<s>
Vector	O
fields	O
can	O
usefully	O
be	O
thought	O
of	O
as	O
representing	O
the	O
velocity	O
of	O
a	O
moving	O
flow	B-Algorithm
in	O
space	O
,	O
and	O
this	O
physical	O
intuition	O
leads	O
to	O
notions	O
such	O
as	O
the	O
divergence	B-Application
(	O
which	O
represents	O
the	O
rate	O
of	O
change	O
of	O
volume	O
of	O
a	O
flow	B-Algorithm
)	O
and	O
curl	O
(	O
which	O
represents	O
the	O
rotation	O
of	O
a	O
flow	B-Algorithm
)	O
.	O
</s>
<s>
The	O
reason	O
for	O
this	O
notation	O
is	O
that	O
a	O
vector	O
field	O
determines	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
the	O
space	O
of	O
smooth	O
functions	O
to	O
itself	O
,	O
,	O
given	O
by	O
differentiating	O
in	O
the	O
direction	O
of	O
the	O
vector	O
field	O
.	O
</s>
<s>
The	O
transformation	O
properties	O
of	O
vectors	O
distinguish	O
a	O
vector	O
as	O
a	O
geometrically	O
distinct	O
entity	O
from	O
a	O
simple	O
list	O
of	O
scalars	O
,	O
or	O
from	O
a	O
covector	B-Algorithm
.	O
</s>
<s>
An	O
alternative	O
definition	O
:	O
A	O
smooth	O
vector	O
field	O
on	O
a	O
manifold	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
such	O
that	O
is	O
a	O
derivation	B-Algorithm
:	O
for	O
all	O
.	O
</s>
<s>
If	O
the	O
manifold	O
is	O
smooth	O
or	O
analytic	B-Language
—	O
that	O
is	O
,	O
the	O
change	O
of	O
coordinates	O
is	O
smooth	O
(	O
analytic	B-Language
)	O
—	O
then	O
one	O
can	O
make	O
sense	O
of	O
the	O
notion	O
of	O
smooth	O
(	O
analytic	B-Language
)	O
vector	O
fields	O
.	O
</s>
<s>
Streamlines	B-Application
,	I-Application
streaklines	I-Application
and	I-Application
pathlines	I-Application
are	O
3	O
types	O
of	O
lines	O
that	O
can	O
be	O
made	O
from	O
(	O
time-dependent	O
)	O
vector	O
fields	O
.	O
</s>
<s>
pathlines	B-Application
:	O
showing	O
the	O
path	O
that	O
a	O
given	O
particle	O
(	O
of	O
zero	O
mass	O
)	O
would	O
follow	O
.	O
</s>
<s>
streamlines	B-Application
(	O
or	O
fieldlines	B-Application
)	O
:	O
the	O
path	O
of	O
a	O
particle	O
influenced	O
by	O
the	O
instantaneous	O
field	O
(	O
i.e.	O
,	O
the	O
path	O
of	O
a	O
particle	O
if	O
the	O
field	O
is	O
held	O
fixed	O
)	O
.	O
</s>
<s>
The	O
fieldlines	B-Application
can	O
be	O
revealed	O
using	O
small	O
iron	O
filings	O
.	O
</s>
<s>
Vector	O
fields	O
can	O
be	O
constructed	O
out	O
of	O
scalar	O
fields	O
using	O
the	O
gradient	O
operator	O
(	O
denoted	O
by	O
the	O
del	B-Device
:	O
∇	O
)	O
.	O
</s>
<s>
The	O
associated	O
flow	B-Algorithm
is	O
called	O
the	O
,	O
and	O
is	O
used	O
in	O
the	O
method	O
of	O
gradient	B-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
We	O
say	O
central	O
fields	O
are	O
invariant	O
under	O
orthogonal	B-Algorithm
transformations	I-Algorithm
around	O
0	O
.	O
</s>
<s>
Since	O
orthogonal	B-Algorithm
transformations	I-Algorithm
are	O
actually	O
rotations	O
and	O
reflections	O
,	O
the	O
invariance	O
conditions	O
mean	O
that	O
vectors	O
of	O
a	O
central	O
field	O
are	O
always	O
directed	O
towards	O
,	O
or	O
away	O
from	O
,	O
0	O
;	O
this	O
is	O
an	O
alternate	O
(	O
and	O
simpler	O
)	O
definition	O
.	O
</s>
<s>
The	O
divergence	B-Application
of	O
a	O
vector	O
field	O
on	O
Euclidean	O
space	O
is	O
a	O
function	O
(	O
or	O
scalar	O
field	O
)	O
.	O
</s>
<s>
The	O
divergence	B-Application
at	O
a	O
point	O
represents	O
the	O
degree	O
to	O
which	O
a	O
small	O
volume	O
around	O
the	O
point	O
is	O
a	O
source	O
or	O
a	O
sink	O
for	O
the	O
vector	O
flow	B-Algorithm
,	O
a	O
result	O
which	O
is	O
made	O
precise	O
by	O
the	O
divergence	B-Application
theorem	O
.	O
</s>
<s>
The	O
divergence	B-Application
can	O
also	O
be	O
defined	O
on	O
a	O
