Module 1 (10 hours):
Review of basic concepts of real numbers (Archimedean property, completeness property etc.), Metric spaces, compactness, connectedness (with emphasis on R^n ), Functions of several-variables, Limit and Continuity.

Module 2 (16 hours):
Directional derivative, Partial derivative, Total derivative, Jacobian, Chain rule and Mean-value theorem, Repeated Partial Derivatives, Higher order derivatives, Taylor's theorem, Extremum problems, Extremum problems with constraints, Lagrange's multiplier method, Inverse function theorem, Implicit function theorem.

Module 3 (6 hours):
Riemann Integral of real-valued functions on Euclidean spaces, Fubini's Theorem.

Module 4 (8 hours):
Sequence and Series of functions, Pointwise Convergence, Uniform convergence, Weierstrass Approximation Theorem, Equicontinuity, Arzela-Ascoli Theorem.

To revisit few basic concepts of real numbers, metric spaces, and learn the topological properties of R^n, such as connectedness, compactness, etc.

To introduce the concepts of limit, continuity, differentiation and integration on higher dimensions in continuation to what we know for the single variable case.

To learn applications of differentiation and integration on higher dimensions.

To introduce the basics of sequence and series of functions and the approximation theorems.

CO1: Students will gain an understanding of the fundamentals of Euclidean spaces and their topological properties, while also learning about limits and continuity in functions of multiple variables.

CO2: Students will explore differentiation in higher dimensions and gain familiarity with the Inverse Function Theorem and the Implicit Function Theorem.

CO3: Students will learn to solve selected optimization problems involving functions of multiple variables.

CO4: Students will develop an understanding of integrating functions of several variables and learn to apply Fubini's Theorem for evaluating integrals.

CO5: Students will become familiar with the basics of sequences and series of functions, along with key approximation theorems.

W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 1984.

S. R. Ghorpade, B. V. Limaye, A course in multivariable calculus and analysis, Springer, New York, 2010.

T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2001. , 5th edition.

R. G. Bartle, The elements of Real Analysis, John-Wiley and Sons, 1967.