In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point. Technically, it represents the volume density of the outward flux of a vector field from an infinitesimal volume around the point.

Notation and Definition

Symbol: The divergence operator is denoted by "div" or by the nabla symbol followed by a dot (∇ ·).

Definition: If F is a three-dimensional vector field with components F = (P, Q, R) then the divergence of F is a scalar field calculated as:

div F = ∇ · F = ∂P/∂x + ∂Q/∂y + ∂R/∂z


  • Sources and Sinks: A point where the divergence is positive is called a "source" of the vector field. Imagine fluid flow: a source would be a point where fluid is being introduced. A point where the divergence is negative is a "sink", where the fluid is being removed.
  • Flux Density: The magnitude of the divergence at a point represents the density of the outward flux of the vector field at that point. A higher magnitude of divergence indicates more flux emanating (or converging) to that specific point.
  • Zero Divergence: A vector field with zero divergence everywhere is called an "incompressible" or "solenoidal" field. This suggests that the amount of field entering any small region exactly equals the amount exiting.

Physical Examples

  • Fluid Flow: Divergence measures whether fluid expands (positive divergence) or compresses (negative divergence) at a point.
  • Electromagnetism: In Gauss's law (Maxwell's Equations), the divergence of the electric field is proportional to the electric charge density at that point. Positive divergence indicates regions of positive charge, and negative divergence implies areas of negative charge.
  • Heat Flow: The divergence of the heat flux field provides information about sources or sinks of heat within a medium.


  • Linearity: The divergence operator is linear (it distributes across sums of vector fields and scalar multiples).
  • Product Rule: There is a product rule for divergence: ∇ · (φF) = φ (∇ · F) + F · ∇φ (where φ is a scalar field and F is a vector field).
  • Divergence Theorem: This crucial theorem (also known as Gauss's Theorem) links the divergence of a vector field in a volume to the field's flux across the boundary of the volume.


Divergence plays a significant role in many areas of physics and engineering:

  • Fluid Dynamics: Analyzing flow patterns, vorticity, and the effects of turbulence.
  • Electromagnetism: Describing electric and magnetic fields, and understanding charge distributions.
  • Continuum Mechanics: Modeling stresses and strains in deformable materials.
  • Image Processing: Used in edge detection and noise reduction algorithms.