Riemannian	B-Architecture
manifold	I-Architecture
,	O
that	O
is	O
,	O
a	O
manifold	O
with	O
a	O
Riemannian	O
metric	O
that	O
measures	O
the	O
length	O
of	O
vectors	O
.	O
</s>
<s>
The	O
curl	O
is	O
defined	O
only	O
in	O
three	O
dimensions	O
,	O
but	O
some	O
properties	O
of	O
the	O
curl	O
can	O
be	O
captured	O
in	O
higher	O
dimensions	O
with	O
the	O
exterior	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
curl	O
measures	O
the	O
density	O
of	O
the	O
angular	O
momentum	O
of	O
the	O
vector	O
flow	B-Algorithm
at	O
a	O
point	O
,	O
that	O
is	O
,	O
the	O
amount	O
to	O
which	O
the	O
flow	B-Algorithm
circulates	O
around	O
a	O
fixed	O
axis	O
.	O
</s>
<s>
In	O
recent	O
decades	O
many	O
phenomenological	O
formulations	O
of	O
irreversible	O
dynamics	O
and	O
evolution	O
equations	O
in	O
physics	O
,	O
from	O
the	O
mechanics	O
of	O
complex	O
fluids	O
and	O
solids	O
to	O
chemical	O
kinetics	O
and	O
quantum	O
thermodynamics	O
,	O
have	O
converged	O
towards	O
the	O
geometric	O
idea	O
of	O
"	O
steepest	O
entropy	O
ascent	O
"	O
or	O
"	O
gradient	B-Algorithm
flow	I-Algorithm
"	O
as	O
a	O
consistent	O
universal	O
modeling	O
framework	O
that	O
guarantees	O
compatibility	O
with	O
the	O
second	O
law	O
of	O
thermodynamics	O
and	O
extends	O
well-known	O
near-equilibrium	O
results	O
such	O
as	O
Onsager	O
reciprocity	O
to	O
the	O
far-nonequilibrium	O
realm	O
.	O
</s>
<s>
Consider	O
the	O
flow	B-Algorithm
of	O
a	O
fluid	O
through	O
a	O
region	O
of	O
space	O
.	O
</s>
<s>
At	O
any	O
given	O
time	O
,	O
any	O
point	O
of	O
the	O
fluid	O
has	O
a	O
particular	O
velocity	O
associated	O
with	O
it	O
;	O
thus	O
there	O
is	O
a	O
vector	O
field	O
associated	O
to	O
any	O
flow	B-Algorithm
.	O
</s>
<s>
The	O
converse	O
is	O
also	O
true	O
:	O
it	O
is	O
possible	O
to	O
associate	O
a	O
flow	B-Algorithm
to	O
a	O
vector	O
field	O
having	O
that	O
vector	O
field	O
as	O
its	O
velocity	O
.	O
</s>
<s>
The	O
curves	O
are	O
called	O
integral	O
curves	O
or	O
trajectories	O
(	O
or	O
less	O
commonly	O
,	O
flow	B-Algorithm
lines	O
)	O
of	O
the	O
vector	O
field	O
and	O
partition	O
into	O
equivalence	O
classes	O
.	O
</s>
<s>
The	O
flow	B-Algorithm
may	O
for	O
example	O
reach	O
the	O
edge	O
of	O
in	O
a	O
finite	O
time	O
.	O
</s>
<s>
In	O
two	O
or	O
three	O
dimensions	O
one	O
can	O
visualize	O
the	O
vector	O
field	O
as	O
giving	O
rise	O
to	O
a	O
flow	B-Algorithm
on	O
.	O
</s>
<s>
If	O
we	O
drop	O
a	O
particle	O
into	O
this	O
flow	B-Algorithm
at	O
a	O
point	O
it	O
will	O
move	O
along	O
the	O
curve	O
in	O
the	O
flow	B-Algorithm
depending	O
on	O
the	O
initial	O
point	O
.	O
</s>
<s>
Typical	O
applications	O
are	O
pathline	B-Application
in	O
fluid	O
,	O
geodesic	O
flow	B-Algorithm
,	O
and	O
one-parameter	O
subgroups	O
and	O
the	O
exponential	O
map	O
in	O
Lie	O
groups	O
.	O
</s>
<s>
By	O
definition	O
,	O
a	O
vector	O
field	O
on	O
is	O
called	O
complete	O
if	O
each	O
of	O
its	O
flow	B-Algorithm
curves	O
exists	O
for	O
all	O
time	O
.	O
</s>
<s>
Given	O
a	O
smooth	O
function	O
between	O
manifolds	O
,	O
,	O
the	O
derivative	B-Algorithm
is	O
an	O
induced	O
map	O
on	O
tangent	O
bundles	O
,	O
.	O
</s>
<s>
Algebraically	O
,	O
vector	O
fields	O
can	O
be	O
characterized	O
as	O
derivations	B-Algorithm
of	O
the	O
algebra	O
of	O
smooth	O
functions	O
on	O
the	O
manifold	O
,	O
which	O
leads	O
to	O
defining	O
a	O
vector	O
field	O
on	O
a	O
commutative	O
algebra	O
as	O
a	O
derivation	B-Algorithm
on	O
the	O
algebra	O
,	O
which	O
is	O
developed	O
in	O
the	O
theory	O
of	O
differential	O
calculus	O
over	O
commutative	O
algebras	O
.	O
</s>
