MATH 2110Q Lecture 22: Curl and Divergence

• School: UConn
• Department: Mathematics
• Course: MATH 2110Q
• Professor: Myron Minn-Thu-Aye

Document Summary

Find div ( ) for = (cid:1766)(cid:1876)(cid:1877),(cid:1877)(cid:1878),(cid:1876)+(cid:1878)(cid:1767: divergence is always a scalar function. Example: vectors flowing in/out at this point are equal length, so div ( ) = 0, for any =(cid:1766)(cid:1842),(cid:1843),(cid:1844)(cid:1767), the curl of is curl ( ) = * Find curl ( ) for = (cid:1766)(cid:1876)(cid:1878),(cid:1877)(cid:2870)+(cid:885),(cid:1876)(cid:3053)(cid:1877)(cid:2870)(cid:1767: = (cid:3051),(cid:3052),(cid:3053) * (cid:1766)(cid:1842),(cid:1843),(cid:1844)(cid:1767, = (cid:3052)(cid:1844) (cid:3053)(cid:1843),(cid:3053)(cid:1842) (cid:3051)(cid:1844),(cid:2870)(cid:3051)(cid:1843) (cid:3052)(cid:1842) , = (cid:1844)(cid:1877) (cid:1843)(cid:3053),(cid:1842)(cid:3053) (cid:1844)(cid:3051),(cid:1843)(cid:1876) (cid:1842)(cid:3052) , curl ( ) = (cid:1766)(cid:884)(cid:1876)(cid:2870)(cid:1877) (cid:882),(cid:1876) (cid:884)(cid:1876)(cid:1877)(cid:2870),(cid:882) (cid:882)(cid:1767, = (cid:1766)(cid:884)(cid:1876)(cid:2870)(cid:1877),(cid:1876) (cid:884)(cid:1876)(cid:1877)(cid:2870),(cid:882)(cid:1767, at each point, curl ( ) is the axis of rotation in f. |curl ( )| gives rotational speed: is a velocity speed. In each part, suppose is a vector field that looks like the given diagram in each horizontal plane (the z-component of every vector is 0). Example (a) (b: no rotation in this vector field, curl ( ) = (cid:882) = (cid:1766)(cid:882),(cid:882),(cid:882)(cid:1767) everywhere.