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Unformatted text preview: Lecture XX
Vector Line Integrals; Conservative Fields 1 Vector line integrals �
Let F be a vector ﬁeld of domain D. Let D be connected, i.e. for any two points
P, P � in D there is a curve C contained in D that goes from P to P � . Let C be
a ﬁnite directed curve contained in D that is rectiﬁable, i.e. it has a length. We
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deﬁne the vector line integral
F · dR as the limit of a Riemann sum:
C � �
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F · dR = C lim
n→∞ n
� �
ˆ
F (Pi∗ ) · T (Pi∗ )Δsi , max Δsi →0 i=1 ˆ
where T (P ) is the unit tanget vector for C at P , and Pi∗ , Δsi are deﬁned in the
same way as for scalar line integrals. Let us give two basic laws for vector line
integrals:
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1. Let F be a vector ﬁeld on a ﬁnite directed curve C , and let C be divided
into curves C1 and C2 , both having the same direction as C . Then:
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� · dR =
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� · dR +
�
�
�
F
F
F · dR.
C C1 C2 �
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2. Let F and G be vector ﬁelds on a ﬁnite, directed curve C . Then:
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aF + bG · dR = a
F · dR + b
G · dR.
C C C There are two ways of evaluating vector line integrals. First, we can evaluate
using the deﬁnition of the integral as the limit of a Riemann sum. We can
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also evaluate by parameter. Let R(t) be a path for C , with t from a to b,
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� ��
ˆ
ˆ
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R(t) = x(t)ˆ + y (t)ˆ + z (t)k. Since T = dR � dR � and ds = � dR �, we get that
i
j
dt
dt
dt
dt
�
C �
�
F · dR = �� � �ˆ
F · T ds = C �
a 1 b � �
� dR
F·
dt �
dt 2 Conservative Fields �
Deﬁnition 1 Let D be a domain for a vector ﬁeld F . If for any points A, B ∈ D
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and any curve C ∈ D from A to B , the integral C F · dR depends only on A
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and B and not on the choice of C , then F is called a conservative ﬁeld.
An equivalent deﬁnition is the following:
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Deﬁnition 2 Let D be a domain for a vector ﬁeld F . If for any simple closed
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curve C ∈ D, C F · dR = 0, then F is called a conservative ﬁeld.
Any conservative ﬁeld has a scalar potential. 2 ...
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This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.
 Two '04
 Duorg
 Math, Integrals

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