c 2 are the two parameters. Two methods for obtaining independent integrals of
equations (9.83) are now presented.
Theory of Proportions
From the theory of proportions one can make use of the following result:
For constants α,β,γ, not all zero, one can write
dx
F 1
=dy
F 2
=dz
F 3
=αdx +βdy +γdz
αF 1 +βF 2 +γF 3
.
In many instances one can choose appropriate values for the constants α,β,γ to
construct equations which can be easily integrated to produce a family of surfaces
representing a solution of the differential equations. Using the method of propor-
tions, by trial and error, one tries to construct two independent family of solutions.
Consider one surface from each family. These surfaces intersect and the curve of
intersection defines a field line. The two family of surfaces intersect in a family of
field lines.
Example9-6. Find the field lines associated with the vector field
F=F(x,y,z )= yˆe 1 +xˆe 2 +zˆe 3
Solution The field lines are obtained from the differential equations
dx
y =
dy
x =
dz
z (9 .84)
In this equation make note that an addition of the numerators and denominators of
the first two fractions produces an exact differential and
dx +dy
x+y =
dz
z =
d(x+y)
x+y.
In this new equation the variables are separated and then an integration produces
ln(x+y)=ln z+ln c 1 where ln c 1 is selected for the constant of integration in order to
simplify the algebra. This result can be expressed as
μ 1 (x,y,z)=
x+y
z =c^1
and represents one family of solution surfaces. Return to the equations (9.84) defin-
ing the field lines and observe that from the first two fractions one can write
dx
y =
dy
x.
