Lecture Notes for 120 - UCLA Department of Mathematics

1.1. CURVES 7
to the differential equation
apple dx
dt
dy
dt
apple
=
@F
@y
@F
@x
(x (t) ,y(t))
(x (t) ,y(t))
In three variables we need two functions as such functions have level sets that are
surfaces:
F 1 (x, y, z) = c 1
F 2 (x, y, z) = c 2
In this case we also have a differential equation approach. Both of the gradients rF 1
and rF 2 are perpendicular to their level sets. Thus the cross product rF 1 ⇥rF 2
is tangent to the intersection of these two surfaces and we can describe the curves
as solutions to
dq
dt =(rF 1 ⇥rF 2 )(q)
It is important to realize that when we are looking for solutions to a first order
system
dq
= F (q (t))
dt
then we geometrically obtain the same curves if we solve
dq
dt =
(q (t)) F (q (t))
where is some scalar function as the directions of the velocities stay the same.
However, the parametrizations of the curves will change.
Classically curves were given descriptively in terms of geometric or even mechanical
constructions. Thus a circle is the set of points in the plane that all have
afixeddistanceR to a fixed center. It then it became more common starting with
Descartes to describe them by equations. Only about 1750 did Euler switch to
considering parametrized curves.
We present a few classical examples of these constructions in the plane.
Example 1.1.1. Consider the equation
F (x, y) =x 2 + y 2 = c
when c>0 this describes a circle of radius p c.Whenc =0we only get the origin,
while when c