Integrals gradient

line integral work DW is the line integral from point a to point b a b

Cartesian coordinates This is a definition! x y 4 a b c

x y 4 a b c

a b  Closed line integral work DW is the line integral from point a to point b - continue on to a a b = 0 ==> conservative field  0 ==> nonconservative field

conservative field V Ra Rb Rc I

Kirchhoff 1824-1887

x y 4 a b c

nonconservative field

x y 4 a b c

x y 4 a b c

x y 4 a b c

Where are the computers? Vectors, phooey! Too much math!

closed surface integral z x y ds F

F x y z Ds Ds cube a3

volume integral x y z

derivative operations directional derivative - value of the rate of change of scalar quantity in a given direction gradient - direction and value of the maximum rate of change of scalar quantity divergence - source or sink of a vector field curl - direction and magnitude of rotation

gradient How do you climb to the top?

directional derivative gradient

gradient

gradient separation of two equipotential contours is Dx DV/Dl along l DV/Dl along normal

Cartesian coordinates

Example start with scalar function V = x2 sin y end up with a vector V = 2x sin y ux + x2 cos y uy

>>[ex,ey]=gradient(V,.2,.2); >>quiver(x,y,ex,ey,2)

at the center of the world Finland at the center of the world 900 km 40 km