Streamlines
A curve is a streamline (flow line, line of force) of a vector field F if at
each point p, F(p) is tangent to the curve through this point. (Remember the
magnet and the iron filings from high school?)
If C is a streamline given parametrically by
x=x(T)
y=y(T)
z=z(T)
or by R(T)={x(T), y(T), z(T)},
the tangent vector to C at a point is given by
R'(T)={x'(T), y'(T), z'(T)}
Since C is a streamline, F(x(T), y(T), z(T)) is also tangent to C at the point.
Therefore, the vectors are parallel, or
R'(T)=t F(x(T), y(T), z(T))
for some scalar t.
This implies that
{dx/dT, dy/dT, dz/dT}={t f1, t f2, t f3}
or, solving for t and equating (if the fi are nonzero)
dx/f1 = dy/f2 = dz/f3.
Solve this system of differential equations to find the streamlines.
We will do this with Mathematica, but to be honest, often it is easier
to do by hand for simple differential equations.
Up to Potential Flow in Two Dimensions
Example 1