Talk:PlanetPhysics/Divergence

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\begin{document}

\section{Divergence}

The divergence of a vector field is defined as

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

This is easily seen from the definition of the \htmladdnormallink{dot product}{http://planetphysics.us/encyclopedia/DotProduct.html} and that of the \emph{del} \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} $$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $$ $$ \nabla = \frac{\partial}{\partial x} {\bf \hat{i}} + \frac{\partial}{\partial y}{\bf \hat{j}} + \frac{\partial}{\partial z}{\bf \hat{z}}$$

carrying out the dot product with ${\bf V}$ then gives (1).

\subsection{Physical Meaning}

(this \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} is a \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} 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 \htmladdnormallink{vector fields}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} is a uniform \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} shown in below figure.

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

Mathematically, this field would be

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

The divergence is then

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

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}

\subsection{Coordinate Systems}

Cartesian Coordinates

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

Cylindrical Coordinates

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

Spherical Coordinates

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

\end{document}