VECTOR CALCULUS
Finally, we note that a curver(u) representing the trajectory of a particle maysometimes 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
dudu
dt.Differentiating again with respect to time gives the acceleration as
a=dv
dt=d
dt(
dr
dudu
dt)
=d^2 r
du^2(
du
dt) 2
+dr
dud^2 u
dt^2.10.4 Vector functions of several argumentsThe 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
∂ui+∂ay
∂uj+∂az
∂uk.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=∑nj=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+∑nj=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 1du 1 +∂a
∂u 2du 2 +···+∂a
∂undun=∑nj=1∂a
∂ujduj. (10.19)As an example, the infinitesimal change in an electric fieldEin moving from a
positionrto a neighbouring oner+dris given by
dE=∂E
∂xdx+∂E
∂ydy+∂E
∂zdz. (10.20)