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Premium experiment • divergence, curl & flow

MATHEMATICS • MULTIVARIABLE CALCULUS

Vector Fields & Particle Flow

Combine radial source or sink strength with rotational swirl, then track a passive particle while reading divergence and curl.

● PARTICLE FLOWField vectors and trajectory
Divergence
Curl
Speed
Particle position

Reasoning inspector

live model state
Source strength a—
Swirl ω—
Divergence—
Curl—
Particle position—
Radius—
Field speed—
Radial rate dr/dt—
Angular rate—

Local field structure

Divergence measures local expansion or contraction while curl measures local rotation; particle paths reveal how both effects combine over time.

LEARN WITH THE SIMULATION

Understand Vector Fields & Particle Flow by experimenting.

START HERE

A vector field assigns a direction and magnitude to every point. This laboratory uses the linear planar field F(x,y) = (ax−ωy, ωx+ay), which cleanly separates radial source or sink behavior from rotation. The parameter a controls divergence, ω controls scalar curl, and a passive tracer reveals how those local properties combine into expanding, contracting or spiraling trajectories. The premium inspector also exposes radius, radial growth rate and angular rate.

BUILD THE IDEA

What you should understand first

  • Read field arrows as local velocity vectors rather than as a pre-drawn particle path.
  • Derive divergence ∇·F = 2a and scalar curl = 2ω for the modeled linear field.
  • Explain why source strength changes radial growth or decay while swirl changes rotation rate.
  • Predict which field invariants stay unchanged when only the tracer's starting position moves.

TRY IT YOURSELF

How can divergence and curl be changed independently in this field?

  1. Set swirl to zero and compare positive, zero and negative source strength.
  2. Return source strength to zero and change swirl through positive and negative values.
  3. Combine nonzero source and swirl and follow the resulting spiral trajectory.
  4. Move the starting point without changing a or ω and verify that the inspector's divergence and curl remain constant.
Read the visual:

In this special linear field, divergence depends only on a and curl only on ω. The tracer location changes its velocity magnitude and position but not those two spatially uniform invariants. Combining nonzero values produces spiraling expansion or contraction.

WHY IT WORKS 01

The field decomposes naturally into radial and rotational pieces

The vector (ax,ay) points radially outward when a>0 and inward when a<0, while (−ωy,ωx) is perpendicular to the position vector and therefore drives rotation. Adding them produces the full field. This decomposition explains the geometry without integrating the trajectory first.

WHY IT WORKS 02

Divergence and curl are constant because the field is linear

Taking partial derivatives gives ∂Fx/∂x = a and ∂Fy/∂y = a, so divergence is 2a everywhere. The planar scalar curl is ∂Fy/∂x − ∂Fx/∂y = ω − (−ω) = 2ω. Neither expression contains x or y, so moving the tracer cannot change these invariants.

WHY IT WORKS 03

Radius and angle obey especially simple differential equations

For this field, the radial distance satisfies dr/dt = ar, while the polar angle changes at dθ/dt = ω. Positive a produces exponential radial growth, negative a produces exponential decay and ω sets a constant angular speed. The resulting path is therefore a logarithmic spiral when both terms are nonzero.

DO NOT MIX THESE UP

Common misconception

Divergence and curl are local field properties, not labels for one particular trajectory. Zero divergence does not mean every vector is zero, and zero curl does not mean the field has no motion. The particle shown is a passive tracer following dx/dt = F(x,y), not a massive object accelerated by a force through Newton's second law.

PRACTICE CHALLENGE

Can you match target divergence and curl, then move the tracer and explain why those values do not change?

  • Choose target divergence and curl values.
  • Adjust source strength a and swirl ω until both targets are reached.
  • Move the tracer starting point while keeping a and ω fixed.
  • Use radius, radial-rate and angular-rate metrics to explain the new trajectory while confirming the same divergence and curl.
Reveal the reasoning

For F=(ax−ωy,ωx+ay), divergence is exactly 2a and curl is exactly 2ω, so the targets determine a and ω independently. Changing the starting point changes the tracer's initial radius and therefore speed and radial rate, but it does not alter the field derivatives that define divergence and curl.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The field is F=(a x - omega y, omega x + a y) in a two-dimensional Cartesian plane.
  • Particle paths are passive-tracer trajectories integrated with fourth-order Runge-Kutta stepping.
  • Divergence is 2a and scalar curl is 2 omega everywhere in the modeled linear field.

Model limitations

  • The field is time-independent and excludes nonlinear spatial terms, boundaries and particle inertia.
  • Displayed arrow lengths are normalized for readability rather than encoding absolute magnitude directly.

CONNECT IT

Where this matters

  • Fluid-flow visualization
  • Electromagnetic field intuition
  • Dynamical systems
  • Multivariable-calculus interpretation of divergence and curl

WORDS THAT MATTER

Key vocabulary

Vector field
Function assigning a vector to every point in a region.
Divergence
Scalar ∂Fx/∂x+∂Fy/∂y measuring local source or sink strength.
Curl
In a planar field, scalar ∂Fy/∂x−∂Fx/∂y measuring local rotational tendency.
Passive tracer
Point that moves with the prescribed field velocity without altering the field.
Radial rate
Rate dr/dt at which the tracer's distance from the origin changes.