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

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

x x o xx ss ss

Cartesian coordinates

source

example

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

0 1 x

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

The start of Katrina

Hurricane from a satellite.

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

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

Cartesian coordinates

right hand rule Flow velocity vorticity Magnetic field current

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

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

y d y a y b y c