MATH 234 Lecture Notes - Lecture 33: Ellipse, Parametrization, Divergence Theorem

• School: UW-Madison
• Department: Mathematics
• Course: MATH 234
• Professor: Betsy Stovall

Document Summary

Is a simply connected plane domain (no holes),(cid:1829) is the boundary (cid:3019) R (cid:1845: flux integral of vector field v. (cid:1842) (cid:1876)+(cid:1843)(cid:1877)= ((cid:1842)(cid:1843)) ((cid:1876)(cid:4666)(cid:4667)(cid:1877)(cid:4666)(cid:4667))=(cid:1876) (cid:4666)(cid:4667) is a parametrization of (cid:1829) (cid:1846) (cid:4666)(cid:4667)= (cid:2869) (cid:3051) (cid:4666)(cid:4667) (cid:1876) (cid:4666)(cid:4667) unit together vector. Is also a solution of (*) . Is the boundary of a simply connected (cid:1844) (cid:1877) (cid:4666)(cid:4667)) (cid:883) (cid:1845) So, proved that gree(cid:374)"s for(cid:373)ula applied to flux i(cid:374)tegral. Definition: if =((cid:1842)(cid:1843)) is a (cid:884)(cid:1830) vector field, then function (cid:1842)(cid:1876)+(cid:1843)(cid:1877) is called the divergence of . Theorem: (cid:884)(cid:1830) divergence theorem: (cid:1844) is a simply connected plane domain so (cid:1829) is its boundary. We choose vector field (cid:4672)(cid:3017)(cid:3018)(cid:4673) satisfying (cid:1843)(cid:3051) (cid:1842)(cid:3052)=(cid:883) (cid:1842)= (cid:1877)(cid:884) (cid:1843)=(cid:1876)(cid:884), Area (cid:1844)= (cid:3019: let (cid:1844) be the domain inside the ellipse y. Boundary is ellipse a b b a x (cid:1829)={(cid:4666)(cid:1876),(cid:1877)(cid:4667)|(cid:1876)(cid:2870)(cid:1853)(cid:2870)+(cid:1877)(cid:2870)(cid:1854)(cid:2870)=(cid:883)} (cid:3029: find (cid:1877)(cid:2870) (cid:1876)+(cid:885)(cid:1876)(cid:1877) (cid:1877) where (cid:1829) is the boundary of (cid:1844) (cid:1876) =(cid:882) for any closed (cid:1829) (cid:3051)(cid:3052)=(cid:3052)(cid:3051) or (cid:1842)(cid:3052)=(cid:1843)(cid:3051)