Begin2.DVI

This differential equation is derived by requiring the direction of the vector field

at an arbitrary point (x, y )have the same direction as the tangent to the field line

curve which passes through the same point (x, y ). An integration of the differential

equation defining the field lines produces

ln x= ln y+ ln c

or the family curves defining the field lines is given by

ψ(x, y ) =

y

x=k,

where kis an arbitrary constant.

The equipotential curves and field lines associated with the given vector field

F
=xˆe 1 +yeˆ 2 are illustrated in figure 9-4. In two dimensions, the vector fields are

best visualized by sketches of the equipotential curves and field lines. In sketching

the vector fields be sure to distinguish the field lines from the equipotential curves by

placing arrows at various points on the field lines. These arrows indicate the direction

of the vector field at various points.

Figure 9-4. Equipotential curves and field lines for F
=xˆe 1 +yˆe 2