calculus

Divergence (Vector Calculus)

The divergence of a vector field measures the net "outflow" at each point. ∇·F > 0 means a source; < 0 a sink. Foundational for fluid dynamics and electromagnetism.

Divergence is a scalar operation on a vector field F⃗=(F1,F2,F3)\vec{F} = (F_1, F_2, F_3) in R3\mathbb{R}^3:

∇⋅F⃗=∂F1∂x+∂F2∂y+∂F3∂z\nabla \cdot \vec{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}

Physical meaning: (∇⋅F⃗)(p)(\nabla \cdot \vec{F})(p) measures the net outflow rate of F⃗\vec{F} per unit volume at point pp.

  • >0> 0: net source (fluid spreading, positive charge density).
  • <0< 0: sink.
  • =0= 0: incompressible field (water flowing without compression).

The divergence theorem (Gauss's) connects divergence over a region to flux through its boundary — one of the four great theorems of vector calculus. Underlies fluid dynamics, electromagnetism (Maxwell's equations), and probability current in QM.