Physics 570
Homework No. 2
Solutions:
due Wednesday, 3 February, 2010
1.
On a 2dimensional manifold, using Cartesian coordinates
{
x,y
}
, we are given the local vector
ﬁelds
e
V
and
e
C
, with components relative to the standard coordinate basis as
V
x
=
xy ,
V
y
=

y
2
,
C
x
=
y ,
C
y
=

x .
a.
For each of these vector ﬁelds, ﬁrst determine the curve Γ(
η
) for which this vector ﬁeld is
the tangent vector, and normalize constants of integration so that for
η
= 0 the curve is at
the initial point on the manifold,
x
= 2
,y
= 1. Then produce a sketch of the curve and
also a few other adjacent curves, i.e., ones with the same tangent vector ﬁeld but having an
initial point somewhere nearby. Such a set of curves is referred to as a
congruence of curves
,
because they are all “congruent” to each other in the sense of the use of the word in plane
geometry.
b.
If the initial point is not particularly nearby, do the curves still look basically the same? Test
this by now choosing an initial point as
x
=

2
,y
=

1. [5 pts]
...........................................................................................
First we consider the curve for which
e
V
is the tangent vector. We know that a tangent vector
has components that are just the derivatives of the components along the curve; therefore, we
may write the following:
dx
dη
=
V
x
=
xy ,
dy
dη
=
V
y
=

y
2
.
We easily integrate the second of these integrations, to give
y
= 1
/
(
η

c
), where
c
is a constant
of integration. Inserting this information into the ﬁrst equation it is also, now, easily integrated
to give
x
=
α
(
η

c
), with
α
a second constant of integration. We may then simply specify the
curve in the way shown on the ﬁrst line of the equations just below, which is satisfactory for
the requested solution to the problem; however, an optimal presentation is, instead, shown by
determining some “meaning” for the constants of integration, by rewriting them in terms of the