Math
中文

Multivariable Calculus · Vector field

Vector field

A vector field attaches an arrow F(x, y) = (P, Q) to every point of its domain; read as a velocity, it carries everything along. Drop ink into it and watch the drops travel, turn, shear and stretch.

01

Vector fields in this visualization

A vector field in the plane assigns to each point (x,y)(x, y) of its domain a vector F(x,y)=(P(x,y), Q(x,y))\mathbf F(x, y) = (P(x, y),\ Q(x, y)). Read as a velocity field, it moves every point along: a particle obeys the autonomous system below, and its path is an integral curve (a streamline, or flow line) of the field. This page draws the field, carries ink and particles along it, and evaluates F wherever you put the probe.

A particle in the field
x′(t)=P(x,y),y′(t)=Q(x,y)x'(t) = P(x, y), \qquad y'(t) = Q(x, y)

An arrow is only the velocity at that instant, so a particle is not carried along a straight arrow but along a curve tangent to the arrows everywhere: in the rotation field (−y,x)(-y, x) the arrow at (1, 0) points straight up, yet the drop's centre goes round the unit circle. The lattice arrows share one scale, written under the picture (a few longer ones are capped and carry a slash); the probe's arrow is F(P) at true length, with its components P and Q dashed.

A small drop of ink shows the field's structure near a point. To first order it is carried by F at its centre and deformed by the field's derivative matrix: its area grows at the relative rate div⁡F\operatorname{div}\mathbf F (for a drop of finite size, the average of the divergence over it), its infinitesimal line elements turn on average at 12curl⁡F\tfrac12\operatorname{curl}\mathbf F, and what is left is shear and stretch, which neither number describes. The two diameters show it: in the shear field one leans 45° by t = 1 while the other stays level.

02

Things to notice

  • The background particles are reborn at random places, so their crowding means nothing; to see expansion, watch a drop's area (divergence, E25). Their short tails fade and are not streamlines — those are a separate key.
  • Each drop starts at t = 0 when placed; the readout follows the most recently placed drop. Forward increases t, backward decreases it, and both can cross 0. You can drop ink while flowing in either direction. Negative time follows the same field backward from the initial circle; it does not require a recorded past. The five linear presets use the matrix exponential; the pendulum and typed fields use numerical integration with error control. Reversibility requires unique solutions that exist throughout, and points that have stopped cannot be recovered.
  • Drops stop where they leave the computed area, where the field blows up or has no value; the outline breaks there and the drop loses its fill. The fields here are two-dimensional and do not change with time.
03

Controls

  • Click the picture to drop ink, or drag for a string of drops (three at most). With Pan on, dragging moves the view. Drag P, or type its coordinates (−100 to 100); it catches on the arrows’ lattice. Type P and Q with + − × ÷ ^, sin, exp, ln, sqrt …
  • Presentation mode opens a full-screen view for projectors: Space runs or stops the flow, → and ← set its direction, H hides F’s value at the probe. Share copies a link that reopens the field, the probe and the ink at their time.
04

Related

Further reading: Wikipedia: Vector field; OpenStax, Calculus Volume 3, 6.1 Vector Fields.