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AMATH 231 COURSE SUMMARY
Curves & Vector Fields
[
C
h
.
1
]
 Curves in R
n
:
n
R
b
a
g
→
]
,
[
:
))
(
),.
..,
(
),
(
(
)
(
2
1
t
g
t
g
t
g
t
g
n
=
(ie path of a particle through space with parameter time
t
)
 Expressing Equations in Parametric Form (above)
 Limits:
L
t
g
t
t
=
→
)
(
lim
0
means that
0

)
(

0
lim
=
−
→
L
t
g
t
t
 Continuity:
g
is continuous at t
0
means that
)
(
)
(
0
0
t
g
t
g
t
t
lim
=
→
where
)
(
0
t
g
is defined
a
n
d
)
(
0
t
t
t
→
lim
g
exists.
 Derivatives:
t
t
g
t
t
g
t
g
t
∆
−
∆
+
=
′
→
∆
)
(
)
(
lim
)
(
0
 Arclength:
∫
′
=
b
a
dt
t
g
s

)
(

 Scalar Field:
R
R
f
n
→
:
)
(
x
f
 Vector Field:
m
n
R
R
F
→
:
))
(
),.
..,
(
),
(
(
)
(
2
1
x
F
x
F
x
F
x
F
m
=
 Sketching Field Lines:
If given
)
,
(
y
x
F
, use
))
(
(
)
(
t
g
F
t
g
=
′
where
))
(
),
(
(
)
(
t
y
t
x
t
g
=
Line Integrals & Green's Theorem
[
C
h
.
2
]
 Line Integral of a Scalar Field:
If curve
C
in R
n
is defined by
]
,
[
),
(
b
a
t
t
g
x
∈
=
where
g
is C
1
and scalar field
f
is continuous on
C
, then the line integral of
f
along
C
is:
∫
∫
′
=
b
a
C
dt
t
g
t
g
f
ds
f

)
(

))
(
(
(total amount of "stuff" along C)
 Line Integral of a Vector Field:
If curve
C
in R
n
is defined by
]
,
[
),
(
b
a
t
t
g
x
∈
=
where
g
is C
1
and vector field
F
is continuous on
C
, then the line integral of
F
along
C
is:
∫
∫
′
⋅
=
⋅
b
a
C
dt
t
g
t
g
F
x
d
F
)
(
))
(
(
(total amount of F
•
x
along C)
 Properties:
∫
∫
⋅
+
⋅
=
⋅
C
C
C
x
d
G
x
d
F
x
d
G
F
)
∫
+
(
∫
∫
⋅
=
⋅
C
C
x
d
F
k
x
d
F
k
)
(
∫
∫
∫
⋅
+
⋅
=
⋅
∪
2
1
2
1
C
C
C
C
x
d
F
x
d
F
x
d
F
∫
∫
⋅
−
=
⋅
−
C
C
x
d
F
x
d
F
 Conservative (Gradient) Fields:
 First Fundamental Theorem:
Let
U
be connected and open. Let
n
R
⊆
n
R
U
F
→
:
be a continuous vector field
with a pathindependent line integral in
U
.
I
f
∫
⋅
x
x
x
d
F
x
0
)
(
=
φ
for some point
x
0
, then
U
x
x
F
x
∈
∀
=
φ
∇
)
(
)
(
.
 Second Fundamental Theorem:
Let
U
be connected and open. Let
n
R
⊆
n
R
U
F
→
:
be a continuous vector field in
U
.
I
f
φ
∇
=
F
, where
is a C
R
C
→
φ
:
1
C
is any curve joining
x
1
x
2
in
U
,
t
h
e
n
)
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View Full Document Simple Closed Curves & PathIndependence:
Let
U
be connected and open. A continuous vector field
n
R
⊆
n
R
U
F
→
:
is
pathindependent in
U
if and only if
0
=
⋅
C
x
d
F
∫
for every simple closed curve in
U
.
 SimplyConnected Sets:
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 Fall '09
 Vrscay

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