<s>
Discrete	B-Algorithm
calculus	I-Algorithm
or	O
the	O
calculus	O
of	O
discrete	O
functions	O
,	O
is	O
the	O
mathematical	O
study	O
of	O
incremental	O
change	O
,	O
in	O
the	O
same	O
way	O
that	O
geometry	O
is	O
the	O
study	O
of	O
shape	O
and	O
algebra	O
is	O
the	O
study	O
of	O
generalizations	O
of	O
arithmetic	O
operations	O
.	O
</s>
<s>
Discrete	B-Algorithm
calculus	I-Algorithm
has	O
two	O
entry	O
points	O
,	O
differential	O
calculus	O
and	O
integral	O
calculus	O
.	O
</s>
<s>
These	O
two	O
points	O
of	O
view	O
are	O
related	O
to	O
each	O
other	O
by	O
the	O
fundamental	O
theorem	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
Informally	O
,	O
the	O
limit	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
as	O
is	O
infinitesimal	O
calculus	O
.	O
</s>
<s>
Even	O
though	O
it	O
serves	O
as	O
a	O
discrete	O
underpinning	O
of	O
calculus	O
,	O
the	O
main	O
value	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
is	O
in	O
applications	O
.	O
</s>
<s>
Discrete	O
differential	O
calculus	O
is	O
the	O
study	O
of	O
the	O
definition	O
,	O
properties	O
,	O
and	O
applications	O
of	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
a	O
function	O
.	O
</s>
<s>
The	O
process	O
of	O
finding	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
called	O
differentiation	O
.	O
</s>
<s>
Given	O
a	O
function	O
defined	O
at	O
several	O
points	O
of	O
the	O
real	O
line	O
,	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
at	O
that	O
point	O
is	O
a	O
way	O
of	O
encoding	O
the	O
small-scale	O
(	O
i.e.	O
,	O
from	O
the	O
point	O
to	O
the	O
next	O
)	O
behavior	O
of	O
the	O
function	O
.	O
</s>
<s>
By	O
finding	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
a	O
function	O
at	O
every	O
pair	O
of	O
consecutive	O
points	O
in	O
its	O
domain	O
,	O
it	O
is	O
possible	O
to	O
produce	O
a	O
new	O
function	O
,	O
called	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
function	O
or	O
just	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
the	O
original	O
function	O
.	O
</s>
<s>
In	O
formal	O
terms	O
,	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
a	O
linear	B-Architecture
operator	I-Architecture
which	O
takes	O
a	O
function	O
as	O
its	O
input	O
and	O
produces	O
a	O
second	O
function	O
as	O
its	O
output	O
.	O
</s>
<s>
The	O
"	O
difference	B-Algorithm
quotient	I-Algorithm
"	O
is	O
the	O
rate	O
of	O
change	O
of	O
the	O
function	O
over	O
one	O
of	O
the	O
intervals	O
defined	O
by	O
the	O
formula	O
:	O
</s>
<s>
As	O
the	O
rate	O
of	O
change	O
is	O
that	O
for	O
the	O
whole	O
interval	O
,	O
any	O
point	O
within	O
it	O
can	O
be	O
used	O
as	O
such	O
a	O
reference	O
or	O
,	O
even	O
better	O
,	O
the	O
whole	O
interval	O
which	O
makes	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
a	O
-cochain	O
.	O
</s>
<s>
The	O
most	O
common	O
notation	O
for	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
:	O
</s>
<s>
If	O
the	O
input	O
of	O
the	O
function	O
represents	O
time	O
,	O
then	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
represents	O
change	O
with	O
respect	O
to	O
time	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
