Divergence Calculator

The Divergence Calculator finds the divergence of a 3D vector field F(x, y, z) = (P, Q, R) at a point — a scalar that measures how much the field is expanding out of (or collapsing into) that point. It shows the simplified symbolic divergence (still a function of x, y, z) alongside the numeric value at your chosen point, plus a plain-language read of whether that point is a source, a sink, or neutral, and a full step-by-step derivation of every partial derivative involved.

It's built for multivariable calculus and vector calculus students checking a homework answer, and for anyone who needs div F without differentiating three partial derivatives by hand.

How to use it

1. Enter the three components of your vector field: P(x, y, z), Q(x, y, z), and R(x, y, z) — standard math notation works: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser.
2. Enter the point (x, y, z) to evaluate the divergence at.
3. Read div F at (x, y, z) for the numeric divergence, Symbolic div F for the simplified expression, and Interpretation for whether that point behaves as a source, a sink, or neither.
4. Step-by-Step Derivation shows all three partial derivatives, the simplified symbolic divergence, and the final numeric evaluation.

Results update automatically as you type. For example, the field F = (x^2*y, y*z, x*z^2) at (1, 2, 3) returns a divergence of 13 — a net source at that point.

What is divergence?

For a vector field F(x, y, z) = (P, Q, R), the divergence is defined as:

div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Physically, divergence measures the net outward flux per unit volume at a point — imagine the field as a fluid's velocity: a positive divergence means fluid is expanding outward from that point (a source), a negative divergence means fluid is converging inward (a sink), and a divergence of zero everywhere means the field is incompressible (also called solenoidal) — fluid neither accumulates nor depletes anywhere.

Why does the calculator show the symbolic divergence too?

Divergence is itself a scalar field — a function of (x, y, z) — not just a number at one point. Seeing the simplified symbolic expression makes it possible to tell whether div F is zero everywhere (incompressible), constant, or varies from point to point, which the single evaluated number alone can't show. A field can have zero divergence at the one point you happened to check while being far from incompressible elsewhere — the symbolic form clears that up immediately.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2*y - 2*z), trigonometric functions (sin(x*z), cos(y)), roots and logarithms (sqrt(x), log(y)), and combinations of these, using x, y, and z as the three variables. Angles in trig functions are in radians by default — write pi for π.

To find the curl (rotation) of the same kind of vector field instead, see the Curl Calculator; to evaluate arbitrary math expressions directly, see the Math Evaluator.

Privacy

This calculator runs entirely in your browser. Your vector field and point are never uploaded, logged, or stored — the computation happens locally on your device.