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Study BSc - Real Analysis

UNIT I The Real and Complex number systems the field axioms, the order anixms, integers, the unique Factorization thorium for intcgcrs, Rational numbers, Irrational numbers --- Upper bounds , Maximum Elements, least upper bound, the completeness axiom, some properties for the sypremum, properties of the integers deduced from the completeness anxiom- The Archimedian property of the real number system Rational numbers with finite decimal representation of real numbers absolute values and the triangle inequality – the Cauchy- Sohewarz,inequality-plus and minus infinity and the extended real number system. Basic notions of asset theory. Notations –ordered pairs – Cartesian product of two sets, Relations and functions further terminology concerning functions one one functions and inverse composite functions- sequences-similar sets – countable and uncountable sets- uncountability of the real number system-set algerbra-countable of collection of countable sets. UNIT II Elements of point set topology: Euclidean space R". The strcture of open Sets in R" – closed sets and adherent points- The Bolzno – Weierstrass theorem – the Cantor intersection theorem. Covering Lindelof covering theorem the Heine Borel covering Compactness in R" Metric Spaces – Point set topology in metric spaces – compact subsets of a metric space – Boundary of a set. Unit III Convergent sequences in a metric space- Cauchy sequences – complete metric Spaces. Limit of a function Continuous functions composite functions. Continuous complex valued functions. Examples of continuous functions - continuity and inverse images of open or closed sets – functions continuous on compact sets – Topological mappings – Bolzano’s theorem. Unit IV Connectedness – components of metric space – Uniform continuity and compact setsfixed point theorem for contractions – monotonic functions.Definition of derivative-Derivative and continuity- Algebra of derivatives- the chain rule – one sided derivatives and infinitives derivatives- functions with non-zero derivatives-zero derivatives and local extrema – Roll’s theorem – the mean value theorem for derivatives- Taylor’s formula with remainder. Unit V Properties of monotonic functions- functions of bounded variation – total Variation – additive properties of total variation on (a,x) as a function of x- functions of bounded variation expressed as the difference of increasing functions- continuous functions of bounded variation. The Riemann – Stieltjes integral : Introduction – Notation – The definition of Riemann stieltjes integral Reduction to a Riemann integral.




BSc Subjects


Applied Electronics
Atomic Physics and Nuclear Physics
Classical Algebra and Calculus
CLIENT SERVER COMPUTING
Climatology and Oceanography
Complex Analysis
DATA COMMUNICATION AND NETWORKS
DATA STRUCTURES AND C PROGRAMMING
Differential Equation Lap lace Transforms
DIGITAL FUNDAMENTALS AND ARCHITECTURE
Economic geography
Geography of India
Geomorphology
Heat Thermodynamics Mechanics Properties of Matter and Sound
Human geography
INTERNET and JAVA PROGRAMMING
MATHEMATICAL FOUNDATION FOR COMPUTER
Mathematical Physics
Mechanics
Modern Algebra
OBJECT ORIENTED PROGRAMMING WITH C++
Optics Electricity and Magnetism
Principles of cartography
Quantum Mechanics and Relativity
Real Analysis
SOFTWARE ENGINEERING
Solid State Physics
Statistics for geographers
Statistics for Mathematics
SYSTEM SOFTWARE AND OPERATING SYSTEM
Trigonometry Vector Calculus and Analytical Geometry
VISUAL PROGRAMMING
WEB TECHNOLOGY
Business Economics
Business Management
Business Organisation and Office Management
Business Statistics
Commercial Law and Company Law
Company law and secretarial practice I
Company law and Secretarial practice II
Corporate Accounting
Cost and Management Accounting
Executive Communication
Financial Accounting
Industrial law
Life Office Management
Managerial Economics
Mathematics for Business
Principles of Accountancy
Principles of Life Insurance
Principles of Management
Principles of Marketing
Priniciples of Reinsurance
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