Divergence Calculator

Divergence measures how much a vector field spreads out from a point.
Positive divergence means the field behaves as a source, with fluid flowing outward. Negative divergence means it behaves as a sink, with fluid flowing inward. Zero divergence is incompressible flow: whatever enters a region also leaves it.

For a 3D vector field F = (P, Q, R), the divergence is:

div F = dP/dx + dQ/dy + dR/dz

This calculator uses component functions that depend on a single variable each: P(x) = ax^m, Q(y) = by^n, R(z) = cz^r. The partial derivatives are:

dP/dx = max^(m-1), dQ/dy = nby^(n-1), dR/dz = rcz^(r-1)

Sum those three values at your evaluation point (x0, y0, z0) to get the scalar divergence.

Worked example: F = (2x^2, 3y, z^3) at point (1, 2, 1). Here a=2, m=2, b=3, n=1, c=1, r=3. dP/dx = 4x = 4(1) = 4 dQ/dy = 3 (constant for linear Q) dR/dz = 3z^2 = 3(1) = 3 div F = 4 + 3 + 3 = 10

A divergence of +10 at that point means the field is expanding outward there, acting as a source.

Every one of those figures is a placeholder in the boxes above, so pressing Calculate on the defaults reproduces this example exactly.

The divergence theorem says the total divergence inside a closed region equals the net flux through its surface. This is how Gauss’s law in electromagnetism relates the charge inside a surface to the electric field passing through it.

For vector fields with cross-terms (P depending on y and z as well as x), you need the partial derivatives of those cross-terms too. The curl calculator on this site handles linear fields with all nine coefficients.

One case worth knowing: an exponent of zero makes that component a constant, and the derivative of a constant is zero no matter where you evaluate it. A component like P(x) = 5 contributes nothing to the divergence at any point, including the origin.