PlanetPhysics/Divergence

Divergence
The divergence of a vector field is defined as

$$\nabla \cdot {\mathbf V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$$

This is easily seen from the definition of the dot product and that of the del operator $$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $$ $$ \nabla = \frac{\partial}{\partial x} {\mathbf \hat{i}} + \frac{\partial}{\partial y}{\mathbf \hat{j}} + \frac{\partial}{\partial z}{\mathbf \hat{z}}$$

carrying out the dot product with $${\mathbf V}$$ then gives (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.

\begin{figure} \caption{Uniform Flow} \includegraphics[scale=1]{UniformFlow.eps} \end{figure}

Mathematically, this field would be

$$ {\mathbf V} = 5 {\mathbf \hat{i}} $$

The divergence is then

$$ \nabla \cdot {\mathbf V} = \frac{\partial}{\partial x} 5 = 0 $$

Source/Sink flow field ( div > 0 / div < 0)

\begin{figure} \caption{Positive Divergence} \includegraphics[scale=1]{PositiveDivergence.eps} \end{figure}

\begin{figure} \caption{Negative Divergence} \includegraphics[scale=1]{NegativeDivergence.eps} \end{figure}

Circular flow with zero divergence

\begin{figure} \caption{Circular Flow} \includegraphics[scale=1]{CircularFlow.eps} \end{figure}

Coordinate Systems
Cartesian Coordinates

$$ \nabla \cdot {\mathbf V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}$$

Cylindrical Coordinates

$$\nabla \cdot {\mathbf V} = \frac{1}{r}\frac{\partial}{\partial r} (r V_r) + \frac{1}{r} \frac{\partial V_{\theta}}{\partial \theta} + \frac{\partial V_z}{\partial z}$$

Spherical Coordinates

$$\nabla \cdot {\mathbf V} = \frac{1}{r^2}\frac{\partial}{\partial r} (r^2 V_r) + \frac{1}{r sin \theta} \frac{\partial}{\partial \theta}(V_{\theta} sin \theta) + \frac{1}{r sin \theta}\frac{\partial V_{\phi}}{\partial \phi}$$