What expressions can I enter for P, Q, R?

Use standard JavaScript math syntax: x, y, z as variables; *, /, +, - operators; Math.sin, Math.cos, Math.exp, Math.log, Math.pow(a,b), Math.sqrt, and numeric constants. Example: x*y + Math.sin(z).

How is the partial derivative computed?

The tool uses a central-difference approximation: ∂f/∂x ≈ (f(x+h,y,z) − f(x−h,y,z)) / (2h) with h = 1e-6. This is accurate to O(h²) for smooth functions.

What does a result of 0 mean?

A divergence of zero at every point means the field is solenoidal (divergence-free), like an incompressible fluid or a magnetic field satisfying ∇·B = 0.

Can I use 2D fields (no z dependence)?

Yes. Set R = 0 and enter expressions for P and Q that may or may not involve z. The calculator still evaluates correctly at the given z coordinate.

Why do I get NaN or Infinity?

Your expression likely involves a singularity at the chosen point (e.g., division by zero). Try a different evaluation point away from undefined regions.

Divergence Calculator

Name: Divergence Calculator
Availability: InStock
Author: Nham Vu

Enter the three components of a vector field and compute its divergence ∂P/∂x + ∂Q/∂y + ∂R/∂z at any point.

Vector Field F = (P, Q, R)

Use x, y, z as variables. JavaScript Math.* functions are supported.

P(x, y, z) — x-component

Q(x, y, z) — y-component

R(x, y, z) — z-component

Evaluation Point

x₀

y₀

z₀

Result

div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z

—

Partial Derivatives

∂P/∂x —

∂Q/∂y —

∂R/∂z —

Quick Examples

Summary

Enter the three components of a vector field and compute its divergence ∂P/∂x + ∂Q/∂y + ∂R/∂z at any point.

How it works

Enter the three component expressions P(x,y,z), Q(x,y,z), and R(x,y,z) of your vector field.
Specify the evaluation point (x₀, y₀, z₀).
The tool computes ∂P/∂x, ∂Q/∂y, and ∂R/∂z at the given point using a central-difference method.
The divergence is displayed as the sum of the three partial derivatives.
A negative result indicates convergence (sink); positive indicates divergence (source); zero is a divergence-free field.

Use cases

Verify textbook divergence problems for vector calculus courses.
Check whether a vector field is incompressible (div F = 0) in fluid mechanics.
Evaluate divergence of electric or magnetic fields in physics problems.
Quickly test numerical partial derivatives before implementing them in code.
Explore how divergence varies by plugging in different evaluation points.

Frequently Asked Questions

Last updated: 2026-07-23 · Reviewed by Nham Vu