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lecture21

# lecture21 - MATH 100 Class 21 Line Integral independent of...

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2006 Fall MATH 100 Lecture 14 1 MATH 100 Class 21 Line Integral independent of path (1,1) to (1,0) then (1,0), to (0,0) from (c) (1,1) to (0,0) from (b) (1,1) to (0,0) from (a) over evaluate , ) , ( : Ex path of t independen integral Line 21. Class 3 x y x y xdy ydx r d F j x i y y x F C C = = + = + = G G G G G

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2006 Fall MATH 100 Lecture 14 2 MATH 100 Class 21 Line Integral independent of path only. endpoints on depends integral the because same, the All 1 1 ) 1 ( ) 0 ( 0 1 0 , , 1 : 1 0 , 0 , : (c) 1 4 3 , (b) 1 2 , (a) 1 0 1 0 2 1 1 0 3 1 0 2 3 3 1 0 1 0 2 1 = + + + = + = = = = = = = + = + = = = = + = + = = ∫∫ dt d t td dt t t y x C t y t x C dt t dt t t dt t xdy ydx t y t x tdt tdt tdt xdy ydx t y t x CC C C C Sol:
2006 Fall MATH 100 Lecture 14 3 MATH 100 Class 21 Line Integral independent of path ) , ( ) , ( ) , ( region, in the entirely lies and ), , ( at and , ) , ( at starts that curve smooth piecewise any for then region, the of point each at ) , ( ) , ( If ). , ( & ) , ( points the continuing region open some in continuous are & where ) , ( that Suppose integral) line of Theorem l fundamenta (The 0 0 1 1 C 1 1 0 0 1 1 0 0 y x y x r d y x F y x y x C y x y x F y x y x g f j g i f y x F φφ φ = = + = G G G G G G Thm:

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2006 Fall MATH 100 Lecture 14 4 MATH 100 Class 21 Line Integral independent of path j y i x y x F b y b x y x a y a x y x b t a t y y t x x C G G G + = = = = = = φφ φ ) , ( Since )). ( ), ( ( ) , ( )), ( ), ( ( ) , ( s.t. ), ( ), ( by e Parmeteriz 1 1 0 0 Proof:
2006 Fall MATH 100 Lecture 14 5 MATH 100 Class 21 Line Integral independent of path [] for function potential : function ve conservati : case In this . if path of t independen -Integral - ) , ( ) , ( )) ( ), ( ( )) ( ), ( ( )) ( ), ( ( )) ( ), ( ( ) , ( So 0 0 1 1 0 C C C C F F F y x y x a y a x b y b x t y t x dt t y t x dt d dt dt dy y dt dx x dy y dx x r d y x F b t G G G G G φ φφ = = = = = + = + = =

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2006 Fall MATH 100 Lecture 14
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lecture21 - MATH 100 Class 21 Line Integral independent of...

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