ASSIGNMENT #5
1.
Let
F
be a
C
2
vector field on
3
R
. Prove that
.
F
F
F
2
div
grad
curl
curl
∇
−
=
2.
Let
and
0
k
j
i
r
≠
+
+
=
z
y
x
r
=
r
. In each case find a constant
k
such that the
expression is turned into an identity. Using some basic identities might help. Calculations
in fact are no longer then one line.
(a)
(b)
kr
r
=
∇
3
2
3
)]
(
[
r
k
r
r
=
∇
⋅
∇
∇
(c)
0
)
(
=
⋅
∇
k
r
r
(d)
4
2
2
)
(
r
k
r
=
⋅
∇
∇
r
3.
The height of a circular fence (radius 10m, centered at origin) at position
is
)
,
(
y
x
given by the function
, so the height varies from 3m to 5m.
)
(
01
.
0
4
)
,
(
2
2
y
x
y
x
h
−
+
=
Suppose that 1L of paint covers 100m
2
. Sketch the fence and determine how much paint
you will need if you paint both sides of the fence.
4.
It is often very useful to parametrize a curve
C
:
b
t
t
≤
≤
=
0
),
(
g
x
with respect to arc length
du
u
t
s
s
∫
′
=
=
0
)
(
)
(
g
t
to get
C
:
)
(
0
),
(
b
s
s
s
≤
≤
=
f
x
.
Consider the parametrization of
C
:
4
0
),
2
sin
,
2
cos
2
,
2
sin
(
)
(
π
≤
≤
−
=
t
t
t
t
t
x
.
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 Fall '09
 RomauldStanczak
 Calculus, Vector Space, Manifold, smooth curve

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