<s>
A	O
field	B-Application
line	I-Application
is	O
a	O
graphical	O
visual	B-Application
aid	I-Application
for	O
visualizing	O
vector	O
fields	O
.	O
</s>
<s>
A	O
diagram	O
showing	O
a	O
representative	O
set	O
of	O
neighboring	O
field	B-Application
lines	I-Application
is	O
a	O
common	O
way	O
of	O
depicting	O
a	O
vector	O
field	O
in	O
scientific	O
and	O
mathematical	O
literature	O
;	O
this	O
is	O
called	O
a	O
field	B-Application
line	I-Application
diagram	I-Application
.	O
</s>
<s>
In	O
fluid	O
mechanics	O
field	B-Application
lines	I-Application
showing	O
the	O
velocity	O
field	O
of	O
a	O
fluid	O
flow	O
are	O
called	O
streamlines	B-Application
.	O
</s>
<s>
A	O
field	B-Application
line	I-Application
is	O
an	O
integral	O
curve	O
for	O
that	O
vector	O
field	O
and	O
may	O
be	O
constructed	O
by	O
starting	O
at	O
a	O
point	O
and	O
tracing	O
a	O
line	O
through	O
space	O
that	O
follows	O
the	O
direction	O
of	O
the	O
vector	O
field	O
,	O
by	O
making	O
the	O
field	B-Application
line	I-Application
tangent	O
to	O
the	O
field	O
vector	O
at	O
each	O
point	O
.	O
</s>
<s>
A	O
field	B-Application
line	I-Application
is	O
usually	O
shown	O
as	O
a	O
directed	O
line	O
segment	O
,	O
with	O
an	O
arrowhead	O
indicating	O
the	O
direction	O
of	O
the	O
vector	O
field	O
.	O
</s>
<s>
For	O
two-dimensional	O
fields	O
the	O
field	B-Application
lines	I-Application
are	O
plane	O
curves	O
;	O
since	O
a	O
plane	O
drawing	O
of	O
a	O
3-dimensional	O
set	O
of	O
field	B-Application
lines	I-Application
can	O
be	O
visually	O
confusing	O
most	O
field	B-Application
line	I-Application
diagrams	I-Application
are	O
of	O
this	O
type	O
.	O
</s>
<s>
Since	O
at	O
each	O
point	O
where	O
it	O
is	O
nonzero	O
and	O
finite	O
the	O
vector	O
field	O
has	O
a	O
unique	O
direction	O
,	O
field	B-Application
lines	I-Application
can	O
never	O
intersect	O
,	O
so	O
there	O
is	O
exactly	O
one	O
field	B-Application
line	I-Application
passing	O
through	O
each	O
point	O
at	O
which	O
the	O
vector	O
field	O
is	O
nonzero	O
and	O
finite	O
.	O
</s>
<s>
Points	O
where	O
the	O
field	O
is	O
zero	O
or	O
infinite	O
have	O
no	O
field	B-Application
line	I-Application
through	O
them	O
,	O
since	O
direction	O
cannot	O
be	O
defined	O
there	O
,	O
but	O
can	O
be	O
the	O
endpoints	O
of	O
field	B-Application
lines	I-Application
.	O
</s>
<s>
Since	O
there	O
are	O
an	O
infinite	O
number	O
of	O
points	O
in	O
any	O
region	O
,	O
an	O
infinite	O
number	O
of	O
field	B-Application
lines	I-Application
can	O
be	O
drawn	O
;	O
but	O
only	O
a	O
limited	O
number	O
can	O
be	O
shown	O
on	O
a	O
field	B-Application
line	I-Application
diagram	I-Application
.	O
</s>
<s>
Therefore	O
which	O
field	B-Application
lines	I-Application
are	O
shown	O
is	O
a	O
choice	O
made	O
by	O
the	O
person	O
or	O
computer	O
program	O
which	O
draws	O
the	O
diagram	O
,	O
and	O
a	O
single	O
vector	O
field	O
may	O
be	O
depicted	O
by	O
different	O
sets	O
of	O
field	B-Application
lines	I-Application
.	O
</s>
<s>
A	O
field	B-Application
line	I-Application
diagram	I-Application
is	O
necessarily	O
an	O
incomplete	O
description	O
of	O
a	O
vector	O
field	O
,	O
since	O
it	O
gives	O
no	O
information	O
about	O
the	O
field	O
between	O
the	O
drawn	O
field	B-Application
lines	I-Application
