Functions of 2 and 3 Real Variables for JAM and MSc Exams

Transitioning from single-variable calculus to functions of multiple variables is one of the most challenging steps in undergraduate mathematics. Understanding how functions behave in two and three dimensions is essential for scoring high on competitive mathematics exams. This text-only course provides a comprehensive, step-by-step guide to mastering functions of two and three real variables. You will build a rock-solid foundation in multivariable calculus, moving from basic definitions to advanced problem-solving techniques. Through detailed written explanations and guided practice problems, you will learn how to analyze limits, evaluate differentiability, and find extreme values with confidence. What you'll learn: 1. Understand the foundational concepts of domains, neighborhoods, and limit points in multi-dimensional space. 2. Evaluate limits and continuity for functions of two and three variables using algebraic and polar coordinates. 3. Calculate partial derivatives, directional derivatives, and total differentials with precision. 4. Analyze differentiability using first principles and standard sufficient conditions. 5. Find local maxima, minima, and saddle points using the Jacobian and Hessian matrices. 6. Practice solving exam-style problems with clear, step-by-step written solutions. The course begins with essential definitions of multi-dimensional spaces, limits, and continuity. You will then progress through partial differentiation, differentiability criteria, and optimization techniques, wrapping up with comprehensive practice sets designed to build exam-day confidence. This course is designed for undergraduate mathematics students, particularly those preparing for competitive university entrance exams, who want to strengthen their multivariable calculus skills. No advanced background is required; a basic understanding of single-variable calculus is recommended. Start reading today to master multivariable calculus and ace your next mathematics exam!