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PHY2061
R. D. Field
Department of Physics
line_1.doc
Univesity of Florida
P
1
Closed Loop
Line Integral of a Vector Function
Let
z
z
y
x
F
y
z
y
x
F
x
z
y
x
F
z
y
x
F
z
y
x
ˆ
)
,
,
(
ˆ
)
.
.
(
ˆ
)
,
,
(
)
,
,
(
+
+
=
!
be a vector
function of position.
In general, the line
integral of
)
,
,
(
z
y
x
F
!
depends of the start
point P
1
, and the end point P
2
and the path
chosen from P
1
to P
2
(curve C):
∫
⋅
=
CurveC
r
d
F
Path
P
P
I
!
!
)
,
,
(
2
1
,
where I(P
1
,P
2
,Path) is the component of
along the path integrated over
the path,
∫∫
∫
+
+
=
=
⋅
CurveC
CurveC
z
y
x
CurveC
dz
F
dy
F
dx
F
dr
F
r
d
F
)
(
cos
θ
!
!
.
Remark:
If
0
=
×
∇
F
!
!
then I(P
1
,P
2
,Path) = I(P
1
,P
2
) and is only a function
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 Spring '08
 FRY
 Physics

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