Plot vector fields with clean interactive controls. Review magnitudes, divergence, curl, and example coordinate values. Export results, save tables, and study flow behavior clearly.
Use explicit multiplication. Write 2*x, x*y, or sin(x).
This example uses the rotational field F(x,y) = <-y, x>.
| x | y | P(x,y) | Q(x,y) | Magnitude |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0.0000 |
| 1 | 0 | 0 | 1 | 1.0000 |
| 0 | 1 | -1 | 0 | 1.0000 |
| 1 | 1 | -1 | 1 | 1.4142 |
| -2 | 1 | -1 | -2 | 2.2361 |
A planar vector field is written as F(x,y) = <P(x,y), Q(x,y)>.
Magnitude: |F(x,y)| = √(P(x,y)2 + Q(x,y)2)
Direction angle: θ = atan2(Q(x,y), P(x,y))
Divergence: div F = ∂P/∂x + ∂Q/∂y
Scalar curl in 2D: curl F = ∂Q/∂x − ∂P/∂y
This calculator estimates divergence and curl numerically near the chosen point. It also samples the field on a grid and draws arrows for local direction and relative strength.
Graphing a vector field turns an abstract formula into visible motion. Each arrow shows direction and strength at one location. This makes multivariable ideas easier to inspect. You can quickly spot circulation, spreading, compression, and symmetry. A clear graph often reveals behavior that raw equations hide.
This graphing vector fields calculator plots the field F(x,y) = <P(x,y), Q(x,y)> on a user defined grid. It samples many coordinate pairs. It then computes the vector components, magnitude, and direction angle. The tool also estimates divergence and curl at a chosen point. These values help explain whether the field expands, contracts, or rotates locally.
Students use vector field graphs in calculus, differential equations, and linear algebra. Engineers use them for flow models, force maps, and signal patterns. Physicists use them in electromagnetism and fluid motion. Data analysts sometimes study gradient style behavior with similar ideas. Seeing the arrows on a plane supports better intuition and faster checking.
A graph alone is helpful, but numbers add precision. The sample table lets you inspect exact values at selected points. Magnitude shows strength. Angle shows orientation. Divergence measures net outward behavior. Curl measures rotational tendency in two dimensions. Together, these outputs create a more complete field analysis workflow.
Look for arrows that point away from the center. That often suggests a source. Arrows pointing inward suggest a sink. Circular patterns often suggest rotation. Saddle patterns show mixed stretching and compression. A normalized plot is useful when direction matters more than size. A scaled plot is useful when relative magnitude matters most.
Use explicit operators in expressions. Enter x*y instead of xy. Test simple presets first. Then move to custom fields. Choose a balanced window and grid size. Very dense grids can crowd the view. Moderate steps usually produce cleaner interpretation and faster export.
A vector field assigns a vector to every point in a region. In two dimensions, each vector has horizontal and vertical components. The graph shows how direction and magnitude change across the plane.
P(x,y) is the horizontal component. Q(x,y) is the vertical component. Together they define the field F(x,y) = <P,Q> at each plotted point.
Normalization makes arrows closer in length. This highlights direction patterns clearly. It is useful when very large vectors hide the shape of smaller nearby vectors.
Divergence estimates local expansion or compression. Positive values often indicate outward flow. Negative values often indicate inward flow. Values near zero suggest little net spreading at that point.
In this two dimensional setting, curl measures local rotation tendency. A positive or negative value indicates rotational behavior around the chosen point, based on component changes.
Yes. You can use functions such as sin, cos, tan, sqrt, abs, log, and exp. Write formulas with explicit multiplication, like 2*sin(x) or x*cos(y).
Small arrows usually mean the vector magnitude is small at those points. If you want a cleaner directional display, enable normalization or adjust the arrow scale.
The CSV file contains sampled coordinates, vector components, magnitudes, and angles. The PDF file captures the result section, including the graph and summary data.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.