Source : ct|01.06.13

< Mathematics and physics

Divergence of a vector field

The concept of divergence is defined in the dictionary as the state of lines which are separating and going in different directions. Also, the "divergence of opinion" is the difference of opinion when two people do not share the same view.
In mathematics and physics, the concept of divergence of a vector field is often encountered. One may ask what is the definition of this concept and what it represente exactly...

1) Definition

Consider a vector field
a(x,y,z) = ax(x,y,z)i + ay(x,y,z)j + az(x,y,z)k

We call divergence of a vector a, the scalar number

which is also noted

with i =(1,0,0), j=(0,1,0), k=(0,0,1), and the nabla operator equal to

2) Meaning

To illustrate what concretely represente the divergence of a vector field, we consider an elementary volume space dv = dx dy dz

nx and nx+dx are unit vectors. They are perpendicular to the left and right faces of this volume and are directed outwardly.

The flows of the vector a through those two elementary faces are respectively equal to a(x,y,z)dS.nx et a(x+dx,y,z)dS.nx+dx. Therefore those flows are equal to ax(x,y,z)dS.i.nx and ax(x+dx,y,z)dS.i.nx+dx. Thus they are equal to -ax(x,y,z).dydz et +ax(x+dx,y,z).dydz because (considering the direction of the unit normal vectors to the faces considered) i.nx=-1 et i.nx+dx=+1.

Therefore, the sum of the flows of a on those both elementary faces is dΦ = [ax(x+dx,y,z) - ax(x,y,z)]dydz and

By the same reasoning on the sufaces dxdz and dxdy, we deduce that the sum of the flows of a on all faces of the elementary volume dV is

We deduce dΦ= (div a)dv with dv=dxdydz and

div a = dΦ/dv

This result shows that the divergence of a vector field at a point M of space represents the flow (per unit volume) of this vector field (through surface defining a unit of volume at this point).

• A positive divergence at a point M (x, y, z) corresponds to a flow coming out of this point (diverging field - expansion fluid)

• A negative divergence at a point M (x, y, z) corresponds to a flow coming in toward this point (converging field - fluid compression)
  

• A divergence equal to 0 at a point M (x, y, z) corresponds to a flow coming out of this point that compensate a flow coming in toward this point (field uniform - incompressible fluid) or corresponds to a swirling field.

Generalisation

More generally, we consider any space volume V composed of an infinite number of volume elements (which are infinitely small). The internal flows through the surfaces of all constituent units of the internal volume of this space, are equal to zero two by two because of the opposite directions of their respective surfaces (opposite unit normal vectors). There remain only flows through the outer surface S of the volume V.

From dΦ = div a . dv we deduce

hence

(Formula Ostrogradsky)

Mikhaïl Vassilievitch Ostrogradsky (1801-1862) - Ukrainian physicist and mathematician
Source commons wikimedia - image public domain

The flow of a field of vectors through a closed surface S which defines a space volume V is equal to the integral of the divergence of this field on this space.

4) Divergence in cylindrical coordinates

We can calculate the divergence of a vector field expressed in cylindrical coordinates.

We consider a vector V(r,θ,z)=MN(r,θ,z) whose origin is located at a point M(r,θ,z), within a fixed coordinate system (O,i,j,k).
In cylindrical coordinates, V(r,θ,z) = Vr(r,θ,z) u + Vθ(r,θ,z) v + Vz(r,θ,z) k.

We consider an infinitesimal volume element dv around the point M.

The flow of vector V through the faces perpendicular to u is
dΦu = Vr(r+dr) [(r+dr)dθdz] - Vr(r) [rdθdz]

The flow of vector V through the faces perpendicular to v is
dΦv = Vθ (θ+dθ) dr dz - Vθ (θ) dr dz]

The flow of vector V through the faces perpendicular to k is
dΦk = Vz(z+dz) r dr dθ - Vz(z) r dr dθ

Hence the total flux (dΦ) of V through the element of volume dv= r dr dθ dz
dΦ= dΦu+ dΦv+ dΦz

This gives the divergence of a vector field expressed in cylindrical coordinates

5) Divergence of a vector field, in spherical coordinates

We consider a vector V(r,θ,φ)=MN(r,θ,φ) whose origin is located at a point M (r, θ, φ), within a fixed coordinate system (O,i,j,k).
In spherical coordinates, V(r,θ,φ)=Vr(r,θ,φ) u + Vθ(r,θ,φ) v + Vφ(r,θ,φ) w

We consider an infinitesimal volume element dv around the point M.

The flow of vector V through the faces perpendicular to u is
dΦu = Vr(r+dr) (r+dr)² sinθ dθ dφ - Vr(r) r²sinθ dθ dφ

The flow of vector V through the faces perpendicular to v is
dΦv = Vθ (θ+dθ) sin(θ+dθ) r dr dφ - Vθ(θ) sinθ r dr dφ, d’où

The flow of vector V through the faces perpendicular to w is
dΦw = Vφ (φ+dφ) r dr dθ - Vφ (φ) r dr dθ d’où

Hence the total flux (dΦ) of V through the element of volume dv= r² sinθ dr dθ dφ
dΦ= dΦu+ dΦv+ dΦw

This gives the divergence of a vector field expressed in spherical coordinates

6) Examples

• Example 1 : Moving particles

1) Verify the Ostrogradsky formula in the case of particles moving along the x axis with a speed a =(x,0,0) which increases linearly as a function of the abscissa of these particles.

2) Verify the Ostrogradsky formula with a =(x²,0,0).

Answer
1) We apply Ostrogradsky formula to any volume V = XLH.

2)

• Example 2 : Gauss theorem

The mathematician, physicist and astronomer German Gauss (1777-1855) showed that the flow of the electric field E through a surface S which defines a volume V containing a total charge Q is equal to Q/ε0

Calculate the electric field E and the divergence of the field created at any point M of space by a sphere of radius R which has a charge density equal to ρ.

Answer
The electric field created by a charged sphere has a spherical symmetry. In spherical coordinates E = Er(r) u The application of Gauss theorem gives :