Vector Field Plotter & Particle Tracer Math

Click to place particles. Drag to pan, scroll to zoom.
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What is the Vector Field Plotter?

Quick Answer: A interactive 2D vector field visualization that draws arrows and animates particle flow for a given vector field F(x,y) = (Fx, Fy). Perfect for exploring gradient fields, fluid flow, electric/magnetic fields, and differential equations.

Theory Behind Vector Fields

A vector field assigns a vector to every point in the plane. For example, F = (-y, x) represents a rotational field around the origin. The arrows show direction and magnitude (scaled for clarity). Streamlines are paths that follow the field; particles placed in the field move along these curves. The tool uses Math.js to evaluate the components at each grid point and for particle advection (Euler integration). The arrow length is scaled by a factor to prevent overlap. This visualization is crucial for understanding Gradient, Divergence, Curl, and Conservative Fields in multivariable calculus and physics.

Step-by-Step Examples

Example 1: Rotational Field (Vortex)

  1. Enter Fx = -y, Fy = x. Update.
  2. Arrows form concentric circles. Click on the canvas to place particles; they orbit around the origin.
  3. Increase particle count and speed to see multiple rotating trajectories.

Example 2: Gradient Field

  1. Set Fx = x, Fy = y. This is the gradient of f(x,y) = (x²+y²)/2.
  2. Arrows point outward from the origin. Particles fly away radially.

Example 3: Electric Dipole Approximation

  1. Enter Fx = (2*x^2 - y^2) / (x^2 + y^2)^(5/2), Fy = (3*x*y) / (x^2 + y^2)^(5/2) (a quadrupole-like field).
  2. Adjust arrow scale to 0.3 for readability. Particles trace curved field lines.

Frequently Asked Questions

What functions can I use in Fx and Fy?

All Math.js functions: sin, cos, tan, exp, log, sqrt, abs, pow, and arithmetic. Variables are x and y. Example: sin(x) * cos(y).

How do I add particles?

Click anywhere on the canvas. A particle will spawn at that world coordinate and start moving with the field. Use Clear Particles to remove all.

Why are some arrows very long?

The arrow length is capped to prevent huge vectors. Reduce Arrow Scale if they still overlap. The color also indicates magnitude (blue = small, red = large).

Can I see streamlines?

Particles leave a fading trail that approximates streamlines. For a static streamline, you would need a dedicated streamline integrator; the particle trace provides a dynamic version.

Does this support 3D vector fields?

Currently 2D only. For 3D fields, use our 3D surface/vector plotter tools.