Vector field streamlines

Parametric equations

Given a vector field $\mathbf{F}(x,y)=(F_x,F_y)$, a streamline is a solution of this ordinary differential equation:

An equivalent form:

A streamline is an integral curve of the direction field. It helps with gradients, curl, and the solutions of first-order differential equations.

The field here is a vortex plus a perturbation:

$\epsilon$ avoids division by zero at the origin. The main vortex term is perpendicular to the radius, so the field rotates overall; the small $0.25$ sine and cosine terms tilt it slightly.

Interaction

• Number of streamlines: add or remove streamlines integrated at the same time
• Integration steps: more steps make each streamline longer and more detailed
• Flow speed: drive the field’s change over time so the streamlines keep updating

What to notice

• A streamline is tangent to the vector field. It can be read as a geometric solution of the differential equation.
• Near the center of the vortex, streamlines circle densely.
• At a fixed moment, streamlines of the field from different starting points never cross (except at singular points); the field here changes with $t$.

Related

• Basic patterns of a vector field
• Differential equations, seen geometrically

Further reading

• Field line (Wikipedia)