3D Vector Field Explorer

Build intuition for 3D vector calculus. Plot any field F(x, y, z), move a probe through it, and read off the divergence and curl numerically. Includes presets for rotational, radial, and dipole patterns.

Probe at (1.000, 1.000, 0.000)

F_x

-1.000

F_y

1.000

F_z

0.000

|F|

1.414

∇ · F (divergence)

0.300

∇ × F (curl)

(0.000, 0.000, 2.000), |curl| = 2.000

Drag to orbit. Scroll to zoom. X axis is red, Y is green, Z is blue. Arrows show the vector field F(x, y, z) at grid points. Edit the component expressions to explore your own fields.

Controls

Field components F(x, y, z)

F_x =

F_y =

F_z =

Display

Grid density

Arrow scale

View extent

m

Color mode

Normalize arrow lengths

Probe

x

y

z

Presets

Reference Guide

Vector Field

A vector field assigns a vector to every point of space.

\vec{F}(x,y,z) = F_x\,\hat{i} + F_y\,\hat{j} + F_z\,\hat{k}

Examples include velocity in a fluid, gravitational acceleration, and the electric field. Use mathjs syntax for expressions (sqrt, sin, exp, ^ for powers).

Divergence

The divergence measures how much the field locally spreads outward (or sinks in).

\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}

Positive divergence indicates a source, negative a sink. Pure rotational fields have zero divergence everywhere.

Curl

The curl measures the local rotation of the field.

\nabla \times \vec{F} = \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\ \frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x},\ \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)

Conservative fields (gradients) have zero curl. The classic F = (-y, x, 0) has constant curl (0, 0, 2).

Helmholtz Decomposition

Any smooth vector field can be written as the sum of a gradient and a curl:

\vec{F} = -\nabla\phi + \nabla \times \vec{A}

This separates the irrotational part from the solenoidal part. It underlies the existence of both V (potential) and A (vector potential) in electromagnetism.

TopicsCalculus Vectors Derivatives Functions Graphs

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