Description about the workshop

This short-term workshop on Analysis-I offers a rigorous introduction to foundational concepts in real analysis, tailored for undergraduate and early graduate students in mathematics and allied disciplines. Topics include Rolles Theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Taylor’s and Maclaurin’s theorem. Limit, continuity, partial derivatives and their geometrical interpretation for functions of several variables, total differential and differentiability, derivatives of composite and implicit functions, derivatives of higher order and their commutativity, Euler’s theorem on homogeneous functions, Taylor’s expansion of functions, maxima and minima ODEs: First order exact differential equations, general linear differential equations with constants coefficients, method of variation of parameters, Cauchy-Euler equations. Improper integrals and tests for convergence, Beta and Gamma functions and their elementary properties. Differentiation under integral sign including variables limits-Leibnitz rule. Double and triple integrals, changing the order of integration, change of variables – Jacobian of a transformation. Definition of vector and scalar fields, level surfaces, limit, continuity, differentiability of vector functions. Directional derivative, gradient, curl, divergence and their geometrical interpretation. 

Participants will engage with curated exercises, historical insights, and intuitive geometric interpretations to build deep conceptual clarity. The program is ideal for those preparing for competitive exams, research careers, or advanced coursework in pure and applied mathematics. Prior exposure to calculus is recommended; mathematical maturity will be nurtured throughout.