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l_notes20 - Lecture XX Vector Line Integrals Conservative...

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�� 1 Lecture XX Vector Line Integrals; Conservative Fields 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 d F R as the limit of a Riemann sum: define the vector line integral · C n d F R = lim F ( P i ) · ˆ T ( P i s i , · C n →∞ max Δ s i 0 i =1 where T ˆ ( P ) is the unit tanget vector for C at P , and P i , Δ s i 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 C 1 and C 2 , both having the same direction as C . Then: d d d F R = F R + F R. · · · C C 1 C 2 F
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