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Vector functions of several arguments

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Vector functions of several arguments
VECTOR CALCULUS
Finally, we note that a curve r(u) representing the trajectory of a particle may
sometimes be given in terms of some parameter u that is not necessarily equal to
the time t but is functionally related to it in some way. In this case the velocity
of the particle is given by
dr du
dr
=
.
v=
dt
du dt
Differentiating again with respect to time gives the acceleration as
2
d dr du
dr d2 u
dv
d2 r du
=
+
.
a=
= 2
dt
dt du dt
du
dt
du dt2
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,
u1 , u2 , . . . , un . In this case, the results of subsection 10.1.1 are still valid, except
that the derivatives become partial derivatives ∂a/∂ui defined as in ordinary
differential calculus. For example, in Cartesian coordinates,
∂a
∂ax
∂ay
∂az
=
i+
j+
k.
∂u
∂u
∂u
∂u
In particular, (10.7) generalises to the chain rule of partial differentiation discussed
in section 5.5. If a = a(u1 , u2 , . . . , un ) and each of the ui is also a function
ui (v1 , v2 , . . . , vn ) of the variables vi then, generalising (5.17),
∂a ∂uj
∂a
∂a ∂u1
∂a ∂u2
∂a ∂un
=
+
+ ···+
=
.
∂vi
∂u1 ∂vi
∂u2 ∂vi
∂un ∂vi
∂uj ∂vi
n
(10.17)
j=1
A special case of this rule arises when a is an explicit function of some variable
v, as well as of scalars u1 , u2 , . . . , un that are themselves functions of v; then we
have
n
da
∂a ∂a ∂uj
=
+
.
(10.18)
dv
∂v
∂uj ∂v
j=1
We may also extend the concept of the differential of a vector given in (10.9)
to vectors dependent on several variables u1 , u2 , . . . , un :
∂a
∂a
∂a
∂a
du1 +
du2 + · · · +
dun =
duj .
∂u1
∂u2
∂un
∂uj
n
da =
(10.19)
j=1
As an example, the infinitesimal change in an electric field E in moving from a
position r to a neighbouring one r + dr is given by
dE =
∂E
∂E
∂E
dx +
dy +
dz.
∂x
∂y
∂z
344
(10.20)
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