This is an equation where the variables can be separated and then an integration
produces another independent family of surfaces
μ 2 (x, y, z ) = x
2
2
−y
2
2
=c 2.
Hence the field lines are the intersection of the family of cylindrical surfaces, defined
by hyperbola with rulings parallel to the z-axis, with the family of planes x+y−c 1 z= 0.
Method of Tangents.
Observe that if the field lines are defined as the intersection of two families of
surfaces
μ 1 (x, y, z) = c 1 and μ 2 (x, y, z ) = c 2 ,
then by differentiation one obtains
∂μ 1
∂x
dx +∂μ^1
∂y
dy +∂μ^1
∂z
dz = grad μ 1 ·dr = 0
In a similar fashion one can show
∂μ 2
∂x dx +
∂μ 2
∂y dy +
∂μ 2
∂z dz = grad μ^2 ·dr = 0.
Note that at a point (x, y, z )on a curve of intersection of two surfaces μ 1 =c 1 and
μ 2 =c 2 ,the tangential direction dr =dx ˆe 1 +dy ˆe 2 +dz ˆe 3 is the same as the direction
of the field line at that point. Therefore dr must be proportional to F . At the
common point (x, y, z )on both surfaces the gradient vectors grad μ 1 and grad μ 2 are
perpendicular to the surfaces μ 1 =c 1 and μ 2 =c 2 respectively. These vectors must
therefore be perpendicular to the vector field F at this common point. Consequently,
one can write grad μ 1 ·F = 0 or
∂μ 1
∂x
F 1 +∂μ^1
∂y
F 2 +∂μ^1
∂z
F 3 = 0
and similarly grad μ 2 ·F= 0 or
∂μ 2
∂x F^1 +
∂μ 2
∂y F^2 +
∂μ 2
∂z F^3 = 0.
These equations are the basis for the method of tangents. One tries to find, by using
a trial and error method, two vector functions V = grad μ 1 and W = grad μ 2 such that
V ·F = 0 and W ·F= 0.Then the equations
V ·dr = grad μ 1 ·dr = 0 and W ·dr = grad μ 2 ·dr = 0
are exact differential equations which are easily integrated. From these integrations
one finds two independent family of surfaces μ 1 =c 1 and μ 2 =c 2.
