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1
Lecture
XX
Vector Line
Integrals; Conservative
Fields
Vector line
integrals
Let
�
F
be
a vector field
of
domain
D
.
Let
D
be connected,
i.e.
for
any two points
P, P
�
in
D
there
is
a curve
C
contained
in
D
that goes
from
P
to
P
�
.
Let
C
be
a finite
directed curve
contained
in
D
that is
rectifiable,
i.e.
it has
a length.
We
�
d
�
F
R
as
the
limit
of
a
Riemann
sum:
define
the
vector line
integral
·
C
n
�
d
�
�
F
R
=
lim
F
(
P
i
∗
)
·
ˆ
T
(
P
i
∗
)Δ
s
i
,
·
C
n
→∞
max
Δ
s
i
→
0
i
=1
where
T
ˆ
(
P
) is
the
unit tanget vector
for
C
at
P
,
and
P
i
∗
,
Δ
s
i
are defined
in
the
same
way as
for scalar line
integrals.
Let us
give two basic laws
for
vector
line
integrals:
1.
Let
�
F
be
a vector field
on
a finite directed
curve
C
,
and
let
C
be divided
into curves
C
1
and
C
2
, both
having the same direction
as
C
.
Then:
�
d
�
�
d
�
�
d
�
F
R
=
F
R
+
F
R.
·
·
·
C
C
1
C
2
F
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 Two '04
 Duorg
 Math, Integrals, Vector Space, vector line, max Δsi

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