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अब औरत को लुगाई पढेंगे बच्चे

अब नई शिक्षा नीति के तहत मातृ भाषा में होगी प्राथमिक स्तर तक की पढ़ाई, औरत को लुगाई पढेंगे बच्चे शेखावटी के प्राथमिक स्कूलों के विद्यार्थी अब औरत को लुगाई, आग को बासते और इंसान को मिनख पढ़ते नजर आएंगे। नई शिक्षा नीति के तहत प्राथमिक कक्षाओं में मातृ भाषा में शिक्षा देने की नीति लागू होने से ऐसा होगा। इसके लिए राज्य शैक्षिक अनुसंधान एवं प्रशिक्षण संस्थान ने प्रदेश में संभागवार भाषाई सर्वेक्षण करवाकर स्थानीय भाषा के शब्द कोष तैयार करवाए हैं। शेखावटी के शब्द कोष में शोर को रोला, आदत को बाण, इधर को अठीने, इंतजार को उडीकणों, इकट्ठा को भैळो एवं सांवठो, अध्ययन को पढ़णो, आवास को ठिकाणो, इतना को अतरो, इनकार को नटणों, इसके लिए को इकेपाई एवं अधिक बोलने को लपर-लपर कहा गया है। नई शिक्षा नीति के लिए बच्चों-शिक्षकों की भाषा से शब्दकोष तैयार किया गया है। नई शिक्षा नीति के तहत हुए सर्वेक्षण में प्राथमिक कक्षा के बच्चों की भाषा एवं पढ़ाने वाले शिक्षक की भाषा की जानकारी ली गई थी। इसमें बच्चों के नाम के साथ उनके घर की भाषा, शिक्षक की भाषा, स्कूल का माध्यम, भाषा को समझने एवं बोलने की विद्यार्थियों की क्षम...
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Real Analysis | Mathematics | BSc

Real Analysis

Real Analysis and Theory of Convergence



Authors: Dr. Vimal Saraswat, Dr. Anil Kumar Menaria, Dr. Gajendrapal Singh Rathore

ISBN : 978-81-7906-338-5

Price: Rs. 395.00

Publisher:Himanshu Publications, Hiran Magri Udaipur; Himanshu Publications Prakash House, Ansari Road, New Delhi

E-mail : info@sacademy.co.in

Phone: +91 9664392614

To buy this book click on the link Real Analysis by Saraswat

This book includes the following topics 

Real Number System

  • Introduction
  • Field axiom
  • Uniqueness property
  • Cancellation law of addition and multiplication
  • Order axiom and ordered field
  • Positive class
  • Boundedness
    • Upper bound, Supremum, Lower bound, Infimum, Bounded set
  • Greatest and least element
  • Completeness axiom
  • Complete ordered field
  • Archimedean property of real numbers
    • Archimedean ordered field
  • Betweenness theorem
  • Dedekind's completeness axiom
  • Irrational numbers
  • Rational density theorem or denseness property
  • Absolute value of a real number or Modulus

Point Set Topology

  • Introduction
  • Neighbourhood (nbd) of a real number
  • Properties of neighbourhood
  • Interior and exterior point of a set
  • Interior of a set
  • Open set
  • Limit point of a set
  • Derived set and closed set
  • Closure
  • Open and closed interval
  • Nested interval property
  • Bolzano-Weierstrass theorem
  • Complement of set
  • Open cover, subcover and compact set
  • Heine Borel theorem
  • Connected and disconnected set

Countable Sets

  • Introduction
  • Equivalent sets
  • Finite and infinite set
  • Countable set
  • Uncountable set
  • Cantor ternary set
  • Binary representation
  • Ternary representation
  • Construction of Cantor ternary set
  • Properties of Cantor ternary set

Real Sequences

  • Introduction
  • Sequence
  • Range of a sequence
  • Bounded and unbounded sequence
  • Supremum and infimum of sequence
  • Monotonic sequence
  • Limit point of a sequence
  • Bolzano-Weierstrass theorem
  • Limit of a sequence
  • Convergent sequence
  • Divergent sequence
  • Oscillatory sequence
  • Theorems on convergence sequences
  • Theorems on convergence of monotonic sequences
  • Algebra of sequences
  • Sandwich theorem
  • Limit superior and limit inferior
  • Sub-sequence
  • Some theorems of sub-sequence
  • Cauchy's sequence or fundamental sequence
  • Some important theorems of Cauchy's sequence
  • Cauchy's general principle of convergence for sequence
  • Cauchy's first theorem on limits
  • Cauchy's second theorem on limits
  • Cesaro's theorem

Infinite Series

  • Introduction
  • Sequence of partial sums of series
  • Nature of an infinite series
  • Some important theorems
    • Cauchy's general principle of convergence Test of the convergence of geometric series
  • Comparative tests of the first type
  • Comparative tests of the second type
    • Ratio-comparison test; D' Alembert's ratio test; Raabe's test; de Morgan's and Bertrand's test; Logarithmic ratio test; Second logarithmic ratio test; Gauss's test
  • Some other useful tests
    • Cauchy's nth root test; Cauchy's condensation test
  • Alternating series
  • Absolute convergence
  • Conditionally convergence

Uniform Convergence

  • Introduction
  • Pointwise convergence of a sequence of functions
  • Uniform convergence
  • Series of functions
  • Cauchy's criterion for uniform convergence
  • Test for uniform convergence of a sequence and series of functions
  • Uniform convergence and continuity
  • Term by term integration
  • A sufficient condition for term by term integration of an infinite series
  • A sufficient condition for term by term differentiation of the series

Improper Integrals

  • Finite and infinite intervals
  • Bounded function
  • Improper integral
  • Types of improper integral
  • Convergence of improper integral of first kind
  • Convergence tests for the improper integral of first kind
  • Convergence of improper integral of second kind
  • Convergence tests for the improper integral of second kind
  • Convergence of improper integrals of third kind

Riemann Integration

  • Introduction
  • Partition of a closed interval
  • Norm of partition
  • Refinement of a partition
  • Supremum and infimum
  • Upper and lower Darboux sum
  • Theorems on Darboux sum
  • Upper and lower Riemann integral
  • Integral function
  • Riemann integral
  • Theorems of Riemann integral
    • Necessary and sufficient condition for a function to be R-integrable
  • Particular classes of Riemann integrable functions
  • Riemann integral as the limit of a sum
  • Properties of Riemann integral function
  • Integral function
  • Properties of integral function
  • Primitive
  • Mean value theorems of integral calculus
  • Fundamental theorem of integral
  • Techniques of integration

Fourier Series

  • Introduction
  • Perodic functions
  • Properties of definite integral
  • Some important definite integrals
  • Fourier series
  • Dirichlet's conditions for the expansion of a Fourier series
  • Even and odd functions
  • Fourier series for even and odd functions
  • Fourier's half range series
  • Other forms of Fourier series
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