Vector Fields

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Uri_dolphin3 Uri Wilensky (Author)


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This is a mathematical model that demonstrates abstract vector fields and integral curves.

Generally speaking, a field is a "region in which a body experiences a force as the result of the presence of some other body or bodies. A field is thus a method of representing the way in which bodies are able to influence each other. For example, a body that has mass is surrounded by a region in which another body that has mass experiences a force tending to draw the two bodies together.... The strength of any field can be described as the ratio of the force experienced by a small appropriate specimen to the relevant property of that specimen, e.g. force/mass for the gravitational field" (Oxford Minidictionary of Physics).

By 'abstract vector fields' we mean that this model is not committed to any specific type of force, such as gravity or magnetism. Rather, it simulates a general field, in which some focal property of influence affects a "small appropriate specimen", or particle, placed in the field.

Normally, if you look at a field with bare eyes, you will not necessarily see the forces. For instance, if you drop an apple it falls down, even though you cannot see the gravitational force. The apple is an object in the gravitational field. You saw how it behaved so you could guess that there is some force that made it go down. Humans do not perceive (visually) forces of gravitation or electro-magnetic forces. However, in a model, we can use little arrows (vectors) to show where, how forceful, and in which direction there are forces in this field.


In this model, the field is plotted using vector graphics: green streaks are individual vectors with yellow turtles serving as arrowheads. The length of each vector is roughly proportional to the magnitude of the vector field at each point. In this model, it is just the distance from the origin: The further away from the origin, the larger the vector. Also, all vectors are aimed clockwise along tangents to circles centered on the origin.

The vectors show you in what direction and how forcefully an appropriate specimen -- here, a 'particle' -- will be "knocked about" once it is placed the field. Once the particle is "knocked" to a new location, it will be knocked yet again by the force there (represented by the vector). Actually, it being "knocked about" continuously, but in this simulation, the "knock" occurs at discrete points in the field. Since the particle does not use up the forces, it will keep being knocked about. The path the particle takes is called its 'trajectory.' You will be able to track this trajectory because the particle will leave a red trail behind it as it moves along its trajectory. Trajectories in vector fields are called 'integral curves.'

Even though behavior of particles can be interesting and possibly unanticipated, owing to forces not being distributed uniformly in the field, or some other factor, we have chosen, for clarity, a vector field with a logical and consistent relation between location in space and size/orientation of the force. The vector field chosen for this particular model is

- y d/dx  +  x d/dy

Ideally, in the particular force field modeled here, the particle trajectories should be concentric circles (that is, the particle should go round and round along the same circular trajectory).


SETUP: Clears the world and computes the vector field. PLACE-PARTICLES: Puts the program into the mode in which you can position red test-particles by clicking anywhere in the View. GO: Runs the simulation continuously to show the integral curves.


Notice that the vectors grow in length as you move away from the origin. What effect do short vectors have on a particle? Long vectors?

The way this model is programmed, each particle moves some finite amount before calculating its new heading. Therefore, the particles do not turn as much as they would if their headings were continuously recalculated. This causes their trajectories to spiral slowly outward. (You have to let the model run for a while before this becomes apparent.) We tried to minimize this by having the particles move forward only a very small amount at each time step (the variable step-size). We couldn't make this amount too small since the model would then run too slowly. If you want the particles to spiral less, or you want the model to run faster, change this value.


Place particles in different parts of the world. Does the particle's position have any effect on the trajectory?


Try a different vector field by changing it in the setup-vector, force-x, and force-y procedures. For instance, if you choose

x d/dx - y d/dy

the integral curves will be hyperbolas.


If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

  • Wilensky, U. (1998). NetLogo Vector Fields model. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
  • Wilensky, U. (1999). NetLogo. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.


Copyright 1998 Uri Wilensky.


This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at

This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2002.

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breed [ particles particle ]  ;; the things that are affected by the field
breed [ vectors vector ]    ;; the vectors that affect the particles

  max-modulus  ;; the maximum modulus of all the vectors
  clicked?     ;; true if we have clicked the mouse button but have not yet placed a particle
  step-size    ;; the amount a particle moves forward

  modulus  ;; the length of the vector

;; set up the field, and create vectors

to setup
  set clicked? false

  ;; have particles move forward a small amount each time so that we
  ;; don't spiral too much but also so that the model doesn't run too slowly
  set step-size 0.0001

  ;; create vectors at regular intervals to see the effect of the force
  ;; at a particular place.
  ask patches
    if (pxcor mod 13 = 0) and (pycor mod 13 = 0)
      sprout-vectors 1
      [ setup-vector ]
  ;; draw vector field
  set max-modulus (max [modulus] of vectors)
  ask vectors [ show-vector ]

;; make the turtle become a vector and initialize the vector's variables

to setup-vector  ;; turtle procedure
  set color green
  if (force-x != 0) or (force-y != 0)
    [ set heading atan force-x force-y ]
  set modulus distancexy 0 0

;; particles update their orientation according to the vector-field force
;; operating on the patch where they are at

to go
  let stop? false
  ask particles
    ;; calculate the heading based on the force where this turtle is
    if force-x != 0 or force-y != 0
      [ set heading (atan force-x force-y) ]

    forward step-size * (distancexy 0 0)
  ;; 100 is an arbitrary factor used to produce a reasonable
  ;; frequency of view updates
  tick-advance 100 * step-size
  ;; if one of the particles was going to wrap around the world, stop.
  if stop?
  [ stop ]

;; report true if we will wrap around if we move forward by step-size

to-report going-to-wrap?  ;; turtle procedure
  let next-patch patch-ahead step-size
  report next-patch = nobody

;; place test particles

to place-particles
  if mouse-down?
  [ set clicked? true ]
  if (not mouse-down?) and clicked?
  [ place-particle mouse-xcor mouse-ycor
    display ]

;; create a particle at (x,y)

to place-particle [x y]
  create-particles 1
    setxy x y
    set size 10
    set heading 0
    set color red
    pd  ;; put the pen down so that we can see where it has traveled
    if force-x != 0 or force-y != 0
      [ set heading (atan force-x force-y) ]
  set clicked? false

;; calculate the horizontal force where the turtle is located

to-report force-x  ;; turtle procedure
  report ycor

;; calculate the vertical force where the turtle is located

to-report force-y  ;; turtle procedure
  report (- xcor)

;; draw the vector using a turtle to display strength and direction of field

to show-vector  ;; turtle procedure
  set modulus (10 * modulus / max-modulus)
  forward modulus
  set color yellow

; Copyright 1998 Uri Wilensky.
; See Info tab for full copyright and license.

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