Monotonically increasing Integrator – upper and lower integrals – Additive and linearity Properties of Upper and Lower integrals – Riemann ‘s condition – Comparison theorem – integrators of bounded Variation – Necessary and sufficient Conditions for existence of Riemann- Stiltjes integral - mean value theorem of Riemann- Stiltjes integral.
Introduction – The directional Derivative – Directional derivatives and continuity – The total Derivative –partial Derivatives – The matrix of a linear function – The Jacobian matrix – The chain rule – Matrix form of the chain rule.
Implicit Functions and Extremum Problems – Introduction – functions with non zero Jacobian Determinant – The inverse Function Theorem – The implicit Function theorem.
Outer measure – Measurable sets and Lebesgue measure – Measurable functions – Littlewood’s Theorem.
The Lebesgue integral of bounded functions over a set of finite measure – integral of a non – negative function – General Lebesgue Integral – convergence in measure.
Reference Book:
1. Real Analysis by Gorden, 2nd Edition, Pearson’s Publication, 2011. 2. Introduction to Real Analysis by R.G.Bartle, Donald.R.Sharbert 3rd Edition, John Wily Student Edition. 3. Real and Complex Analysis by W.Rudin, 3rd Edition, McGraw-Hill, New York, 1986.
Text Book:
1. Mathematical Analysis by Tom .M Apostol, Narosa publishing House , Second Edition, Twentieth Reprint, 2002. Unit I –Chapter7 Sections: 7.11 to 7.18. Unit II- Chapter 12 Sections: 12.1 to 12.5 and 12.7 to 12.10. Unit III- Chapter 13 Sections: 13.1 to 13.4. 2. Real Analysis by H.L. Roydon, Third Edition, Macmillan, New York, 1988. Unit IV – V: Chapters 3 and 4.
