Begin2.DVI

If these direction are the same, then by equating like components one must have

dx =ky and dy =kx or

dx

y =

dy

x =k (6 .79)

The equation (6.79) requires that x dx =y dy and if one integrates both sides of this

equation one obtains the family of field lines

x^2

2

−y

2

2

=C

2

or x^2 −y^2 =C (6 .80)

where C/ 2 is selected as the constant of integration to make all terms have a factor

of 2 in the denominator. Plotting these curves over the region Rfor various values

of the constant Cgives the field lines illustrated in the figure 6-16. The final figure

in figure 6-16 illustrates the field lines atop the vector field plot in order that you

can get a comparison of the two techniques.

Example 6-25.

An example of a two-dimensional scalar field

is a scalar function φ=φ(x, y ) representing the

temperature at each point (x, y )inside some spec-

ified region. The scalar field can be visualized by

plotting the family of curves φ(x, y ) = Cfor vari-

ous values of the constant C.

The resulting family of curves are called level curves and represent curves where

the temperature has a constant value. If the scalar field φ=φ(x, y ) represented

height of the water above some reference point, then one can think of say an island

where at different times the level of the water makes a contour of the island shape.

In this case the family of curves φ(x, y ) = C, for various values of the constant C,

are called level curves or contour plots since at various heights Cthe contour of the

island is given. Example contour plots are illustrated in figures 6-16 and 6-17.

Note that there are many computer programs capable of drawing contour plots

or level curves associated with a given scalar function. The figure 6-17 illustrates

contour plots or level curves for several different two-dimensional scalar functions as

the level Cchanges.