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Unformatted text preview: Lecture VII Paths and Curves First we go through several basic notions about paths. Let R ( t ) on [ a, b ] be a given path. Definition 1 R ( t ) is called elementary if for every pair ( t 1 , t 2 ) , with t 1 and t 2 distinct in [ a, b ] , R ( t 1 ) = R ( t 2 ) . Definition 2 R ( t ) is called simple if for every pair ( t 1 , t 2 ) , with t 1 and t 2 distinct in [ a, b ] , except possibly for the pair ( a, b ) , R ( t 1 ) = R ( t 2 ) . R ( a ) = R ( b ) . Definition 3 R ( t ) is called closed if A closed and simple path is called a loop . A loop in E 2 is called a Jordan curve . Given a Jordan curve C , E 2 can be divided into three regions, two bounded and one unbounded. The bounded regions are C and R i , the interior of C . The unbounded region is R e , the exterior of C . Definition 4 A directed curve is a curve along which we have specified a di- rection. Given a curve C , finding a path for C lets us apply calculus techniques. If C is R ( t ) = R cos t ˆ i + R sin t ˆ the circle of radius...
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- Two '04
- Math, dt, Differential geometry of curves, dR ds