MATH 241 Lecture Notes - Lecture 28: Line Integral, Antiderivative

• School: University of Maryland
• Department: Mathematics
• Course: MATH 241
• Professor: Christian Rosendal

Document Summary

(4/16) 15. 2 continued (2) line integrals of vector fields: (a) suppose f is a vf representing a force field, fluid flow, etc. Suppose c is an oriented curve (has direction) that describes the route of an object through . A: the total work is given by the line integral of f over c, denoted. F (cid:1856)(cid:1870) (cid:3030) (b) evaluation (the basic method) then (cid:1856)(cid:1872) = f ((cid:1876)(cid:4666)(cid:1872)(cid:4667),(cid:1877)(cid:4666)(cid:1872)(cid:4667),(cid:1878)(cid:4666)(cid:1872)(cid:4667)) (cid:1870) (cid:4666)(cid:1872)(cid:4667) (cid:2912)(cid:2915)(cid:2913)(cid:2925)(cid:2923)(cid:2915)(cid:2929) (cid:2929)(cid:2913)(cid:2911)(cid:2922)(cid:2911)(cid:2928) (cid:2916)(cid:2931)(cid:2924)(cid:2913)(cid:2930)i(cid:2925)(cid:2924) (cid:3030) (cid:3028) note: the dot product arises as a result of projecting the f vector onto the direction of motion ex. Find the total work done by f an the object as it moves along c. Seln f (cid:1856)(cid:1870) (cid:3030) (3) alternate notation sometimes we write (cid:4666) (cid:4667) (cid:1876)(cid:1877)(cid:1856)(cid:1876)+(cid:1877)(cid:2870)(cid:1856)(cid:1877)+(cid:1876)(cid:1878)(cid:1856)(cid:1878) (cid:3030) ex. if c is param. by r (cid:4666)(cid:1872)(cid:4667)=(cid:1872)(cid:2871)(cid:2835) +cos(cid:1872)(cid:2836) +(cid:1857)(cid:2872)(cid:1863) (cid:882) (cid:1872) (cid:889) (cid:1876)(cid:1877)(cid:1856)(cid:1876)+(cid:1878)(cid:1856)(cid:1877)+(cid:1876)(cid:1878)(cid:1856)(cid:1878) (cid:3030) the previous example could have been. Again the notation tells us what to do! (cid:4666) (cid:4667) then consider: note same as to evaluate: