Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR CALCULUSFinally, we note that a curve r(u) representing the trajectory of a particle maysometimes be given in terms of some parameter u that is not necessarily equal tothe time t but is functionally related to it in some way. In this case the velocityof the particle is given byv = drdt = dr dudu dt .Differentiating again with respect to time gives the acceleration asa = dvdt = d ( ) ( ) 2 dr du= d2 r dudt du dt du 2 + dr d 2 udt du dt 2 .10.4 Vector functions of several argumentsThe concept of the derivative of a vector is easily extended to cases where thevectors (or scalars) are functions of more than one independent scalar variable,u 1 ,u 2 ,...,u n . In this case, the results of subsection 10.1.1 are still valid, exceptthat the derivatives become partial derivatives ∂a/∂u i defined as in ordinarydifferential calculus. For example, in Cartesian coordinates,∂a∂u = ∂a x∂u i + ∂a y∂u j + ∂a z∂u k.In particular, (10.7) generalises to the chain rule of partial differentiation discussedin section 5.5. If a = a(u 1 ,u 2 ,...,u n ) and each of the u i is also a functionu i (v 1 ,v 2 ,...,v n ) of the variables v i then, generalising (5.17),∂a∂v i= ∂a∂u 1∂u 1∂v i+ ∂a∂u 2∂u 2∂v i+ ···+ ∂a∂u n∂u n∂v i=n∑j=1∂a∂u j∂u j∂v i. (10.17)A special case of this rule arises when a is an explicit function of some variablev, as well as of scalars u 1 ,u 2 ,...,u n that are themselves functions of v; thenwehavedadv = ∂a∂v +n∑j=1∂a∂u j∂u j∂v . (10.18)We may also extend the concept of the differential of a vector given in (10.9)to vectors dependent on several variables u 1 ,u 2 ,...,u n :da = ∂a du 1 + ∂a du 2 + ···+ ∂an∑ ∂adu n = du j . (10.19)∂u 1 ∂u 2 ∂u n ∂u jAs an example, the infinitesimal change in an electric field E in moving from aposition r to a neighbouring one r + dr is given bydE = ∂E∂xdx +∂E∂y344j=1∂Edy + dz. (10.20)∂z