Department

MathematicsCourse Code

MAT237Y1Professor

John BlandStudy Guide

FinalThis

**preview**shows pages 1-3. to view the full**10 pages of the document.**•

≤

•

•

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

•

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

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

.

•

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

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

.

###### You're Reading a Preview

Unlock to view full version