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241w-f13_16.1-16.5 (1)

# Examples 1 find 2 let c be graph of 5 3

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Unformatted text preview: les (1) ∫( ) . Find ∫ (2) Let C be graph of 5 . (3) Compute ∫ where C is the line segment from (1, 1, 0) to (2, 3, -1). 〈 Let 〉 be a vector field and C is given by () . The line integral of F along C is ∫( ∫ ( ∫ ) ∫ ( )dt ) 6 Remark If F is a force applied to move an object along a curve C from a point A to a point B, then the work done is ∫ Examples (1) If C is the line segment from (1, 1, 2) to (2, 3, 1), evaluate ∫ 7 (2) If ( ) 〈 〉 and C is given by , evaluate ∫ Remarks: ∫ ∫ ∫ ∫ 8 ∫ . 16.3 The fundamental theorem for line integrals Theorem If C: ( ) () () , is a curve, ( ( )) ∫ ( ( )) , then ∫ If C is a closed curve, i.e., Example Let ( ) || Let C be any curve from ( || ( ) ) 9 ( ( ( ). . . Then √ () Then ∫ () ). ) Theorem Let F be a vector field on an open connected region D (in plane or in the space). Then the following 3 conditions are equivalent: (1) (2) ∫ for some , i.e., F is conservative on D. for all closed paths in D. (3) ∫ is path inde...
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