Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

Finally, we note that a curver(u) representing the trajectory of a particle may

sometimes be given in terms of some parameteruthat is not necessarily equal to

the timetbut is functionally related to it in some way. In this case the velocity

of the particle is given by

v=

dr

dt

=

dr

du

du

dt

.

Differentiating again with respect to time gives the acceleration as

a=

dv

dt

=

d

dt

(

dr

du

du

dt

)

=

d^2 r

du^2

(

du

dt

) 2

+

dr

du

d^2 u

dt^2

.

10.4 Vector functions of several arguments

The concept of the derivative of a vector is easily extended to cases where the

vectors (or scalars) are functions of more than one independent scalar variable,

u 1 ,u 2 ,...,un. In this case, the results of subsection 10.1.1 are still valid, except

that the derivatives become partial derivatives∂a/∂uidefined as in ordinary

differential calculus. For example, in Cartesian coordinates,

∂a

∂u

=

∂ax

∂u

i+

∂ay

∂u

j+

∂az

∂u

k.

In particular, (10.7) generalises to the chain rule of partial differentiation discussed

in section 5.5. If a=a(u 1 ,u 2 ,...,un) and each of theuiis also a function

ui(v 1 ,v 2 ,...,vn) of the variablesvithen, generalising (5.17),

∂a

∂vi

=

∂a

∂u 1

∂u 1

∂vi

+

∂a

∂u 2

∂u 2

∂vi

+···+

∂a

∂un

∂un

∂vi

=

∑n

j=1

∂a

∂uj

∂uj

∂vi

. (10.17)

A special case of this rule arises whenais an explicit function of some variable

v, as well as of scalarsu 1 ,u 2 ,...,unthat are themselves functions ofv;thenwe

have

da

dv

=

∂a

∂v

+

∑n

j=1

∂a

∂uj

∂uj

∂v

. (10.18)

We may also extend the concept of the differential of a vector given in (10.9)

to vectors dependent on several variablesu 1 ,u 2 ,...,un:

da=

∂a

∂u 1

du 1 +

∂a

∂u 2

du 2 +···+

∂a

∂un

dun=

∑n

j=1

∂a

∂uj

duj. (10.19)

As an example, the infinitesimal change in an electric fieldEin moving from a

positionrto a neighbouring oner+dris given by

dE=

∂E

∂x

dx+

∂E

∂y

dy+

∂E

∂z

dz. (10.20)