Course Details
Subject {L-T-P / C} : MA4105 : Calculus of Several Variables { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Dr. Hiranmoy Pal
Syllabus
Real Analysis: Review of basic concepts of real numbers (Archimedean property, completeness property etc.), Metric spaces, compactness, connectedness (with emphasis on R^n ).
Sequence and Series of functions: Pointwise Convergence, Uniform convergence, Weierstrass Approximation Theorem, Equicontinuity, Arzela-Ascoli Theorem.
Mutivariable Calculus: Functions of several-variables, Limit and Continuity, 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. Riemann Integral of real-valued functions on Euclidean spaces, Fubini's Theorem.
Course Objectives
- Revisit few basic concepts of real numbers, metric spaces, and learn the topological properties of R^n, such as connectedness, compactness, etc.
- Introduce the basics of sequence and series of functions, and then study the approximation theorems.
- Introduce the concepts of limit, continuity, differentiation and integration on higher dimensions in continuation to what we know for the single variable case.
- Learn applications of differentiation and integration on higher dimensions.
Course Outcomes
CO1: Students have the basic knowledge about the Euclidean space R^n and its topological properties. <br />CO2: They become familiar with the approximation theorems, such as the Weierstrass Theorem on polynomial approximations. <br />CO3: Students have more general concepts about limit, continuity, differentiation and integration in higher dimensions. <br />CO4: They also learn to solve few optimization problems on the functions of several variables.
Essential Reading
- 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.
Supplementary Reading
- T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2001. , 5th edition
- R. G. Bartle, The elements of Real Analysis, John-Wiley and Sons, 1967.