Page 80 - CatalogNEP-PS

Vector field, Divergence and curl, Line integrals and applications to mass and work, Fundamental
theorem for line integrals, Conservative vector fields, Green's theorem, Area as
a line integral, Surface integrals, Stokes' theorem, Gauss divergence theorem.

*TUTORIAL (15 Hours (1 Hour per week))

SUGGESTED READINGS:

1. Strauss, Monty J., Bradley, Gerald L., & Smith, Karl J. (2007). Calculus (3rd ed.). Dorling
27. Kindersley (India) Pvt. Ltd. Pearson Education. Indian Reprint.
2. Marsden, J. E., Tromba, A., & Weinstein, A. (2004). Basic Multivariable Calculus.
Springer (SIE). Indian Reprint.

HIGER LEVEL DISCIPLINE SPECIFIC COURSES:

Math.411 Advanced Real Analysis 3+1*

LEARNING OBJECTIVES:

The primary objective of this course is to introduce:
 Reimann Stieltjes integrals to relate Riemann-integral and Reimann Stieltjes integral and
understand the partial integration theorem to evaluate R-S integrals of functions.
 Knowledge about term-by-term integration and term by term differentiation
 for evaluating uniform convergence of series of real valued functions.
 The methods to examine uniform convergence of sequences and series of real valued
functions with idea about the uniform convergence of sequence and series of functions.
 The concepts of complete metric space, perfect set and connected set.

LEARNING OUTCOMES:

On completion of the Course, students will be able to
 Develop the understanding of Reimann Stieltjes integrals to relate Riemann-integral and
Reimann Stieltjes integral. Also understand the partial integration theorem to evaluate R-S
integrals of functions.
 Acquire knowledge about term-by-term integration and term by term differentiation
 for evaluating uniform convergence of series of real valued functions.
 Have the knowledge of methods to examine uniform convergence of sequences and series
of real valued functions with idea about the uniform convergence of sequence and series of
functions.
 Acquire knowledge of the concepts of complete metric space, perfect set and connected
set.

THEORY (45 Hours)

UNIT I: (15 Hours)

Convergent Sequences. Sub-sequences. Cauchy Sequences (in metric spaces). Absolute
Convergence. Addition and Multiplication of Series. Rearrangements of Series of Real Number.
Power series, Uniqueness Theorem for Power Series. Abel‘s and Taylor‘s Theorems. Continuity,
Limits of Functions (in Metric Spaces). Continuous Functions, Continuity, Uniform Continuity
and Compactness. Limit Inferior and Limit Superior. Integral Test. Comparison Test.

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