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Properties of numbers : Number Theory #Properties #of #numbers : #Number #Theory

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        So far we have seen about the historical background to Number Theory.   Now we enter into subject starting with  the basic definitions & properties of Numbers. Contents   : * The Principle of Induction * The well - Ordering Principle * Definition of Divisibility * Properties of Divisibility  * Greatest Common Divisor * Relatively Prime Numbers  etc. The Principle of Induction   :              If Z is the set of all integers such that      i) 1 πœ– Z    ii) n πœ– Z implies n+1 πœ– Z  then   iii) all integers ≥ 1 belong to Z. In another manner            The principle of induction is useful to define a statement p(n) is exists for all integers n  which are to be proved in the following steps.  i) We have to prove P(1) is true i.e. the statement is true for n=1 ii) Assume P(k) is true i.e. the statement is true for n=k...
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Partial Differentiation of vector functions : vector calculus #Partial #Differentiation #of #vector #functions : #vector calculus

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          Partial Differentiation of vector function :                 In the previous section we  have studied the differentiation of  vector functions in one variable.  But a vector may be a function of more than one scalar variable.             partial differentiation arise in geometry, physiscs and applied mathematics  when the number of independent variables in the problem  under consideration is two or more. Under such a  situation, any dependent variable will be a function of more than one variable and hence it possesses not ordinary derivatives with respect to a single variable, but partial derivatives with respect to several independent variables .             Let f be the vector function of scalar variables p,q,r over a domain S, then we write f = f ( p,q,r ).     ...
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problems on derivative of vector functions : vector Calculus #vector #calculus #few #problems #& #solutions #on

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***Few problems & Solutions on derivatives of a vector functions *** Problem :               If r = a cost i + a sint j + at tanΞΈ k then find (i) |(dr/dt)  x (d 2 r/dt 2 ) |                                                                                    (ii)  [ (dr/dt)    (d 2 r/dt 2 )   (d 3 r/dt 3 )  ] Solution : Problem :      If r= e t (c cos2t+ d sin2t) where c and d are constant vectors, then show that d 2 r/dt 2 -2 dr/dt  +5r = 0. Solution :  Problem :            If r = t 2  i -t j + (2t+1) k, find the values of dr/dt ,d 2 r/dt 2   , |dr/dt| and | d 2 r/dt 2 | at t=0. Solution :             ...
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Derivative of a Vector Function : vector Calculus #Derivative #of #a #Vector #Function : #vector Calculus

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                                                                                                    Contents :                                                                                  1. Vector function of a scalar variable          2. Limit of a vector function          3. Continuity of a vector function          4. Derivative of a vector function          5. Higher order derivatives          6. Propert...
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Conclusion to introduction of Number Theory : Number Theory #number #theory

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  Conclusion   to introduction of  Number Theorem :                     ( part 5 continuation )                      We conclude this introduction with a brief mention of some outstanding unsolved problems concerning prime numbers. 1            Goldbach’s problem : Is there an even number >2 which is not the sum of two                                                      primes?              *   Is there an even number >2 which is not the difference   of two primes 3.          *  Are there infinitely many twin primes? 4.          *  Are there infinitely many Mersenne primes , that is, p...
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