,	O
and	O
the	O
choice	O
of	O
how	O
many	O
and	O
which	O
lines	O
to	O
show	O
determines	O
how	O
much	O
useful	O
information	O
the	O
diagram	O
gives	O
.	O
</s>
<s>
An	O
individual	O
field	B-Application
line	I-Application
shows	O
the	O
direction	O
of	O
the	O
vector	O
field	O
but	O
not	O
the	O
magnitude	O
.	O
</s>
<s>
In	O
order	O
to	O
also	O
depict	O
the	O
magnitude	O
of	O
the	O
field	O
,	O
field	B-Application
line	I-Application
diagrams	I-Application
are	O
often	O
drawn	O
so	O
that	O
each	O
line	O
represents	O
the	O
same	O
quantity	O
of	O
flux	B-General_Concept
.	O
</s>
<s>
Then	O
the	O
density	O
of	O
field	B-Application
lines	I-Application
(	O
number	O
of	O
field	B-Application
lines	I-Application
per	O
unit	O
perpendicular	O
area	O
)	O
at	O
any	O
location	O
is	O
proportional	O
to	O
the	O
magnitude	O
of	O
the	O
vector	O
field	O
at	O
that	O
point	O
.	O
</s>
<s>
Areas	O
in	O
which	O
neighboring	O
field	B-Application
lines	I-Application
are	O
converging	O
(	O
getting	O
closer	O
together	O
)	O
indicates	O
that	O
the	O
field	O
is	O
getting	O
stronger	O
in	O
that	O
direction	O
.	O
</s>
<s>
In	O
vector	O
fields	O
which	O
have	O
nonzero	O
divergence	B-Application
,	O
field	B-Application
lines	I-Application
begin	O
on	O
points	O
of	O
positive	O
divergence	B-Application
(	O
sources	O
)	O
and	O
end	O
on	O
points	O
of	O
negative	O
divergence	B-Application
(	O
sinks	O
)	O
,	O
or	O
extend	O
to	O
infinity	O
.	O
</s>
<s>
For	O
example	O
,	O
electric	O
field	B-Application
lines	I-Application
begin	O
on	O
positive	O
charges	O
and	O
end	O
on	O
negative	O
charges	O
.	O
</s>
<s>
In	O
fields	O
which	O
are	O
divergenceless	O
(	O
solenoidal	O
)	O
,	O
such	O
as	O
magnetic	O
fields	O
,	O
field	B-Application
lines	I-Application
have	O
no	O
endpoints	O
;	O
they	O
are	O
either	O
closed	O
loops	O
or	O
are	O
endless	O
.	O
</s>
<s>
In	O
physics	O
,	O
drawings	O
of	O
field	B-Application
lines	I-Application
are	O
mainly	O
useful	O
in	O
cases	O
where	O
the	O
sources	O
and	O
sinks	O
,	O
if	O
any	O
,	O
have	O
a	O
physical	O
meaning	O
,	O
as	O
opposed	O
to	O
e.g.	O
</s>
<s>
For	O
example	O
,	O
Gauss	O
's	O
law	O
states	O
that	O
an	O
electric	O
field	O
has	O
sources	O
at	O
positive	O
charges	O
,	O
sinks	O
at	O
negative	O
charges	O
,	O
and	O
neither	O
elsewhere	O
,	O
so	O
electric	O
field	B-Application
lines	I-Application
start	O
at	O
positive	O
charges	O
and	O
end	O
at	O
negative	O
charges	O
.	O
</s>
<s>
A	O
gravitational	O
field	O
has	O
no	O
sources	O
,	O
it	O
has	O
sinks	O
at	O
masses	O
,	O
and	O
it	O
has	O
neither	O
elsewhere	O
,	O
gravitational	O
field	B-Application
lines	I-Application
come	O
from	O
infinity	O
and	O
end	O
at	O
masses	O
.	O
</s>
<s>
A	O
magnetic	O
field	O
has	O
no	O
sources	O
or	O
sinks	O
(	O
Gauss	O
's	O
law	O
for	O
magnetism	O
)	O
,	O
so	O
its	O
field	B-Application
lines	I-Application
have	O
no	O
start	O
or	O
end	O
:	O
they	O
can	O
only	O
form	O
closed	O
loops	O
,	O
extend	O
to	O
infinity	O
in	O
both	O
directions	O
,	O
or	O
continue	O
indefinitely	O
without	O
ever	O
crossing	O
itself	O
.	O
</s>
<s>
However	O
,	O
as	O
stated	O
above	O
,	O
a	O
special	O
situation	O
may	O
occur	O
around	O
points	O
where	O
the	O
field	O
is	O
zero	O
(	O
that	O
cannot	O
