Problem Set 10
MH2100/MTH211
Solutions
1 Questions on understanding the lecture
E±²³´µ¶² · The NewtonLeibniz formula
¸
b
a
±²
=
²
(
b
)
²
(
a
) was proved in lectures only
for a smooth curve
C
. Does the formula hold for piecewise smooth curves?
S¹º»¼µ¹½
Yes, we can just sum it over all smooth pieces.
:)
E±²³´µ¶² ¾ Prove that the work of a conservative vector ﬁeld over any piecewise smooth
closed curve is zero.
S¹º»¼µ¹½
From Exercise 1, as we can assume the vector ﬁeld equals grad(
²
) because it’s
conservative. Then the work integral over a piecewise smooth closed curve
C
can be
computed as
¿
C
±²
= 0
³
:)
E±²³´µ¶² À How can one parametrize a plane
´µ
+
¶·
+
¸¹
+
±
= 0
³
S¹º»¼µ¹½
If
´ º
= 0, we can let
·
=
»
,
¹
=
¼
and get
µ
=

¶
´
» 
¸
´
¼ 
±
´
. If
´
= 0 then either
¶ º
= 0 or
¸
=
º
= 0 and we solve for that corresponding variable instead.
:)
1
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View Full Document2 Questions on calculation
E±²³´µ¶² · Evaluate
¸
(
±
2
²³
2
²´
2
)
(
±
1
²³
1
²´
1
)
±µ±
+
³µ³
+
´µ´
√
±
2
+
³
2
+
´
2
, where the point (
±
1
²³
1
²´
1
) lies on the sphere
±
2
+
³
2
+
´
2
=
¶
2
and the point (
±
2
2
2
) is on the sphere
±
2
+
³
2
+
´
2
=
·
2
for
¶ >
0
and
· >
0.
S¹º»¼µ¹½
Note that this integral can be regarded as a line integral of the vector ﬁeld
v
=
±
√
±
2
+
³
2
+
´
2
i
+
³
√
±
2
+
³
2
+
´
2
j
+
´
√
±
2
+
³
2
+
´
2
k
. Based on the expression of the integral, it
suggests that the integral in pathindependent hence we shall be able to ﬁnd the potential
of the given ﬁeld. Let
¸
be the potential of the given vector ﬁeld. Then
¸
±
=
±
¾
±
2
+
³
2
+
´
2
¹
¸
(
±²³²´
) =
¾
±
2
+
³
2
+
´
2
+
º
(
³²´
)
»
Notice that
º
(
) = 0 works and hence
¸
(
) =
¾
±
2
+
³
2
+
´
2
is the potential. By
NewtonLeibniz’s formula, the integral is
¸
(
±
2
2
2
)
 ¸
(
±
1
1
1
) =
·  ¶»
:)
E±²³´µ¶² ¿
(a) Let
C
be a closed curve. Prove that the area inside
C
equals
À
C
±µ³
=

À
C
³µ±
=
1
2
À
C
±µ³  ³µ±²
(b) Apply this formula to ﬁnd the area enclosed by the astroid
±
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 Spring '13
 Cos, Manifold, Vector field

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