhi (1996).
• J. A. Gallian, Contemporary Abstract Algebra, D. C. Heath Canada (1986).

#### MTH102: Analysis in one variable

[Cr:3, Lc:2, Tt:1, Lb:0]

Course Outline

• The real number system, completeness axiom, complex numbers.
• Sequences, limits, convergence, series.
• Polynomials, rational functions, continuous functions.
• Trigonometric, exponential, logarithmic and hyperbolic functions.
• Differentiation, mean value theorem, Taylor’s theorem.
• Uniform convergence, power series.
• Riemann integral, fundamental theorem of calculus.
• Fourier series.

• R. R. Goldberg, Methods of Real Analysis, Wiley (1970).
• K. A. Ross, Elementary Analysis, The Theory of Calculus, Springer (2004).
• T. M. Apostol, Calculus, Blaisdell Publishing Company, 1961.
• S. Shirali and H. L. Vasudeva, Mathematical Analysis, Alpha Science International Ltd. (2006).

#### MTH201: Curves and surfaces

[Cr:3, Lc:2, Tt:1, Lb:0]

Course Outline

• Differentiation of vectors.
• Curves in the plane and in space, arc length, reparametrization.
• Curvature, torsion, Serret-Frenet formulae.
• Fundamental theorem of curves in plane and space.
• Surfaces in three dimension (2-manifolds), smooth surfaces.
• Change of variable formula, surfaces of revolution.
• First and second fundamental forms, isometries, conformal mappings.
• Normal and principal curvatures, Gaussian curvature and the Gauss map.
• Geodesics, geodesic curvature, Gauss’ theorema egregium.

• Isoperimetric inequality, four vertex theorem.
• Area and volume integrals, surface area.

• L. Brand, Vector Analysis, Dover Publications (2006).
• A. Pressley, Elementary Differential Geometry, SUMS, Springer (2001).

#### MTH202: Probability and statistics

[Cr:3, Lc:2, Tt:1, Lb:0]

Course Outline

• Recapitulation: Counting (urn, coins, cards).
• Axiomatic approach to probability, conditional probability, independence of events.
• Discrete random variables, probability mass function, some standard discrete distributions and examples.
• Continuous random variables, probability density function, some standard continuous distributions and examples.
• Bivariate distributions (discrete and continuous), marginal and conditional distributions, covariance, correlation coefficient.
• Moments, Markov’s inequality, Chebychev’s inequality.
• Sums of independent random variables, law of large numbers, central limit theorem
• A glimpse into estimation theory (maximum likelihood estimation, method of moments) and testing of hypothesis.