Begin2.DVI

and then integrate both sides to obtain

∫ 6 −x 2

3 −x

dx=

∫ 4 −y

4 +y^2

dy (8.3)

Use a table of integrals and evaluate the integrals and then collect all the constants

of integration and combine them into just one arbitrary constantC to obtain the

result

3 x+^1

2

x^2 + 3 ln|x− 3 |= 2 tan−^1 (y/2)−^1

2

ln(4 +y^2 ) +C (8.4)

The equation (8.4) representsa one-parameter family of curveswhich describe the

field lines associated with the given vector field. Assign values to the constantCand

sketch the corresponding field line. Place arrows on the curves to show the direction

of the vector field.

Figure 8-2.

Vector plot and field lines forV~(x, y) = (3−x)(4−y)ˆe 1 + (6−x^2 )(4 +y^2 )ˆe 2

The figure 8-2 illustrates three graphs created by a computer. The figure 8-2(a)

represents a vector field plot over the region− 2 ≤x≤ 2 and− 2 ≤y≤ 2. The figure

8-2(b) is a graph of the field lines associated with the vectorfield with arrows placed

on the field lines. The figure 8-2(c) is the vectors of figure 8-2(a) placed on top of

the field lines of figure 8-2(b) to compare the different representations.

Divergence of a Vector Field

The study of field lines leads to the concept ofintensity of a vector fieldorthe

density of the field lines in a region. To visualize this, place an imaginary surface