Seeing Vector Fields
A vector field assigns an arrow, a magnitude and direction, to every point of the plane: F(x, y) = ⟨P(x, y), Q(x, y)⟩. Wind maps, ocean-current charts, magnetic-field diagrams, and the phase portraits of differential equations are all vector fields, and the standard way to see one is the arrow grid this tool draws: sample the field at regular points, draw each sample as an arrow scaled to its strength, and let the eye assemble the flow.
Reading the picture is a skill with a short vocabulary: arrows radiating from a point mark a source; converging, a sink; circling, a vortex; sweeping past a point that pushes flow out along one axis and in along another, a saddle. Real fields are patchworks of these local behaviors, and the two derivative diagnostics, divergence (spreading rate) and curl (spinning rate), quantify what the arrows show.
How to Use This Plotter
Enter the two component formulas. The tool draws a 13×13 grid of arrows, direction true, length scaled between a floor and cap so both weak and strong regions stay legible, and computes the divergence and curl symbolically alongside, connecting picture to calculus at a glance. The chips tour the classic portraits: a nonlinear oscillator's closed swirls, a saddle's hyperbolic sweep, a periodic lattice of counter-rotating cells, and a spiral source combining rotation with expansion.
Worked Reading: the Saddle Field
F = ⟨x, −y⟩ pushes flow away from the y-axis horizontally (P = x grows with x) and toward the x-axis vertically (Q = −y). The arrow plot shows the signature X-pattern: trajectories approach along the vertical axis, veer, and escape along the horizontal, the saddle of dynamical systems, with div F = 1 + (−1) = 0 and curl F = 0. Everything expands in one direction exactly as fast as it contracts in the other: an incompressible, irrotational field that nonetheless reshapes every region it touches (pure shear-like deformation, the third ingredient beyond spreading and spinning).
Vector Fields as Differential Equations
Interpret the field as a velocity rule, "at point (x, y), move with velocity ⟨P, Q⟩", and the arrow plot becomes a phase portrait: solutions of the system x′ = P, y′ = Q are curves threading the arrows tangentially. The oscillator chip is a pendulum-like system whose closed orbits are visible as rings; equilibria sit wherever both components vanish (arrows shrink to dots), and their local type, source, sink, saddle, center, determines the system's stability. This picture-first approach to differential equations often reveals more than formulas: long-run destinations, basins of attraction, and cycles are all visible before a single equation is solved.
Common Mistakes to Avoid
- Reading arrow length as exact magnitude. Plotters (this one included) compress lengths so strong regions don't bury weak ones, direction is exact, length is indicative. For precise magnitudes, evaluate √(P² + Q²) at the point.
- Confusing field arrows with trajectories. Arrows are instantaneous velocities at fixed points; trajectories are curves that flow through the arrows. Related, not identical, an arrow plot is the derivative of the motion picture, so to speak.
- Missing equilibria between grid points. A zero of the field can hide between samples; solve P = 0 and Q = 0 simultaneously to locate equilibria exactly rather than trusting the grid.
- Judging div and curl purely by eye. The shear-flow lesson from the curl page cuts both ways: intuition misreads subtle fields. The plotter computes both symbolically precisely so the eye can be calibrated against the truth.
Real-World Applications
Every weather forecast graphic with wind barbs is a vector field plot; oceanographers chart currents, aerodynamicists visualize airflow around prototypes, and geophysicists map Earth's magnetic field the same way. In engineering software, CFD post-processors and electromagnetic simulators present their solutions as exactly these arrow grids (plus streamlines), because human insight into a flow still starts with looking at it.
The phase-portrait interpretation powers qualitative science: ecologists plot predator-prey dynamics as fields over population space and read cycles straight from the swirls; epidemiologists visualize S-I-R dynamics; control engineers inspect stability of feedback systems by the portrait around equilibria; and neuroscience models neuron firing with two-variable fields whose saddles and cycles are the spike mechanism. Wherever a system's state is a point and its law is "velocity depends on state," this plot is the theory's face.
Frequently Asked Questions
What are streamlines, and how do they relate to the arrows?
Streamlines are curves everywhere tangent to the field, the paths dust motes would trace if carried by the flow. They're the integral curves of the arrow plot, computable by ODE solvers like Euler's method using the field as the right-hand side.
How do I find where the field is zero?
Solve the system P(x, y) = 0, Q(x, y) = 0. Each solution is an equilibrium, visually, a grid region where arrows dwindle. Their classification (via the Jacobian's eigenvalues at the point) is the local dictionary: sink, source, saddle, spiral, or center.
Why do some fields form closed rings of arrows?
Closed orbits, the system returns to its starting state and repeats, as in idealized pendulums and predator-prey cycles. Centers (rings around an equilibrium) are delicate: they survive in conservative systems but usually break into spirals when damping or forcing is added.
Can this plotter show 3D fields?
This tool is 2D, the plane is where arrow plots stay readable. 3D fields are usually visualized by streamlines, slices, or isosurfaces instead; the mathematics (div, curl, line integrals) extends componentwise as covered in the sibling tools' FAQs.
What's the relationship between this and a gradient field plot?
A gradient field ∇f is a special vector field: arrows point uphill on f's landscape, perpendicular to its contours, with zero curl always. Plot one (try P = 2x, Q = 2y, the gradient of x² + y²) and the radiating, non-swirling signature is unmistakable, the visual meaning of "conservative."
How is the arrow grid's density chosen?
A 13×13 lattice, dense enough for flow structure to emerge, sparse enough that arrows don't overlap into noise. Professional tools let density and region vary; the constant here keeps every field comparable at a glance, and the symbolic div/curl readouts compensate for anything the resolution hides.