हार्मोनल परिवर्तन के कारण कम उम्र में लड़कियों में हो रहा शारीरिक परिवर्तन प्रदुषण और जंक फूड घटा रहा मुश्किल भरे दिन की उम्र लड़कियों में समय से पहले हार्मोनल चेंज के मामले बढ़ रहे हैं। शारीरिक बदलाव के लिए औसत उम्र 13 से 14 साल मानी जाती है, अब 8 से 11-12 साल में वजन बढ़ने जैसी समस्या आ रही है। कम उम्र में मुश्किल भरे दिनों का सामना करना पड़ रहा है। विशेषज्ञों की मानें तो समस्या प्रदुषण और जंक फूड के अत्यधिक सेवन से बढ़ रही है। शरीर में हार्मोनल परिवर्तन, किसी प्रकार की सिस्ट और ट्यूमर जैसे कारण सामने आ रहे हैं। इनके लिए मुख्य कारण निम्न हैं- लड़कियों में हार्मोनल परिवर्तन से पीरियड जल्दी आते हैं। आनुवंशिक समस्या इसके लिए जिम्मेदार हो सकती है। तनाव से भी हार्मोनल परिवर्तन होते हैं। आयरन तथा विटामिन-डी जैसे पोषण तत्वों की कमी भी हार्मोनल परिवर्तन के लिए जिम्मेदार हैं। प्रदुषण के सम्पर्क में रहने पर भी ये समस्या होती है। शरीर में कही भी सिस्ट या ट्यूमर होने पर भी ये सम्भव है। अगर 8 साल से 12 साल की उम्र के बीच किसी बालिका के शरीर में तेजी से परिवर्तन हो तो मुश...
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
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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