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 /∂...

          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 1 :               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 2 :      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 3 :            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 :           ...

                                                                                                    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...