Graphics Reference
In-Depth Information
Finally, in the above discussion we have assumed that f was defined everywhere.
However, if f has a restricted domain
D
, then additional problems arise, namely, what
do we do when the new point
x
k+1
falls outside this domain? One needs to “clip” the
point somehow. A guideline for handling such a situation when the domain is a rec-
tangle is the following:
If our current guess is
on
the boundary of
D
and the next forces us to go outside,
then one should suspect a critical point
on
the boundary and deal with this as a
special case. One uses as initial guess the corners and center of the rectangle.
4.8 Integration
This section will sketch how one can define integrals for functions of several variables
and state several of the most important theorems. Proofs are omitted and can be found
in [Spiv65] or [Buck78].
Let
A
Õ
R
n
and f :
A
Æ
R
. To define the integral of f we shall follow the sequence
of steps below:
(1) We define the integral for the case where
A
is a rectangular set by using upper
and lower sums for partitions of that set.
(2) We prove that the integral exists if the points of discontinuity of f are a set of
measure zero.
(3) We extend the definition of the integral to arbitrary bounded sets
A
whose
boundaries are sets of measure zero.
Definition.
A subset
A
of
R
n
of the form [a
1
,b
1
] ¥ [a
2
,b
2
] ¥ ...¥ [a
n
,b
n
] with b
i
> a
i
,
i = 1, 2,..., n, is called an
n-rectangle
or simply a
rectangle
if n is clear from the
context. The boundary of an n-rectangle consists of 2n planar pieces called the
faces
of the n-rectangle. The
volume
of
A
, denoted by vol(
A
), is defined by
n
'
1
()
=
(
)
vol
b
-
a
.
i
i
i
=
If b
1
- a
1
= b
2
- a
2
= ...= b
n
- a
n
, then
A
is called an
n-dimensional cube
or
n-cube
or
simply
cube
.
Definition.
A
partition
of a rectangle
A
= [a
1
,b
1
] ¥ [a
2
,b
2
] ¥ ...¥ [a
n
,b
n
] in
R
n
is a
sequence P = (P
1
,P
2
,...,P
n
) where P
i
is a partition of the interval [a
i
,b
i
]. A
subrectan-
gle
of the partition P is a rectangle [c
1
,d
1
] ¥ [c
2
,d
2
] ¥ ...¥ [c
n
,d
n
] where [c
i
,d
i
] is a
subinterval of the partition P
i
. The
norm
of the partition P, denoted by |P|, is defined
by
{
}
P
=
max
P
i
=
12
, ,...,
n
.
i
