notes001 (27)

# notes001 (27) - = x y Thus Z C F& d r = Z C r f& d r...

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November 20, 2009 NAME________________________________ Let F be the vector ±eld F ( x; y ) = i + j and let C be the directed line segment beginning at the point (0 ; 0) and ending at the point (1 ; 1) . Evaluate the line integral Z C F d r . You must include all details of your work. Solution: The easiest way to do this is to use the Fundamental Theorem of Line Integrals: The vector ±eld F ( x; y ) = i + j is conservative and has potential function f ( x; y
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Unformatted text preview: ) = x + y . Thus Z C F & d r = Z C r f & d r = f (1 ; 1) ± f (0 ; 0) = 2 ± 0 = 2 . Now we show how to do the integral using the de±nition of line integrals: The curve C can be parameterized as r ( t ) = t i + t j ² t ² 1 from which we see that r ( t ) = i + j and F ( r ( t )) & r ( t ) = ( i + j ) & ( i + j ) = 2 . Thus Z C F & d r = Z C F ( r ( t )) & r ( t ) dt = Z 1 2 dt = 2 . 1...
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