be	O
intersected	O
by	O
field	B-Application
lines	I-Application
,	O
because	O
their	O
direction	O
would	O
not	O
be	O
defined	O
)	O
and	O
the	O
simultaneous	O
begin	O
and	O
end	O
of	O
field	B-Application
lines	I-Application
takes	O
place	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
in	O
the	O
transverse	O
plane	O
passing	O
through	O
the	O
middle	O
point	O
,	O
an	O
infinite	O
number	O
of	O
field	B-Application
lines	I-Application
diverge	O
radially	O
.	O
</s>
<s>
The	O
concomitant	O
presence	O
of	O
the	O
lines	O
that	O
end	O
and	O
begin	O
preserves	O
the	O
divergence-free	O
character	O
of	O
the	O
field	O
in	O
the	O
point	O
.	O
</s>
<s>
The	O
electric	O
field	B-Application
lines	I-Application
in	O
this	O
case	O
are	O
straight	O
lines	O
that	O
emanate	O
from	O
the	O
charge	O
uniformly	O
in	O
all	O
directions	O
in	O
three-dimensional	O
space	O
.	O
</s>
<s>
However	O
,	O
if	O
the	O
electric	O
field	B-Application
lines	I-Application
for	O
this	O
setup	O
were	O
just	O
drawn	O
on	O
a	O
two-dimensional	O
plane	O
,	O
their	O
two-dimensional	O
density	O
would	O
be	O
proportional	O
to	O
,	O
an	O
incorrect	O
result	O
for	O
this	O
situation	O
.	O
</s>
<s>
Given	O
a	O
vector	O
field	O
and	O
a	O
starting	O
point	O
a	O
field	B-Application
line	I-Application
can	O
be	O
constructed	O
iteratively	O
by	O
finding	O
the	O
field	O
vector	O
at	O
that	O
point	O
.	O
</s>
<s>
Then	O
the	O
field	O
at	O
that	O
point	O
is	O
found	O
and	O
moving	O
a	O
further	O
distance	O
in	O
that	O
direction	O
the	O
next	O
point	O
of	O
the	O
field	B-Application
line	I-Application
is	O
found	O
.	O
</s>
<s>
By	O
repeating	O
this	O
and	O
connecting	O
the	O
points	O
,	O
the	O
field	B-Application
line	I-Application
can	O
be	O
extended	O
as	O
far	O
as	O
desired	O
.	O
</s>
<s>
This	O
is	O
only	O
an	O
approximation	O
to	O
the	O
actual	O
field	B-Application
line	I-Application
,	O
since	O
each	O
straight	O
segment	O
is	O
n't	O
actually	O
tangent	O
to	O
the	O
field	O
along	O
its	O
length	O
,	O
just	O
at	O
its	O
starting	O
point	O
.	O
</s>
<s>
But	O
by	O
using	O
a	O
small	O
enough	O
value	O
for	O
,	O
taking	O
a	O
greater	O
number	O
of	O
shorter	O
steps	O
,	O
the	O
field	B-Application
line	I-Application
can	O
be	O
approximated	O
as	O
closely	O
as	O
desired	O
.	O
</s>
<s>
The	O
field	B-Application
line	I-Application
can	O
be	O
extended	O
in	O
the	O
opposite	O
direction	O
from	O
by	O
taking	O
each	O
step	O
in	O
the	O
opposite	O
direction	O
by	O
using	O
a	O
negative	O
step	O
.	O
</s>
<s>
If	O
the	O
vector	O
field	O
describes	O
a	O
velocity	O
field	O
,	O
then	O
the	O
field	B-Application
lines	I-Application
follow	O
stream	B-Application
lines	I-Application
in	O
the	O
flow	O
.	O
</s>
<s>
Perhaps	O
the	O
most	O
familiar	O
example	O
of	O
a	O
vector	O
field	O
described	O
by	O
field	B-Application
lines	I-Application
is	O
the	O
magnetic	O
field	O
,	O
which	O
is	O
often	O
depicted	O
using	O
field	B-Application
lines	I-Application
emanating	O
from	O
a	O
magnet	O
.	O
</s>
<s>
Field	B-Application
lines	I-Application
can	O
be	O
used	O
to	O
trace	O
familiar	O
quantities	O
from	O
vector	O
calculus	O
:	O
</s>
<s>
Divergence	B-Application
may	O
be	O
easily	O
seen	O
through	O
field	B-Application
lines	I-Application
