lect14_2

# lect14_2 - Line Integrals(14.2 Let C be a curve in R 2 or...

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Line Integrals - (14.2) Let C be a curve in R 2 or in R 3 . The curve C is said to be oriented if there is a direction to it. For example, C : r ! ! t " "# 2cos t , 2sin t \$ ,0 ! t ! ! 2 . The curve C is oriented since it goes from the point 2, 0 to 0, 2 counterclockwise as t increases from 0 to ! 2 . Line Integrals Let C be an oriented curve which is represented by parametric representations: r ! ! t " "# x ! t " , y ! t " , z ! t " \$ , a ! t ! b where functions x ! t " , y ! t " and z ! t " have continuous first derivatives. Let f ! x , y , z " be continuous in a region D which contains the curve C . 1. The line integral of f ! x , y , z " with respect to arc length along the oriented curve C : " C f ! x , y , z " ds " " a b f ! x ! t " , y ! t " , z ! t "" ! x % ! t "" 2 & ! y % ! t 2 & ! z % ! t 2 dt " " a b f ! r ! ! t || r ! ! t " || dt Similarly, we have, " C f ! x , y " ds " " a b f ! x ! t " , y ! t x % ! t 2 & ! y % ! t 2 dt 2. The line integral of f ! x , y , z " with respect to x along the oriented curve C : " C f ! x , y , z " dx " " a b f ! x ! t " , y ! t " , z ! t x % ! t " dt Similarly, the line integral of f ! x , y , z " with respect to x along the oriented curve C : " C f ! x , y , z " dy " " a b f ! x ! t " , y ! t " , z ! t y % ! t " dt 3. The line integral of F !

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## This note was uploaded on 02/05/2012 for the course MATH 2142 taught by Professor Lerna during the Fall '10 term at FIU.

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lect14_2 - Line Integrals(14.2 Let C be a curve in R 2 or...

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