Graphics Reference
In-Depth Information
4.9.4.
If
w=
a dx
+
b dy
+
c dy
is a 1-form on
R
3
, show that
Ê
Á
∂
∂
b
x
∂
∂
a
y
ˆ
˜
Ê
Á
∂
∂
c
x
∂
∂
a
z
ˆ
˜
Ê
Á
∂
∂
c
y
∂
∂
b
z
ˆ
˜
d
w
=
-
dx
Ÿ+
dy
-
dx
Ÿ+
dz
-
dy
Ÿ
dz
.
4.9.5.
If
w=
F dy
Ÿ
dz
+
F dz
Ÿ
dx
+
F dx
Ÿ
dy
1
2
3
is a 2-form on
R
3
, show that
w=
(
)
d
div F dx
ŸŸ,
dy
dz
where F = (F
1
,F
2
,F
3
).
Section 4.9.1
4.9.1.1.
Compute the line integral
Ú
xy dx
+
y dy
c
along the curve c when
c is the part of the parabola y = x
2
from (0,0) to (1,1).
(a)
(b)
c is the polygonal path with vertices (0,0), (1,0), (1,1), (0,1), and (0,0).