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 hte divergence can be interpreted with the species 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 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.

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 given below.s

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.