3. divergence  given a vector field, the divergence operation tells if there is a source or sink useful for relating electric fields to charges vector field ==> scalar

4. x x o xx ss ss two surfaces  s surround the point x o vector field enters from left and leaves from right evaluate integrals at the 2 surfaces Taylor series about the center point x o

5. x x o xx ss ss

6. Cartesian coordinates

7. source

8. example

9. Gauss’s law or divergence theorem convert from volume integral to surface integral

12. 0 1 x

13. curl or rot place paddle wheel in a river no rotation at the center rotation at the edges

15. The start of Katrina

16. Hurricane from a satellite.

17. the vector u n is out of the screen right hand rule  s is surface enclosed within loop closed line integral

18. four lines  x and  y surround the point x o, y o vector field follows closed loop evaluate integrals Taylor series about the center point

21. Cartesian coordinates

22. right hand rule Flow velocity vorticity Magnetic field current

27. repeated vector operations curl (grad a) = 0 divcurl A = 0 div (grad a) = Laplacian (a)

34. Phasors v = V cos (  t + ø) v = Re [V e j  t ] where V = V e jø

37. 0.01 1100 1 y d 0.01 1100 1 y a 0.01 1100 1 y b 0.01 1100 1 y c