a	O
function	O
that	O
takes	O
a	O
time	O
as	O
input	O
and	O
gives	O
the	O
position	O
of	O
a	O
ball	O
at	O
that	O
time	O
as	O
output	O
,	O
then	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
is	O
how	O
the	O
position	O
is	O
changing	O
in	O
time	O
,	O
that	O
is	O
,	O
it	O
is	O
the	O
velocity	O
of	O
the	O
ball	O
.	O
</s>
<s>
If	O
a	O
function	O
is	O
linear	O
(	O
that	O
is	O
,	O
if	O
the	O
points	O
of	O
the	O
graph	B-Application
of	O
the	O
function	O
lie	O
on	O
a	O
straight	O
line	O
)	O
,	O
then	O
the	O
function	O
can	O
be	O
written	O
as	O
,	O
where	O
is	O
the	O
independent	O
variable	O
,	O
is	O
the	O
dependent	O
variable	O
,	O
is	O
the	O
-intercept	O
,	O
and	O
:	O
</s>
<s>
The	O
difference	B-Algorithm
quotient	I-Algorithm
give	O
an	O
exact	O
meaning	O
to	O
the	O
notion	O
of	O
change	O
in	O
output	O
with	O
respect	O
to	O
change	O
in	O
input	O
.	O
</s>
<s>
is	O
a	O
point	O
on	O
the	O
graph	B-Application
of	O
the	O
function	O
.	O
</s>
<s>
Here	O
is	O
a	O
particular	O
example	O
,	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
the	O
squaring	O
function	O
.	O
</s>
<s>
In	O
technical	O
language	O
,	O
integral	O
calculus	O
studies	O
a	O
certain	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
The	O
Riemann	O
sum	O
inputs	O
a	O
function	O
and	O
outputs	O
a	O
function	O
,	O
which	O
gives	O
the	O
algebraic	O
sum	O
of	O
areas	O
between	O
the	O
part	O
of	O
the	O
graph	B-Application
of	O
the	O
input	O
and	O
the	O
x-axis	O
.	O
</s>
<s>
More	O
precisely	O
,	O
it	O
relates	O
the	O
difference	B-Algorithm
quotients	I-Algorithm
to	O
the	O
Riemann	O
sums	O
.	O
</s>
<s>
The	O
fundamental	O
theorem	O
of	O
calculus	O
:	O
If	O
a	O
function	O
is	O
defined	O
on	O
a	O
partition	O
of	O
the	O
interval	O
,	O
,	O
and	O
if	O
is	O
a	O
function	O
whose	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
,	O
then	O
we	O
have	O
:	O
</s>
<s>
Difference	O
equations	O
relate	O
an	O
unknown	O
function	O
to	O
its	O
difference	O
or	O
difference	B-Algorithm
quotient	I-Algorithm
,	O
and	O
are	O
ubiquitous	O
in	O
the	O
sciences	O
.	O
</s>
<s>
The	O
early	O
history	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
is	O
the	O
history	O
of	O
calculus	O
.	O
</s>
<s>
Such	O
basic	O
ideas	O
as	O
the	O
difference	B-Algorithm
quotients	I-Algorithm
and	O
the	O
Riemann	O
sums	O
appear	O
implicitly	O
or	O
explicitly	O
in	O
definitions	O
and	O
proofs	O
.	O
</s>
<s>
During	O
the	O
20th	O
century	O
discrete	B-Algorithm
calculus	I-Algorithm
remains	O
interlinked	O
with	O
infinitesimal	O
calculus	O
especially	O
differential	O
forms	O
but	O
also	O
starts	O
to	O
draw	O
from	O
algebraic	O
topology	O
as	O
both	O
develop	O
.	O
</s>
<s>
The	O
recent	O
development	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
,	O
starting	O
with	O
Whitney	O
,	O
has	O
been	O
driven	O
by	O
the	O
needs	O
of	O
applied	B-Algorithm
modeling	I-Algorithm
.	O
</s>
<s>
Discrete	B-Algorithm
calculus	I-Algorithm
is	O
used	O
for	O
modeling	O
either	O
directly	O
or	O
indirectly	O
as	O
a	O
discretization	O
of	O
infinitesimal	O
calculus	O
in	O
every	O
branch	O
of	O
the	O
physical	O
sciences	O
