Real Analysis Reading Course

This course is specifically designed for students seeking to bridge the gap between computational calculus and the formal proof-based thinking required for higher level mathematical analysis. Course Overview The curriculum is primarily modeled after the first volume of the definitive classic by Vladimir A. Zorich. The course emphasizes a synthesis of formal rigor and geometric intuition, exploring the interplay between classical analysis and the sciences (physics, mechanics, and biology). Academic Modules The course is structured into eight comprehensive modules: I. Foundations: Mathematical Logic, Set Theory, and Functions. II. The Real Number System: Axiomatics, Completeness, and Cardinality. III. Theory of Limits: Sequences, Series, and the Cauchy Criterion. IV. Continuity: Global properties, Discontinuities, and Inverse Functions. V. Single-Variable Differentiation: Differentials and Taylor’s Formula. VI. Integration: The Riemann Integral and the Newton-Leibniz Formula. VII. Multivariable Calculus (I): Limits and Continuity in R^m VIII. Multivariable Calculus (II): Jacobians, Implicit Functions, and Constrained Extrema. Scientific Applications - Beyond abstract theory, the course explores the mathematical modeling of reality, including: - Relativistic Mechanics: Galilean and Lorentz transformations. - Physics: Radioactive decay, rocket motion, and atmospheric pressure. - Oscillations: Simple harmonic and damped motion. Course Structure (Weekly) - Theory Session: Formal lectures focusing on proofs and conceptual development. - Problem Solving: Rigorous training in non-standard problems from competitive examinations. - Reading Session: Guided reading to cultivate the habit of interpreting primary mathematical literature. Course Logistics Tuition Fee: ₹3,000 per month for 5 classes per week. Certification: An official Proof of Completion will be awarded upon successful fulfillment of the course requirements, suitable for inclusion in academic CVs and resumes.