How this calculation works
Evaluate the field on a regular grid and use one shared arrow-length multiplier, retaining relative directions and sampled magnitudes. Calculate divergence and scalar curl analytically at a separate evaluation point. Field arrows describe vectors at positions, not tracked trajectories.
div F=Pₓ+Qᵧ; planar scalar curl=Qₓ−Pᵧ; magnitude=√(P²+Q²)A worked example
For F=(−y,x) at (1,0), the vector is (0,1) with magnitude 1. Divergence is zero and scalar curl is 2, consistent with a rotational field rather than an outward source.
Good to know
Two polynomial components degree ≤6 on a 3–15 by 3–15 grid. The finite grid can miss features. Equal coordinate scales and globally scaled arrows do not provide physical units, streamlines or a flow simulation. Local zero divergence does not by itself establish zero boundary flux for an arbitrary region.
Reference: OpenStax Calculus: divergence and curl (opens in a new tab) ↗
Arrows are sampled vectors rather than paths
At a grid point, the arrow shows the two components assigned by the field. The plot uses a common positive multiplier for all arrow lengths and equal coordinate scales, so relative sampled magnitudes and directions remain comparable.
The head of an arrow is not the future position of a particle. A trajectory requires solving a differential equation through the field. A grid can also miss local behavior between its samples, particularly when a polynomial changes quickly.
The polynomial ODE calculator follows a scalar initial-value slope rule with equal numerical steps. Sampled field arrows alone are not trajectories.
Divergence and curl ask different local questions
In two dimensions, divergence adds ∂P/∂x and ∂Q/∂y. Scalar curl subtracts ∂P/∂y from ∂Q/∂x. These are local differential summaries, not the magnitude of the field vector.
For F=(−y,x), the vector at (1,0) is (0,1), with magnitude 1. Divergence is zero everywhere, while scalar curl is 2 everywhere. The sampled rotational pattern illustrates why a zero divergence need not mean a zero vector field.
Keep a point calculation separate from an integral
The evaluation point need not be a particular grid sample. Its derivatives are computed analytically from the same two polynomial components. Moving or refining the plot grid therefore does not change the point’s true polynomial derivative values.
The Green’s theorem calculator integrates scalar curl over its supported region to obtain an oriented boundary integral. This page computes the local field and derivative summaries instead. A polynomial Jacobian calculator places the same partial derivatives into a full two-output derivative matrix.
A few questions, answered
Are the arrows normalized to unit length?
No. They share one multiplier based on the sampled maximum, retaining relative magnitudes. A zero vector is shown as a dot.
Do more grid points make the derivative calculation more accurate?
The derivatives are analytic polynomial derivatives, independent of grid density. A finer grid changes only the sampled visual coverage.
Does zero curl at one point prove the field is conservative?
No. A single-point observation cannot establish that property across a domain. Domain conditions and the field over that domain matter.