,	O
actuarial	O
science	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
statistics	O
,	O
engineering	O
,	O
economics	O
,	O
business	O
,	O
medicine	O
,	O
demography	O
,	O
and	O
in	O
other	O
fields	O
wherever	O
a	O
problem	O
can	O
be	O
mathematically	O
modeled	O
.	O
</s>
<s>
Physics	O
makes	O
particular	O
use	O
of	O
calculus	O
;	O
all	O
discrete	O
concepts	O
in	O
classical	O
mechanics	O
and	O
electromagnetism	O
are	O
related	O
through	O
discrete	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
The	O
mass	O
of	O
an	O
object	O
of	O
known	O
density	O
that	O
varies	O
incrementally	O
,	O
the	O
moment	O
of	O
inertia	O
of	O
such	O
objects	O
,	O
as	O
well	O
as	O
the	O
total	O
energy	O
of	O
an	O
object	O
within	O
a	O
discrete	O
conservative	O
field	O
can	O
be	O
found	O
by	O
the	O
use	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
An	O
example	O
of	O
the	O
use	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
in	O
mechanics	O
is	O
Newton	O
's	O
second	O
law	O
of	O
motion	O
:	O
historically	O
stated	O
it	O
expressly	O
uses	O
the	O
term	O
"	O
change	O
of	O
motion	O
"	O
which	O
implies	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
saying	O
The	O
change	O
of	O
momentum	O
of	O
a	O
body	O
is	O
equal	O
to	O
the	O
resultant	O
force	O
acting	O
on	O
the	O
body	O
and	O
is	O
in	O
the	O
same	O
direction	O
.	O
</s>
<s>
Commonly	O
expressed	O
today	O
as	O
Force	O
=	O
Mass×Acceleration	O
,	O
it	O
invokes	O
discrete	B-Algorithm
calculus	I-Algorithm
when	O
the	O
change	O
is	O
incremental	O
because	O
acceleration	O
is	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
velocity	O
with	O
respect	O
to	O
time	O
or	O
second	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
the	O
spatial	O
position	O
.	O
</s>
<s>
Maxwell	O
's	O
theory	O
of	O
electromagnetism	O
and	O
Einstein	O
's	O
theory	O
of	O
general	O
relativity	O
have	O
been	O
expressed	O
in	O
the	O
language	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
used	O
to	O
efficiently	O
calculate	O
sums	O
of	O
rectangular	O
domains	O
in	O
images	O
,	O
to	O
rapidly	O
extract	O
features	O
and	O
detect	O
object	O
;	O
another	O
algorithm	O
that	O
could	O
be	O
used	O
is	O
the	O
summed	B-Algorithm
area	I-Algorithm
table	I-Algorithm
.	O
</s>
<s>
Discrete	B-Algorithm
calculus	I-Algorithm
can	O
be	O
used	O
in	O
conjunction	O
with	O
other	O
mathematical	O
disciplines	O
.	O
</s>
<s>
If	O
a	O
function	O
(	O
a	O
-cochain	O
)	O
is	O
defined	O
at	O
the	O
nodes	O
of	O
a	O
graph	B-Application
:	O
</s>
<s>
then	O
its	O
exterior	O
derivative	O
(	O
or	O
the	O
differential	O
)	O
is	O
the	O
difference	O
,	O
i.e.	O
,	O
the	O
following	O
function	O
defined	O
on	O
the	O
edges	O
of	O
the	O
graph	B-Application
(	O
-cochain	O
)	O
:	O
</s>
<s>
If	O
is	O
a	O
-cochain	O
,	O
then	O
its	O
integral	O
over	O
a	O
sequence	O
of	O
edges	O
of	O
the	O
graph	B-Application
is	O
the	O
sum	O
of	O
its	O
values	O
over	O
all	O
edges	O
of	O
(	O
"	O
path	O
integral	O
"	O
)	O
:	O
</s>
<s>
is	O
the	O
linear	B-Architecture
operator	I-Architecture
defined	O
by	O
:	O
</s>
<s>
A	O
cubical	O
complex	O
is	O
a	O
set	O
