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divergence

(Definition)

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.

Figure 1:Uniform Flow
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Mathematically, this field would be

V = 5ˆi

The divergence is then

-∂- ∇ ⋅ V = ∂x 5 = 0

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

Figure 2:Positive Divergence
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Figure 3:Negative Divergence
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Circular flow with zero divergence

Figure 4:Circular Flow
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1.2 Coordinate Systems

Cartesian coordinates

∇ ⋅ V = ∂Vx- + ∂Vy-+ ∂Vz- ∂x ∂y ∂z

Cylindrical coordinates

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

Spherical coordinates

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

"divergence" is owned by bloftin.
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See Also: curl, gradient, Gauss's Law, gradient

Other names:  divergence of a vector field

Attachments:
sources and sinks of vector field (Topic) by pahio

Cross-references: spherical coordinates, cylindrical coordinates, Cartesian coordinates, velocity field, vector fields, work, section, operator, dot product
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This is version 2 of divergence, born on 2006-08-28, modified 2006-08-29.
Object id is 221, canonical name is Divergence.
Accessed 4582 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)

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