Lecture
XII
Terminology for PointSets in
Euclidean
Spaces and
MinimumMaximum Theorems
First let us
take
a short look at a problem that was
on
the exam.
We are
given
a level
curve
(in
E
2
) or
a surface(in
E
3
) and
a point
P
on
that curve or
surface.
How
to find
a vector
normal
to that curve or
surface at that point
P
?
Let us
consider the
case
of
E
3
.
The answer
lies
in
the gradient of
the function
defining the
surface.
The
graph
of
the surface is
given
by
z
=
f
(
x,
y
),
hence
z
−
f
(
x, y
) =
g
(
x, y, z
) = 0.
The gradient of
g
at
P
is
a vector
normal
to the
surface
at
P
:
∂f
∂f
�
P
=
P
j
+
ˆ
ˆ
k.
�
g

−
∂x

P
ˆ
i
−
∂y

Now
let us
take
a look at some basic notions
of
pointset topology of
Eu
clidean
spaces.
Definition 1
Given a point
P
, we
define
a
neighborhood
of
P
in the
following
manner:
(i) For the
1dimensional
space,
a neighborhood
is
an
interval
of
the form
[
c
−
r, c
+
r
]
for some
r >
0,
where
c
is
the point
P
.
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 Two '04
 Duorg
 Math, Critical Point, Sets, global minimum point, global maximum point

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