Vector Field Plotter Calculator

Vector Field Results

Results update from the current expressions and controls.

Field at test point

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Magnitude

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Direction

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Classification

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Divergence

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2D curl

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Critical points

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Line integral

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Parsed field: —

Building the field plot…

Calculator Controls

Enter a two-dimensional field and customize its analysis.

Vector field definition

Preset field

Horizontal component P(x,y)

Example: -y + 0.2*x

Vertical component Q(x,y)

Example: x + 0.2*y

Coordinate range and sampling

Minimum x

Maximum x

Minimum y

Maximum y

Grid columns

Grid rows

Vector appearance

Scaling mode

User scale factor

Color mode

Single vector color

Arrow thickness

2 pixels

Arrowhead size

0.22 relative units

Arrow opacity

0.90

Minimum displayed magnitude

Maximum displayed magnitude

Analysis point and display

Test point x

Test point y

Tick spacing

Scalar overlay

Derivative step

Critical-point tolerance

Show grid

Show axes

Show labels

Equal axes

Origin marker

Streamlines, nullclines, and particles

Streamline seeds

Integration step size

Maximum streamline steps

Streamline direction

Particle count

Particle speed

Path and line integral

Path x(t)

Path y(t)

Minimum t

Maximum t

Integration steps

Show parametric path

Formula Used

A vector field assigns a vector to every coordinate. The calculator evaluates horizontal and vertical field components. It then combines them into one directed result.

F(x,y) = P(x,y)i + Q(x,y)j
|F| = sqrt(P² + Q²)
θ = atan2(Q,P)

div F = ∂P/∂x + ∂Q/∂y
curl F = ∂Q/∂x - ∂P/∂y
Work = ∫ F(r(t)) · r'(t) dt

How to Use This Calculator

Choose a preset or enter both component expressions.
Set coordinate limits, density, scaling, and colors.
Select streamlines, overlays, nullclines, or particle motion.
Enter a test point and optional parametric path.
Press Plot and analyze to update results.
Export the graph, data, report, or shared URL.

Supported Expression Syntax

Use variables x and y inside field expressions. Path expressions may also use the variable t. Multiplication signs should be written explicitly for reliability.

Supported examples include x^2, sqrt(x^2+y^2), sin(x), exp(-x), log(x), abs(y), pi, and e.

Example Vector Fields

Field type	P(x,y)	Q(x,y)	Behavior
Constant	1	0.5	Uniform direction and magnitude
Radial source	x	y	Vectors point away from origin
Radial sink	-x	-y	Vectors point toward origin
Rotation	-y	x	Circular counterclockwise flow
Saddle	x	-y	Expansion and contraction axes
Spiral	-y + 0.2*x	x + 0.2*y	Rotating outward motion

Frequently Asked Questions

What is a vector field?

A vector field assigns direction and magnitude everywhere. Each plotted arrow represents one local vector. Nearby arrows reveal the field’s overall flow pattern.

Why are some arrows omitted?

Undefined or nonfinite evaluations cannot be displayed safely. Magnitude filters can also hide selected vectors. Review warnings when unexpected gaps appear on plots.

What does normalized scaling mean?

Normalized arrows use nearly equal displayed lengths. Their directions remain faithful to the original field. Color can still represent the actual magnitudes accurately.

How is divergence calculated?

Divergence uses centered numerical partial derivatives here. Positive values suggest local source-like behavior. Negative values suggest local sink-like behavior instead.

How is curl calculated?

Two-dimensional curl measures local rotational tendency numerically. Positive signs indicate counterclockwise local rotation. Negative signs indicate clockwise local rotation instead.

Are critical points exact?

Critical points are located using numerical refinement. Their accuracy depends on range and tolerance. Increase density when closely spaced equilibria are expected.

What are nullclines?

Nullclines approximate locations where one component vanishes. Their intersections may indicate equilibrium points. Higher sampling density produces cleaner nullcline estimates overall.

When is a field conservative?

A conservative field usually has negligible curl. Domain holes can still prevent global conservativeness. Always consider singularities and the selected plotting region.

What does the line integral represent?

The line integral estimates work along a path. Direction changes can alter the resulting value. Reverse path limits to test orientation effects directly.

Mathematical Disclaimer

Numerical derivatives and integrations are approximations. Singular fields may produce unstable values near undefined points. Verify sensitive results with independent mathematical software before decisions.