divergence | Physics Library

1 Divergence

The divergence of a vector field is defined as

∂Vx ∂Vy ∂Vz ∇ ⋅ V = ---- + ----+ ---- ∂x ∂y ∂z

This is easily seen from the definition of the dot product and that of the del operator

A ⋅ B = AxBx + AyBy + AzBz

∂ ∂ ∂ ∇ = --ˆi +---ˆj + --ˆz ∂x ∂y ∂z

carrying out the dot product with V then gives (1).

1.1 Physical Meaning

(this section is a work in progress)

Building physical intuition about the divergence of a vector field can be gained by considering the flow of a fluid. One of the most simple vector fields is a uniform velocity field shown in below figure.

Mathematically, this field would be

V = 5ˆi

The divergence is then

-∂- ∇ ⋅ V = ∂x 5 = 0

Source/Sink flow field ( div ¿ 0 / div ¡ 0)

Circular flow with zero divergence

1.2 Coordinate Systems

Cartesian Coordinates

∂Vx- ∂Vy- ∂Vz- ∇ ⋅ V = ∂x + ∂y + ∂z

Cylindrical Coordinates

1-∂- 1-∂V𝜃- ∂Vz- ∇ ⋅ V = r∂r (rVr) + r ∂𝜃 + ∂z

Spherical Coordinates

-1 ∂-- 2 --1----∂- --1---∂Vϕ- ∇ ⋅ V = r2 ∂r(r Vr) + rsin 𝜃∂𝜃 (V𝜃sin𝜃) + rsin𝜃 ∂ ϕ

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