,	O
assuming	O
the	O
lines	O
are	O
drawn	O
such	O
that	O
the	O
density	O
of	O
field	B-Application
lines	I-Application
is	O
proportional	O
to	O
the	O
magnitude	O
of	O
the	O
field	O
(	O
see	O
above	O
)	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
divergence	B-Application
may	O
be	O
seen	O
as	O
the	O
beginning	O
and	O
ending	O
of	O
field	B-Application
lines	I-Application
.	O
</s>
<s>
If	O
the	O
vector	O
field	O
is	O
the	O
resultant	O
of	O
radial	O
inverse-square	O
law	O
fields	O
with	O
respect	O
to	O
one	O
or	O
more	O
sources	O
then	O
this	O
corresponds	O
to	O
the	O
fact	O
that	O
the	O
divergence	B-Application
of	O
such	O
a	O
field	O
is	O
zero	O
outside	O
the	O
sources	O
.	O
</s>
<s>
In	O
a	O
solenoidal	O
vector	O
field	O
(	O
i.e.	O
,	O
a	O
vector	O
field	O
where	O
the	O
divergence	B-Application
is	O
zero	O
everywhere	O
)	O
,	O
the	O
field	B-Application
lines	I-Application
neither	O
begin	O
nor	O
end	O
;	O
they	O
either	O
form	O
closed	O
loops	O
,	O
or	O
go	O
off	O
to	O
infinity	O
in	O
both	O
directions	O
.	O
</s>
<s>
If	O
a	O
vector	O
field	O
has	O
positive	O
divergence	B-Application
in	O
some	O
area	O
,	O
there	O
will	O
be	O
field	B-Application
lines	I-Application
starting	O
from	O
points	O
in	O
that	O
area	O
.	O
</s>
<s>
If	O
a	O
vector	O
field	O
has	O
negative	O
divergence	B-Application
in	O
some	O
area	O
,	O
there	O
will	O
be	O
field	B-Application
lines	I-Application
ending	O
at	O
points	O
in	O
that	O
area	O
.	O
</s>
<s>
The	O
Kelvin	O
–	O
Stokes	O
theorem	O
shows	O
that	O
field	B-Application
lines	I-Application
of	O
a	O
vector	O
field	O
with	O
zero	O
curl	O
(	O
i.e.	O
,	O
a	O
conservative	O
vector	O
field	O
,	O
e.g.	O
</s>
<s>
In	O
other	O
words	O
,	O
curl	O
is	O
always	O
present	O
when	O
a	O
field	B-Application
line	I-Application
forms	O
a	O
closed	O
loop	O
.	O
</s>
<s>
It	O
may	O
be	O
present	O
in	O
other	O
situations	O
too	O
,	O
such	O
as	O
a	O
helical	B-Application
shape	O
of	O
field	B-Application
lines	I-Application
.	O
</s>
<s>
While	O
field	B-Application
lines	I-Application
are	O
a	O
"	O
mere	O
"	O
mathematical	O
construction	O
,	O
in	O
some	O
circumstances	O
they	O
take	O
on	O
physical	O
significance	O
.	O
</s>
<s>
In	O
fluid	O
mechanics	O
,	O
the	O
velocity	O
field	B-Application
lines	I-Application
(	O
streamlines	B-Application
)	O
in	O
steady	O
flow	O
represent	O
the	O
paths	O
of	O
particles	O
of	O
the	O
fluid	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
plasma	O
physics	O
,	O
electrons	O
or	O
ions	O
that	O
happen	O
to	O
be	O
on	O
the	O
same	O
field	B-Application
line	I-Application
interact	O
strongly	O
,	O
while	O
particles	O
on	O
different	O
field	B-Application
lines	I-Application
in	O
general	O
do	O
not	O
interact	O
.	O
</s>
<s>
The	O
iron	O
filings	O
in	O
the	O
photo	O
appear	O
to	O
be	O
aligning	O
themselves	O
with	O
discrete	O
field	B-Application
lines	I-Application
,	O
but	O
the	O
situation	O
is	O
more	O
complex	O
.	O
</s>
<s>
Because	O
the	O
intrinsic	O
magnetism	O
of	O
the	O
filings	O
modifies	O
the	O
field	O
,	O
the	O
lines	O
shown	O
by	O
the	O
filings	O
are	O
only	O
an	O
approximation	O
of	O
the	O
field	B-Application
lines	I-Application
of	O
the	O
original	O
magnetic	O
field	O
.	O
</s>
