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Final

MAT237Y1 Final: Notes8Exam


Department
Mathematics
Course Code
MAT237Y1
Professor
John Bland
Study Guide
Final

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2.1.
Summary.
We have considered several kinds of geometric objects:
curves in
R
2
surfaces in
R
3
more generally,
k
-dimensional objects in
R
n
, for 1
k < n
. We considered this
general case more briefly.
Of course this general case includes, as special cases, the examples considered
above
curves in
R
2
:
k
= 1 and
n
= 2
surfaces in
R
3
:
k
= 2 and
n
= 3
In every case, the geometric object can be represented in several different ways.
as the graph of a function
as a zero locus (
i.e.
as a level set of a function)
parametrically, that is, as the image of a function.
In all of the above, what we mean by function (for example, what is the domain and the
range) depends on both the character of the geometric object and on the representation we
are considering (that is, graph
vs.
zero locus
vs.
level set.) See below for many examples.
In every case,
a geometric object that can be represented as a graph can
always
be represented as
a zero locus or parametrically.
1

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2
OR
{ }
but geometric objects that can be represented parametrically or as a zero locus
cannot always
be represented as a graph.
Because of this, we are interested in knowing:
given a geometric object represented as
a zero locus or via a parametrization, when can it be written (locally) as the graph of a
function of class C
1
?
This may seem like a dry question, but it is of basic importance for differential geometry
the study of geometric objects using methods of differential (and integral) calculus.
In particular, to the question
when can the zero locus of a function be written locally as the graph of a C
1
function?
is connected to the
geometric content
of the Implicit Function Theorem
2.2.
Curves in
R
2
.
There are 3 natural ways to represent curve in
S
R
2
:
as a
graph
:
(1)
S
=
{
(
x, y
)
R
2
:
y
=
f
(
x
) for
x
I
for some open interval
I
R
and some
f
:
I
R
.
S
=
{
(
x, y
)
R
2
:
x
=
f
(
y
) for
y
I
}
As a
level set
, for example
(2)
S
=
{
(
x, y
)
U
:
F
(
x, y
)
=
0
}
for some open
U
R
2
and some
F
:
U
R
.
This
is also called the zero locus of F , if
it
is the level set where F
=
0, as above.
parametrically
:
(3)
Theorem 1.
If S is a curve in
R
2
that can be represented as the graph of a fuction
f
:
I
R
of class C
1
(for some open interval I
R
), then
S can be represented as a zero locus of a function F of class C
1
, and
S can also be represented parametrically, as the image of a function
f
of class C
1
.
Proof.
If
S
=
{
(
x, y
)
R
2
:
y
=
f
(
x
) for
x
I
}
, then it can be written in the form (2) for
U
=
{
(
x, y
) :
x
I
}
, F
(
x, y
)
=
y
f
(
x
)
.
Clearly
F
is of class
C
1. And it also can be written in the form (3), for
I
:= the domain
of
f
, and
f
(
t
) = (
t,
f
(
t
))
.
Clearly
f
is of class
C
1.
Similar considerations apply if
S
=
(
x, y
)
R
2
:
x
=
f
(
y
) for
y I
, You may find it
useful to write out the details.
Q
S
=
{
f
(
t
) :
t
I
}
for some interval
I
R
and some
f
:
I
R
2
.

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3
.
Σ
{
}
.
Σ
On the other hand, if a curve is represented as a zero locus, or parametrically, then it
cannot always be represented as a graph. For example, the curve below shows the set
S
=
{
(
x, y
)
R
2
:
y
2
=
x
2(4
x
2)
}
.
The curve can also be represented parametrically as
S
=
f
(
t
) :
t
R
,
for
f
(
t
)
=
2 cos
t .
2 sin 2
t
But near the origin, it cannot be represnted as the graph of a
C
1 function
3
2
1
5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
Similarly, the curve below is (a part of)
S
=
{
(
x, y
) :
x
5
y
2
=
0
}
t
2
=
{
f
(
t
) :
t
R
}
for
f
(
t
)
=
t
5
.
3
2
1
5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
This too cannot be written as the graph of a
C
1 function near the origin.
The following theorems tell us conditions under which a zero locus or a parametric curve
can be written locally as the graph of a function of class
C
1.
Theorem 2.
Assume that U is an open subset of
R
2
and that F
:
U
R
is a function of
class C
1
. Let
S
:=
{
(
x, y
)
U
:
F
(
x,
y
)
=
0
}
.
If
F
(
a, b
)
ƒ
=
0
at some point
(
a, b
)
U, then there is an open set V containing
(
a, b
)
such that S
V can be represnted as the graph of a function of class C
1
.
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