line_1 - PHY2061 R. D. Field Line Integral of a Vector...

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PHY2061 R. D. Field Department of Physics line_1.doc Univesity of Florida P 1 Closed Loop Line Integral of a Vector Function Let z z y x F y z y x F x z y x F z y x F z y x ˆ ) , , ( ˆ ) . . ( ˆ ) , , ( ) , , ( + + = ! be a vector function of position. In general, the line integral of ) , , ( z y x F ! depends of the start point P 1 , and the end point P 2 and the path chosen from P 1 to P 2 (curve C): = CurveC r d F Path P P I ! ! ) , , ( 2 1 , where I(P 1 ,P 2 ,Path) is the component of along the path integrated over the path, ∫∫ + + = = CurveC CurveC z y x CurveC dz F dy F dx F dr F r d F ) ( cos θ ! ! . Remark: If 0 = × F ! ! then I(P 1 ,P 2 ,Path) = I(P 1 ,P 2 ) and is only a function
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