Vector Field Divergence Calculator
The divergence of a vector field measures how strongly the field spreads out from a point, the net outward flux per unit volume. A positive value marks a source, a negative value a sink, and zero a balanced flow. It is the dot product of the del operator with the field and is computed from just the three diagonal partial derivatives. This calculator takes those derivatives, evaluated at your point of interest, and returns the scalar divergence exactly. Divergence is central to the divergence theorem, fluid dynamics, and electromagnetism.
Divergence formula
div F = del . F
div F = dFx/dx + dFy/dy + dFz/dz
div F > 0 means a source
div F < 0 means a sink
div F = 0 means solenoidal
The divergence is the sum of the three diagonal partial derivatives, the dot product of del with the field. Its sign classifies the point as a source, a sink, or balanced.
Notes on divergence
- The three inputs are partial derivatives already evaluated at your chosen point.
- Positive divergence indicates net outward flow, a source.
- Negative divergence indicates net inward flow, a sink.
- Zero divergence everywhere marks a solenoidal, incompressible field.
- The divergence theorem relates divergence to total flux through a closed surface.
Divergence: frequently asked questions
What is the divergence of a vector field?
The divergence measures the net rate at which a vector field flows outward from a point. It is a scalar: positive divergence means the point acts as a source, negative means a sink, and zero means flow in equals flow out.
What is the formula for divergence?
For a field F = (Fx, Fy, Fz), the divergence is dFx/dx + dFy/dy + dFz/dz, the sum of the three diagonal partial derivatives. It is the dot product of the del operator with the field.
What does zero divergence mean?
A field with zero divergence everywhere is called solenoidal or incompressible. Such fields conserve flux: whatever flows into any closed region also flows out. Magnetic fields and incompressible fluid flows are examples.
How does divergence relate to flux?
The divergence theorem links the two: the total flux of a field out of a closed surface equals the integral of the divergence over the enclosed volume. Divergence is the local flux density per unit volume.
Why do I enter partial derivatives?
Divergence is defined by the partial derivatives of the field components evaluated at your point. Supplying those three derivatives keeps the calculation exact and deterministic without parsing arbitrary symbolic expressions.
Official sources
- NIST Digital Library of Mathematical Functions: Vectors and vector-valued functions.
- NASA Glenn Research Center: Vector basics.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.