Calculator Inputs
Formula used
A two dimensional vector field is written as F(x,y) = (Fₓ(x,y), Fᵧ(x,y)). The plot shows arrows sampled over a grid in the chosen domain.
For the custom polynomial model:
- Fₓ=a₀+a₁x+a₂y+a₃x²+a₄y²+a₅xy
- Fᵧ=b₀+b₁x+b₂y+b₃x²+b₄y²+b₅xy
Local measures are computed at (x₀,y₀):
- div F = ∂Fₓ/∂x + ∂Fᵧ/∂y
- curl_z F = ∂Fᵧ/∂x − ∂Fₓ/∂y
How to use this calculator
- Pick a field model, or keep the polynomial option.
- Set domain bounds for x and y to frame motion.
- Choose grid sizes; higher values add more arrows.
- Enable normalization to show directions without magnitude bias.
- Set a point (x₀,y₀) to inspect divergence and curl.
- Press Plot Vector Field to render results.
- Use CSV or PDF buttons to export your results.
Example data table
Example for the custom polynomial defaults on domain [-2,2] with a 5×5 grid. Values are illustrative of typical outputs.
| x | y | Fₓ | Fᵧ | |F| |
|---|---|---|---|---|
| -2 | -2 | -2.0 | 2.0 | 2.8284 |
| -2 | 0 | -2.0 | 0.0 | 2.0000 |
| -2 | 2 | -2.0 | -2.0 | 2.8284 |
| 0 | -2 | 0.0 | 2.0 | 2.0000 |
| 0 | 0 | 0.0 | 0.0 | 0.0000 |
| 0 | 2 | 0.0 | -2.0 | 2.0000 |
| 2 | -2 | 2.0 | 2.0 | 2.8284 |
| 2 | 0 | 2.0 | 0.0 | 2.0000 |
| 2 | 2 | 2.0 | -2.0 | 2.8284 |
Vector field plotter article
1) Purpose of a vector field plot
A vector field plot turns an abstract function F(x,y) into a readable map of directions and strength. In physics it helps inspect velocity fields in fluids, force patterns in mechanics, and phase portraits of dynamical systems. This tool samples the field on a grid and draws arrows, giving a quick sense of where motion points and where it intensifies or weakens.
2) Sampling density and workload
Sampling density is controlled by nx and ny. The total number of arrows equals nx multiplied by ny, so a 17 by 17 grid draws 289 vectors, while the maximum 61 by 61 draws 3721. Higher density reveals structure near critical regions, but it also increases visual clutter. Use a coarser grid for exploratory work and refine after choosing good bounds.
3) Scaling, normalization, and readability
Arrow length can be normalized or left proportional. Normalization fixes arrow lengths and emphasizes direction, which is useful when magnitudes vary widely. With normalization off, strong regions dominate visually. Autoscale adjusts arrow lengths using the largest sampled magnitude so arrows fit the canvas, while manual scaling lets you exaggerate or compress the field.
4) Divergence and curl at a point
Divergence and curl are local diagnostics computed at the chosen point (x0,y0). Divergence measures net outflow; positive values indicate a local source and negative values indicate a sink. The z component of curl measures local rotation in the plane. For the polynomial model, derivatives are analytic; for other models, central differences estimate partial derivatives using the step size h.
5) Field models with physical intuition
The built in models represent common physical patterns. Uniform fields model constant drift. Source and sink fields emulate radial flow, with a softening epsilon preventing singular behavior at the origin. Vortex fields produce circulation around the origin. Saddle fields show stretching along one axis and compression along the other. The oscillator model interprets y as velocity and x as position to visualize damped harmonic motion.
6) Choosing effective domain bounds
Domain bounds determine what you learn. A tight window around the origin reveals separatrices, centers, and sharp gradients, while a wider window shows global trends and asymptotic direction. When exploring, start with symmetric bounds like -5 to 5 and then zoom into features of interest. Changing bounds also changes the sampled maximum magnitude used for autoscale.
7) Numerical stability and parameter tuning
Numerical stability matters when fields change rapidly. Very small epsilon can create huge values near r equals 0, and very small h can amplify floating point noise in numerical derivatives. If divergence and curl look erratic for non polynomial models, increase h modestly, for example from 0.001 to 0.01, and avoid evaluating exactly at the singular point.
8) Exporting data for analysis
Export features support reporting and validation. The CSV file contains x, y, Fx, Fy, and magnitude for every sampled grid point, making it easy to reproduce plots in other software or compute statistics. The PDF button captures the result panel including the canvas in a print friendly page, useful for lab notes, homework, or design reviews.
FAQs
1) What does normalization do?
Normalization draws arrows with equal length and keeps only direction. Use it when magnitudes vary strongly across the domain and you still want to see directional structure everywhere.
2) Why do the arrows look too crowded?
Reduce nx and ny, or enlarge the domain. A dense grid can hide patterns because many arrows overlap, especially when autoscale produces longer vectors.
3) What does a positive divergence mean?
Positive divergence indicates local outflow, like a source. Negative divergence indicates local inflow, like a sink. The value is computed at the selected evaluation point.
4) What does the curl value represent here?
The reported curl is the z component for a planar field. It measures local tendency to rotate about the point. Large magnitude suggests strong local circulation.
5) How do I avoid blow ups near the origin?
Increase epsilon for the source or vortex models and avoid setting the evaluation point exactly at x0 equals 0 and y0 equals 0. This softens the singularity.
6) Can I reproduce the exact same grid later?
Yes. Keep the same model, bounds, nx, ny, and parameters. Export CSV to preserve the sampled values and use it as a fixed reference dataset.
7) What is the custom polynomial model used for?
It provides a controllable local approximation to many fields using constant, linear, and quadratic terms. It is useful for teaching, regression style fitting, and testing divergence and curl behavior.