Vector Field Plotter

Math Calculus • Multivariable Calculus

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

6. Vector Field Plotter

Plot \( \mathbf{F}=\langle P,Q,R\rangle \), compute divergence and curl, and preview source/sink/rotation behavior.

Mode: Slice plane (3D): P component: Q component:

Input supports pi, e, sqrt(2), trig (sin, cos), etc. Drag the graph to pan, mouse wheel to zoom. Hover to probe \((x,y,z)\), \(\mathbf{F}\), \(\nabla\cdot \mathbf{F}\), and \(\nabla\times \mathbf{F}\) (at the hover point).

Evaluation point (and slice-through point in 3D)

x₀: y₀:

In 3D, the plotted plane passes through \((x_0,y_0,z_0)\):
• xy: \(z=z_0\) • xz: \(y=y_0\) • yz: \(x=x_0\).

Plot window (for the displayed axes)

x min: x max: y min: y max:

Field & graph settings (optional)

Arrow density: Arrow scale: Normalize arrows: Show grid: Magnitude heatmap: Zero tolerance:

Ready

Enter a vector field and click “Plot & Compute”.

Graph

Quiver plot with axes + tick values

Drag to pan, mouse wheel to zoom, double-click to reset. Hover to see values at the cursor. The evaluation point \((x_0,y_0,z_0)\) is marked.

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Theory

Vector fields

A vector field assigns a vector to every point in space. In 2D: \(\mathbf{F}(x,y)=\langle P(x,y),\,Q(x,y)\rangle\). In 3D: \(\mathbf{F}(x,y,z)=\langle P(x,y,z),\,Q(x,y,z),\,R(x,y,z)\rangle\).

This calculator plots a quiver (arrow) representation. In 3D, it plots a 2D slice through a chosen plane.

Divergence: sources and sinks

The divergence measures net “outflow” from a point:

\[ \nabla\cdot\mathbf{F}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}. \]

\(\nabla\cdot\mathbf{F} > 0\): local source behavior (flow spreads out).
\(\nabla\cdot\mathbf{F} < 0\): local sink behavior (flow converges).
\(\nabla\cdot\mathbf{F} \approx 0\): incompressible-like behavior (no net expansion).

Curl: rotation / circulation

The curl measures local rotation:

\[ \nabla\times\mathbf{F}= \left\langle \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\ \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right\rangle. \]

In 2D, you often focus on the out-of-plane component:

\[ (\nabla\times\mathbf{F})_z=\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}, \]

which indicates counterclockwise vs. clockwise rotation (sign) and rotation strength (magnitude).

Conservative (irrotational) check

A vector field is conservative if it is the gradient of a potential function, \(\mathbf{F}=\nabla \phi\). In many common settings (simply connected domains), a practical test is:

\[ \nabla\times\mathbf{F}=\mathbf{0}. \]

This calculator reports whether the curl is approximately zero at the chosen evaluation point (within a tolerance). A full conservative test can also require domain conditions and global checks.

Worked example: \(\mathbf{F}=\langle -y, x, 0\rangle\)

Here \(P=-y\), \(Q=x\), \(R=0\). Compute divergence:

\[ \nabla\cdot\mathbf{F}=\frac{\partial(-y)}{\partial x}+\frac{\partial x}{\partial y}+\frac{\partial 0}{\partial z}=0+0+0=0. \]

Compute curl:

\[ \nabla\times\mathbf{F}= \left\langle 0-0,\ 0-0,\ \frac{\partial x}{\partial x}-\frac{\partial(-y)}{\partial y} \right\rangle = \langle 0,0,1-(-1)\rangle = \langle 0,0,2\rangle. \]

So the field has no net source/sink (\(\mathrm{div}=0\)) but has constant rotation (\(\mathrm{curl}\neq 0\)).

Frequently Asked Questions

What does divergence tell you about a vector field?

Divergence measures local net outflow from a point. For F = <P,Q,R>, div F = dP/dx + dQ/dy + dR/dz, where positive values indicate source-like behavior and negative values indicate sink-like behavior.

What does curl measure in a vector field?

Curl measures local rotation or circulation. For F = <P,Q,R>, curl F = <dR/dy - dQ/dz, dP/dz - dR/dx, dQ/dx - dP/dy>, and in 2D you often focus on the z-component dQ/dx - dP/dy.

Why should the direction arrows sometimes be normalized?

Normalization makes arrows the same length so the plot emphasizes direction rather than magnitude. Without normalization, arrow lengths scale with |F|, which helps visualize where the field is stronger or weaker.

How does the 3D slice plot work in this vector field plotter?

A 3D field is displayed on a chosen coordinate plane that passes through the evaluation point. For example, an xy slice uses z = z0, an xz slice uses y = y0, and a yz slice uses x = x0.