devotion & mathematics

        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 * Divisibility * Properties of Divisibility  * Greatest Common Divisor (gcd) * properties of gcd The Principle of Induction   :              If Z is the set of all integers such that      i) 1 𝜖 Z    ii) n 𝜖 Z implies n+1 𝜖 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 iii) Again we have to pro...

          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 ).     ...

***Few problems & Solutions on derivatives of a vector functions *** Problem :               If r = a cost i + a sint j + at tanθ 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 :             ...