Chapter 9B - l is a closed curve l If a curve l is closed,...

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If a curve is closed, we write the line integral as l l l d F r is a closed curve l
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l l d F r is a closed curve l Fundamental Theorem for Line Integrals (( ) ) )) ) ) )) 0 lC f d fd fb fa fa fa =∇ ⋅ = ∇⋅ =- = ∫∫ F F rr & is if for some ( is called a function for ). ff f FF F conservative potential is a closed curve, so have the same initial point and terminal point. l ( ): terminal point b r ( ): initial point a r
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Implications of Conservative Field Fundamental Theorem for Line Integrals (( ) ) )) CC f d fd fb fa =∇ ⋅ = ∇⋅ =- ∫∫ F F rr is independent of path C d F r l d l path closed any for 0 = r F ve conservati is F
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Example circle. unit the (ii) ]; , 0 [ , sin cos ) ( by given (i) is where curve over the integral this evaluate and path of t independen is integral line that the Show . ) 2 ( ) 3 ( ) , ( Let 2 2 p + = + + = t t e t t C C d xy x y y x t C j i r r F j i F
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23 By our earlier example, where (, ) is the potential function of . So is conservative. Hence, the line integral is independent of path. CC f f x y x yx d fd = =+ ⋅ = ∇⋅ ∫∫ F FF F rr ve. conservati is 2 . 2 and 3 Here, 2 2 F = = = + = y P y x Q xy Q x y P 22 (,) ( 3) ( 2 x yy x xy = ++ F ij To show that the line integral is independent of path. C d Fr
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(i) ( ) co s si n , 0 t t te tt p = + ≤≤ r ij ( ( )) ( ( )) ( 1,0) (1,0) 2 CC d fd fb fa ff ⋅ = ∇⋅ =- - F r r r r Fundamental Theorem for Line Integrals (( ) ) )) f d =∇ ∫∫ F F rr 22 (,) ( 3) ( 2 ) x yy x xy = ++ F ij 23 where f f x y x yx = =+ F () (co s ) ( si n ) 0 ( 1,0) e p p pp = + + →- r i jij 0 (0 ) (cos0 sin0 ) e = + = +→ j
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(ii) Since the unit circle is a closed path and is conservative, so we have 0 C d ⋅= F Fr Fundamental Theorem for Line Integrals (( ) ) )) ) ) )) 0 lC f
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This note was uploaded on 11/01/2011 for the course MATH 1505 taught by Professor Yap during the Spring '11 term at National University of Singapore.

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Chapter 9B - l is a closed curve l If a curve l is closed,...

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