mathematics-physics-wiki/docs/en/physics/mathematical-physics/vector-analysis/divergence.md

The divergence of a vector field

Flux densities

Considering a medium with a mass density \rho: \mathbb{R}^4 \to \mathbb{R} and a velocity field \mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3 consisting of a orientable finite sized surface element d\mathbf{A} \in \mathbb{R}^3.

Definition: a surface must be orientable for the surface integral to exist. It must be able to move along the surface continuously without ending up on the "other side".

We then have a volume dV \in \mathbb{R} defined by the parallelepiped formed by dV = \langle d\mathbf{x}, d\mathbf{A} \rangle with the vector d\mathbf{x} = \mathbf{v} dt, for a time interval dt \in \mathbb{R}. The mass flux d\Phi per unit of time through the surface element d\mathbf{A} may then be given by

d \Phi = \rho \langle \mathbf{v}, d\mathbf{A} \rangle.

The mass flux \Phi: \mathbb{R} \to \mathbb{R} through a orientable finite sized surface A \subseteq \mathbb{R}^3 is then given by

\Phi(t) = \int_A \Big\langle \rho(\mathbf{x}, t) \mathbf{v}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,

for all t \in \mathbb{R}.

Definition: let \mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3 be the (mass) flux density given by

\mathbf{\Gamma}(\mathbf{x},t) := \rho(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),

for all (\mathbf{x},t) \in \mathbb{R}^4.

The (mass) flux density is a vector-valued function of position and time that expresses the rate of transport of a quantity per unit of time of area perpendicular to its direction.

The mass flux \Phi through A may then be given by

\Phi(t) = \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,

for all t \in \mathbb{R}.

Definition of the divergence

Definition: the divergence of a flux density \mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3 is given by

\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \lim_{V \to 0} \frac{1}{V} \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle,

for all (\mathbf{x}, t) \in \mathbb{R}^4 for a volume V \subset \mathbb{R}^3 with closed orientable boundary surface A \subset V.

Note that this "dot product" between the nabla operator and the flux density \mathbf{\Gamma} does not imply anything and is only there for notational sake. An alternative to this notation is using \text{div } \mathbf{\Gamma} to denote the divergence.

The definition of the divergence can be interpreted with the particle mass balance for a medium with a particle density n: \mathbb{R}^4 \to \mathbb{R} and a velocity field \mathbf{v}: \mathbb{R}^4 \to \mathbb{R}^3. Furthermore we have that the particles are produced at a rate S: \mathbb{R}^4 \to \mathbb{R}^3.

We then have the particle flux density \mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3 given by

\mathbf{\Gamma}(\mathbf{x},t) = n(\mathbf{x},t) \mathbf{v}(\mathbf{x},t),

for all (\mathbf{x},t) \in \mathbb{R}^4.

For a volume V \subseteq \mathbb{R}^3 with a closed orientable boundary surface A \subseteq \mathbb{R}^3 we have that the amount of particles inside this volume for a specific time is given by

\int_V n(\mathbf{x}, t) dV,

for all t \in \mathbb{R}. We have that the particle flux through A is given by

\int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle,

for all t \in \mathbb{R} and we have that the particle production rate in this volume V is given by

\int_V S(\mathbf{x}, t)dV,

for all t \in \mathbb{R}. We conclude that the sum of the particle flux through A and the time derivative of the particles inside the volume V must be equal to the production rate inside this volume V. Therefore we have

d_t \int_V n(\mathbf{x}, t) dV + \int_A \Big\langle \mathbf{\Gamma}(\mathbf{x},t), d\mathbf{A} \Big\rangle = \int_V S(\mathbf{x}, t)dV,

for all t \in \mathbb{R}.

Assuming the system is stationary the time derivative of the particles inside the volume V must vanish. The divergence is then defined to be the total production for a position \mathbf{x} \in V.

Divergence in curvilinear coordinates

Theorem: the divergence of a flux density \mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3 for a curvilinear coordinate system is given by

\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{\sqrt{g(\mathbf{x})}} \partial_i \Big(\Gamma^i(\mathbf{x},t) \sqrt{g(\mathbf{x})} \Big)

for all \mathbf{x} \in \mathbb{R}^3 and i \in \{1, 2, 3\}.

??? note "Proof:"

Will be added later.

We may also give the divergence for ortho-curvilinear coordinate systems.

Corollary: the divergence of a flux density \mathbf{\Gamma}: \mathbb{R}^3 \to \mathbb{R}^3 for a ortho-curvilinear coordinate system is given by

\nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) = \frac{1}{h_1h_2h_3} \partial_i \Big(\Gamma^{(i)}(\mathbf{x},t)\frac{1}{h_i} h_1 h_2 h_3 \Big)

for all \mathbf{x} \in \mathbb{R}^3 and i \in \{1, 2, 3\}.

??? note "Proof:"

Will be added later.

Please note that the scaling factors may also depend on the position \mathbf{x} \in \mathbb{R}^3 depending on the coordinate system.

It has been found that the volume integral over the divergence of a vector field is equal to the integral of the vector field itself over the surface that bounds the volume. It is known as the divergence theorem and is given below.

Theorem: for a volume V \subset \mathbb{R}^3 with a closed and orientable boundary surface A \subset V with a continuously differentiable flux density \mathbf{\Gamma}: \mathbb{R}^4 \to \mathbb{R}^3 we have that

\int_A \Big\langle \mathbf{\Gamma}(\mathbf{x}, t), d\mathbf{A} \Big\rangle = \int_V \nabla \cdot \mathbf{\Gamma}(\mathbf{x}, t) dV,

for all t \in \mathbb{R}.

??? note "Proof:"

Will be added later.