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GAPHS, LEVEL SETS, SECTIONS
This is a geometrical look at functions of several variables. Many of you do much better in the
course, if on the top of analytical development presented in the text, some geometrical view is
given. For calculus of one variable it has been easy, the piece of paper is xyplane and a graph
of a function
was the set of points
(
x
,
). Its different story in higher dimensions,
)
(
x
f
y
=
)
(
x
f
although the ideas are similar.
Let
T
h
e
s
e
t
U
is sometimes called a “natural” domain of the function
f.
R
R
U
f
n
→
⊂
:
(for example a natural domain of the function
2
2
1
1
)
,
(
y
x
y
x
f
−
−
=
is
)
}
1
:
)
,
{(
2
2
<
+
=
y
x
y
x
U
Def
.
The
graph
of
f
is the subset of
R
n+1
consisting of all the points
))
...
,
,
(
,
,
....
,
,
(
2
1
2
1
n
n
x
x
x
f
x
x
x
for
(
)
in
U.
n
x
x
x
...,
,
,
2
1
graph
f
={
))
...
,
,
(
,
,
....
,
,
(
2
1
2
1
n
n
x
x
x
f
x
x
x
1
+
∈
n
R
: (
)
n
x
x
x
...,
,
,
2
1
U
∈
}
For
n = 2 , the graph is a surface
S in
R
3
,
S = {
}
U
y
x
R
y
x
f
y
x
∈
∈
)
,
(
:
))
,
(
,
,
(
3
Def
.
The
level set of value
is the set of those points
x
R
c
∈
U
∈
at which
.
c
f
=
)
(
x
L
c
=
n
R
c
f
U
⊂
=
∈
}
)
(
:
{
x
x
Note that the level set is always in the domain space.
If
n = 2, we speak of a
level curve
(of value
c
). If
n = 3, we speak of a
level surface.
Example 1
. Describe the level curves of the quadratic function
.
2
2
2
)
,
(
,
:
y
x
y
x
R
R
f
+
→
a
Solution
: We may rewrite
f
as
. The level sets of value
c
are
.
2
2
y
x
z
+
=
c
y
x
=
+
2
2
The level curve of value
c
is empty for
c
< 0, for
c >
0
it is a circle of radius
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View Full Document Very often this information is enough to sketch a graph of a function, in the same way as looking
at the geographical
map of mountains give an idea about their shape. Below are the level curves of
the Example 1, raised to the graph:
But in many cases the sections may help.
Def
. A
section
of the graph of
f
is the intersection of the graph and a “vertical” hyperplane
P
i
in R
n+1
S
i
= P
i
∩
graph
f
In
R
3
those hyperplanes are usually plains in space
x
=
a
,
y
=
b
or
y =
kx
for some
a
,
b
,
k
R
∈
.
Example 2
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto Toronto.
 Fall '09
 RomauldStanczak
 Calculus, Sets

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