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lecture20 - MATH 100 Class 20 Line Integral Definition: let...

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2006 Fall MATH 100 Lecture 14 1 MATH 100 Class 20 Line Integral Definition: let C be a smooth curve in 2-space x x(t), y y(t) (a t b) and let f(x,y) & g(x,y) be continuous on some open region containing C, then (,) (() ,() )' () b Ca fx y d x t y t xt d t ==≤ = ) ' line integral of or along with respect to or . b gxyd y gxt yt ytd t f gC x y = ∫∫
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2006 Fall MATH 100 Lecture 14 2 MATH 100 Class 20 Line Integral form. al differenti - ) , ( ) , ( ) , ( ) , ( integral line combined the with deal to need we n, applicatio In gdy fdx dy y x g dx y x f dy y x g dx y x f CC C + + + ∫∫
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2006 Fall MATH 100 Lecture 14 3 MATH 100 Class 20 Line Integral 22 C Ex: 2 ( ) over circular x cos t, y sin t (0 t ) 2 xydx x y dy π ++ ==
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2006 Fall MATH 100 Lecture 14 4 MATH 100 Class 20 Line Integral 22 2 2 2 00 1 2 0 0 0 2 2cos sin ( sin ) 2 sin (sin ) 2 2 3 ()1 c o s s i n 1 C C xydx tt t d t t d t udu xy d y t d t t ππ π =− = += = = ∫∫ : Sol
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2006 Fall MATH 100 Lecture 14 5 MATH 100 Class 20 Line Integral 3 1 1 3 2 ) ( 2 C 2 2 = + = + + dy y x xydx Remark: 1. Independence of parameterization 2 0 , sin cos 4 0 , 2 sin 2 cos 2 0 , sin cos 2 2 π = = = = = = t t y t x t t y t x t t y t x all produce 1/3
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2006 Fall MATH 100 Lecture 14 6 MATH 100 Class 20 Line Integral 2. Reversal of orientation If we reverse the orientation of the line integral, the line integral is the negation of the original result.
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This note was uploaded on 09/17/2011 for the course MATH 100 taught by Professor Qt during the Fall '09 term at HKUST.

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lecture20 - MATH 100 Class 20 Line Integral Definition: let...

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