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).
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