MATH 241 Lecture Notes - Lecture 31: Surface Integral, Unit Vector

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

Document Summary

A surface is oriented if we have chosen/assigned a direction through the surface no orientation oriented up oriented down ex. ex. no orient. oriented right oriented left: (cid:1845) Formal defn. it"s a co(cid:374)ti(cid:374)uous assig(cid:374)(cid:373)e(cid:374)t of a u(cid:374)it vector to every poi(cid:374)t o(cid:374) the surface (2)intro imagine represents the flow of a fluid in 3d. sps is a surface in 3d, oriented. This is represented by the surface integral of over , denoted. Note: the represents the orientation but typically its hard to know exactly what is think. Parameterize by r (cid:4666)(cid:1873). (cid:1874)(cid:4667)=(cid:1876)(cid:4666)(cid:1873). (cid:1874)(cid:4667)(cid:2835) +(cid:1877)(cid:4666)(cid:1873). (cid:1874)(cid:4667)(cid:2836) +(cid:1878)(cid:4666)(cid:1873). (cid:1874)(cid:4667) where u,v are restricted by r, typically by inequalities. notation for row (3) evaluation basic way then. F n (cid:1845)= f((cid:1876)(cid:4666)(cid:1873),(cid:1874)(cid:4667),(cid:1877)(cid:4666)(cid:1873),(cid:1874)(cid:4667),(cid:1878)(cid:4666)(cid:1873),(cid:1874)(cid:4667)) (cid:4666)r (cid:3048) r (cid:3049)(cid:4667: to determine , check r (cid:3048) r (cid:3049). Either match s orientation or go in opp. dir: note: , is not r (cid:3048) r (cid:3049) even though it looks that way! ex) suppose f (cid:4666)(cid:1876),(cid:1877),(cid:1878)(cid:4667)=(cid:1876)(cid:1877)(cid:2835) +(cid:1878)(cid:2836) +(cid:1877)(cid:2870) .