devotion & mathematics

problems on partial differential of vector functions : Vector Calculus:                                                                 Problem :                               If f = ( 2x 2 y-x 4 ) i + ( e xy – ysinx ) j + (x 2 cosy ) k   then find ∂ 2 f/ ∂x 2                                         and  ∂ 2 f/∂x∂y. Solution :                  Problem :   If A = 2x 2 i -3yz j +xz 2 k and ⲫ = 2z-x 3 y then find  A. [ i ∂ ⲫ /∂x + j ∂ ⲫ /∂y + k ∂ ⲫ /∂z ]              and   A x   [ i ∂ ⲫ /∂x + j ∂ ⲫ /∂y + k ∂ ⲫ /∂...

                                      DIFFERENTIAL OPERATORS                                                                            Contents :  1. Scalar  Point Function  2. Vector Point Function            3. Delta Neighborhood  4. Limit  5. Continuity  6. Directional Derivative at a point   7. Level Surface 1. Scalar Point Funtion :              Let S be a domain in space. If to each point p 𝛜 S there corresponds a scalar  f( p ) then  is called  a scalar point function over the domain S.          ...

        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...