Divergence Calculator

Category: Calculus
- April 29, 2025
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Calculate the divergence of a vector field. The divergence measures the rate at which "density" exits a given region of space and is a key concept in vector calculus.

div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Vector Field Input

Display Options

About Divergence

Divergence is a scalar quantity that measures the tendency of a vector field to "flow" away from or towards a point. It is denoted by ∇·F (del dot F) or div F.

Mathematical Definition:

For a vector field F = (F₁, F₂, F₃) in 3D Cartesian coordinates, the divergence is defined as:

div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
Physical Interpretation:

Divergence measures the net flow of a field out of a small volume around a point. At a given point:

  • Positive divergence: The field is flowing outward (source)
  • Negative divergence: The field is flowing inward (sink)
  • Zero divergence: The field has no net outflow or inflow (incompressible)
Divergence in Different Coordinate Systems:

Cartesian Coordinates (x, y, z):

∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Cylindrical Coordinates (r, θ, z):

∇·F = (1/r) · ∂(r·Fr)/∂r + (1/r) · ∂Fθ/∂θ + ∂Fz/∂z

Spherical Coordinates (ρ, θ, φ):

∇·F = (1/ρ²) · ∂(ρ²·Fρ)/∂ρ + (1/(ρ·sin(φ))) · ∂(sin(φ)·Fθ)/∂θ + (1/(ρ·sin(φ))) · ∂Fφ/∂φ
Applications:
  • Electromagnetics: Gauss's law relates electric field divergence to charge density
  • Fluid Dynamics: The divergence of velocity field indicates compressibility of fluids
  • Heat Transfer: The divergence of heat flux describes heat sources or sinks
  • Quantum Mechanics: Current density divergence relates to charge conservation

What Is the Divergence Calculator?

The Divergence Calculator is an interactive tool that helps you calculate the divergence of a vector field in 2D or 3D space. Divergence is a concept from vector calculus that indicates how much a vector field spreads out or converges at a given point. This tool supports Cartesian, Cylindrical, and Spherical coordinate systems.

It's especially useful for students, engineers, and anyone working with physical fields like fluid flow, electromagnetism, or thermodynamics.

Divergence Formula in Cartesian Coordinates:
div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z

Key Features

  • Supports 2D and 3D vector fields
  • Choose between Cartesian, Cylindrical, and Spherical coordinates
  • Option to evaluate divergence at a specific point
  • Step-by-step breakdown of partial derivatives
  • Option to simplify results and control decimal precision

How to Use the Calculator

  1. Select the number of dimensions (2D or 3D).
  2. Choose the coordinate system that matches your problem.
  3. Enter the vector field components (e.g., F₁, F₂, F₃).
  4. Optionally, choose to evaluate at a specific point by toggling "Evaluate at point."
  5. Adjust display settings such as decimal places or angle units if needed.
  6. Click Calculate Divergence to view results and optional solution steps.

Why Is This Useful?

Understanding divergence helps interpret how a field behaves in space. This tool allows you to:

  • Compute divergence visually and accurately
  • Explore field behavior in physics, engineering, and mathematics
  • Practice multivariable differentiation and partial derivative steps
  • Verify work done by hand or during problem-solving sessions

It's particularly helpful when paired with tools like the Partial Derivative Calculator, Directional Derivative Calculator, or Curl Calculator, especially for learning multivariable calculus.

Related Tools

This calculator fits into a family of math solvers that include:

  • Partial Derivative Calculator – compute partial derivatives step-by-step
  • Directional Derivative Calculator – analyze vector field changes in a specific direction
  • Second Derivative Calculator – evaluate second-order changes for deeper analysis
  • Curl Calculator – calculate the curl of vector fields to examine rotational behavior

FAQ

What is divergence?
Divergence measures how much a vector field spreads out (positive divergence) or converges (negative divergence) at a point.

Can this tool evaluate partial derivatives?
Yes. This calculator performs multivariable differentiation behind the scenes, computing partial derivatives of each vector field component.

What is the difference between Cartesian, Cylindrical, and Spherical coordinates?
These are different ways to describe positions in space. Cartesian uses (x, y, z), Cylindrical uses (r, θ, z), and Spherical uses (ρ, θ, φ). Choose the one that best matches your field or problem.

Do I need to enter numbers or functions?
You can enter expressions like x*y*z, sin(θ), or r*cos(θ). The calculator handles symbolic math and can also evaluate values at specific points if requested.

Does it work for 2D fields?
Yes. Simply choose 2D and the interface will adjust accordingly. The third component is removed.

Can I learn from the steps?
Absolutely. If "Show solution steps" is enabled, the tool breaks down how partial derivatives are computed and how they combine into the final divergence expression.