Figure 8-6. Vector field V =√xkx (^2) +y 2 ˆe 1 +√xky (^2) +y 2 ˆe 2 ,k > 0 constant.
Observe that the magnitude of the vector field at any point (x, y )= (0 ,0) is given
by |V|=k. In polar coordinates (r, θ),where x=rcos θand y=rsin θ, the vector field
can be represented by
V =k(cosθˆe 1 + sin θˆe 2 ).Thus, on the circle r=Constant, the vector field may be thought of as tiny needles
of length |k|,which emanate outward (inward if kis negative) and are orthogonal to
the circle r=constant.The divergence of this vector field is
div V =∇·V = ky
2
(x^2 +y^2 )^3 /^2+ kx2
(x^2 +y^2 )^3 /^2=k
r,where r=
√x^2 +y^2 .The divergence of this field is positive if k > 0 and negative if
k < 0 .If the vector field V represents a velocity field and k > 0 ,the flow is said to
emanate from a source at the origin. If k < 0 ,the flow is said to have a sink at the
origin.
Green’s Theorem in the Plane
Let Cdenote a simple closed curve enclosing a region Rof the xy plane. If M(x, y )
and N(x, y )are continuous function with continuous derivatives in the region R, then
Green’s theorem in the plane can be written as
∫C©M(x, y )dx +N(x, y )dy =∫∫R(
∂N
∂x −∂M
∂y)
dx dy, (8 .13)