Begin2.DVI

Vector Fields Irrotational and Solenoidal

If in addition to being conservative, the two-dimensional vector field given by

F
=M(x, y )ˆe 1 +N(x, y )ˆe 2 is also solenoidal and

div F
=

∂M

∂x

+

∂N

∂y

= 0,

then

(i) The equipotential curves φ(x, y ) = c,c constant, are obtained from the exact

differential equation

dφ = grad φ·d
r or dφ =∂φ

∂x

dx +∂φ

∂y

dy = 0 or dφ =M(x, y )dx +N(x, y )dy = 0

(ii) The family of field lines ψ(x, y ) = c∗,c∗constant, are obtained from the differential

equation

d
r =kF
or dx

M

=dy

N

=k or −N dx +M dy = 0

The solution of the differential equation defining the field lines is easily obtained

since it also is an exact differential equation. The solution can be represented as a

line integral in either of the forms

ψ(x, y ) =

∫x

x 0

−N(x, y 0 )dx +

∫y

y 0

M(x, y )dy =c

or ψ(x, y ) =

∫x

x 0

−N(x, y )dx +

∫y

y 0

M(x 0 , y )dy =c,

(9 .68)

depending upon the choice of the path connecting the end points. These curves

represent the field lines associated with the vector field F ,
where

∂ψ

∂x =−N and

∂ψ

∂y =M.

It follows that if the vector field is both irrotational and solenoidal, then the equipo-

tential curves φ(x, y ) = cand the field lines ψ(x, y ) = c∗ are such that

∂φ

∂x

=∂ψ

∂y

and ∂φ

∂y

=−∂ψ

∂x

. (9 .69)

These equations are called the Cauchy–Riemann equations. In vector form these

equations may be expressed as

grad φ= (grad ψ)׈e 3.