l_notes20 - Lecture XX Vector Line Integrals; Conservative...

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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 � � � define the vector line integral F · dR as the limit of a Riemann sum: C � � � 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: � 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: � � � � · dR = � � · dR + � � � F F F · dR. C C1 C2 � � 2. Let F and G be vector fields on a finite, directed curve C . Then: �� � � � � � � � � � � 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 � also evaluate by parameter. Let R(t) be a path for C , with t from a to b, �� � �� � � �� ˆ ˆ � 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 � � � and any curve C ∈ D from A to B , the integral C F · dR depends only on A � and B and not on the choice of C , then F is called a conservative field. An equivalent definition is the following: � Definition 2 Let D be a domain for a vector field F . If for any simple closed � � � � 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.

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l_notes20 - Lecture XX Vector Line Integrals; Conservative...

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