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PHY2061
R. D. Field
Department of Physics
line_2.doc
Univesity of Florida
P
1
Closed Loop
Question:
When can the vector function
)
,
,
(
z
y
x
F
!
be written as the gradient of a
scalar function
f(x,y,z)
?
We know that
∫
⋅
∇
=
−
2
1
)
(
)
(
1
2
P
P
r
d
f
P
f
P
f
!
!
.
This implies that
∫
=
⋅
∇
=
−
Lood
Closed
r
d
f
P
f
P
f
0
)
(
)
(
1
1
!
!
.
Thus if we demand that
f
F
∇
=
!
!
,
then
∫
=
⋅
Loop
Closed
Any
r
d
F
0
!
!
.
But Stokes’ Theorem tell us that
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 Spring '08
 FRY
 Physics

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