composed	O
of	O
points	O
,	O
line	O
segments	O
,	O
squares	O
,	O
cubes	B-Application
,	O
and	O
their	O
n-dimensional	O
counterparts	O
.	O
</s>
<s>
An	O
elementary	O
cube	B-Application
is	O
the	O
finite	O
product	O
of	O
elementary	O
intervals	O
,	O
i.e.	O
</s>
<s>
Equivalently	O
,	O
an	O
elementary	O
cube	B-Application
is	O
any	O
translate	O
of	O
a	O
unit	O
cube	B-Application
embedded	O
in	O
Euclidean	O
space	O
(	O
for	O
some	O
with	O
)	O
.	O
</s>
<s>
A	O
set	O
is	O
a	O
cubical	O
complex	O
if	O
it	O
can	O
be	O
written	O
as	O
a	O
union	O
of	O
elementary	O
cubes	B-Application
(	O
or	O
possibly	O
,	O
is	O
homeomorphic	O
to	O
such	O
a	O
set	O
)	O
and	O
it	O
contains	O
all	O
of	O
the	O
faces	O
of	O
all	O
of	O
its	O
cubes	B-Application
.	O
</s>
<s>
A	O
chain	O
complex	O
is	O
a	O
sequence	O
of	O
vector	O
spaces	O
connected	O
by	O
linear	B-Architecture
operators	I-Architecture
(	O
called	O
boundary	O
operators	O
)	O
,	O
such	O
that	O
the	O
composition	O
of	O
any	O
two	O
consecutive	O
maps	O
is	O
the	O
zero	O
map	O
.	O
</s>
<s>
It	O
generates	O
a	O
linear	B-Architecture
map	I-Architecture
,	O
called	O
a	O
chain	O
map	O
,	O
from	O
the	O
chain	O
complex	O
of	O
to	O
the	O
chain	O
complex	O
of	O
.	O
</s>
<s>
It	O
consists	O
of	O
a	O
sequence	O
of	O
vector	O
spaces	O
connected	O
by	O
linear	B-Architecture
operators	I-Architecture
satisfying	O
.	O
</s>
<s>
The	O
elements	O
in	O
the	O
kernel	B-Algorithm
of	O
are	O
called	O
cocycles	O
(	O
or	O
closed	O
elements	O
)	O
,	O
and	O
the	O
elements	O
in	O
the	O
image	O
of	O
are	O
called	O
coboundaries	O
(	O
or	O
exact	O
elements	O
)	O
.	O
</s>
<s>
Stokes	O
 '	O
theorem	O
is	O
a	O
statement	O
about	O
the	O
discrete	O
differential	O
forms	O
on	O
manifolds	B-Architecture
,	O
which	O
generalizes	O
the	O
fundamental	O
theorem	O
of	O
discrete	B-Algorithm
calculus	I-Algorithm
for	O
a	O
partition	O
of	O
an	O
interval	O
:	O
</s>
<s>
Stokes	O
 '	O
theorem	O
says	O
that	O
the	O
sum	O
of	O
a	O
form	O
over	O
the	O
boundary	O
of	O
some	O
orientable	O
manifold	B-Architecture
is	O
equal	O
to	O
the	O
sum	O
of	O
its	O
exterior	O
derivative	O
over	O
the	O
whole	O
of	O
,	O
i.e.	O
,	O
</s>
<s>
The	O
essential	O
idea	O
can	O
be	O
understood	O
by	O
the	O
diagram	O
on	O
the	O
left	O
,	O
which	O
shows	O
that	O
,	O
in	O
an	O
oriented	O
tiling	O
of	O
a	O
manifold	B-Architecture
,	O
the	O
interior	O
paths	O
are	O
traversed	O
in	O
opposite	O
directions	O
;	O
their	O
contributions	O
to	O
the	O
path	O
integral	O
thus	O
cancel	O
each	O
other	O
pairwise	O
.	O
</s>
<s>
In	O
discrete	B-Algorithm
calculus	I-Algorithm
,	O
this	O
is	O
a	O
construction	O
that	O
creates	O
from	O
forms	O
higher	O
order	O
forms	O
:	O
adjoining	O
two	O
cochains	O
of	O
degree	O
and	O
to	O
form	O
a	O
composite	O
cochain	O
of	O
degree	O
.	O
</s>
<s>
For	O
cubical	O
complexes	O
,	O
the	O
wedge	O
product	O
is	O
defined	O
on	O
every	O
cube	B-Application
seen	O
as	O
a	O
vector	O
space	O
of	O
the	O
same	O
dimension	O
.	O
</s>
<s>
The	O
Laplace	O
operator	O
represents	O
the	O
flux	O
density	O
of	O
the	O
gradient	B-Algorithm
flow	I-Algorithm
of	O
a	O
function	O
.	O
</s>
