PlanetPhysics/Divergence

Divergence

The divergence of a vector field is defined as

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

This is easily seen from the definition of the [[../DotProduct/|dot product]] and that of the del [[../QuantumSpinNetworkFunctor2/|operator]] $𝐀\cdot 𝐁=A_x{B}_x+A_y{B}_y+A_z{B}_z$ $\mathrm{\nabla }=\frac{\partial }{\partial x}\widehat{𝐢}+\frac{\partial }{\partial y}\widehat{𝐣}+\frac{\partial }{\partial z}\widehat{𝐳}$

carrying out the dot product with $𝐕$ then gives (1).

Physical Meaning

(this [[../IsomorphicObjectsUnderAnIsomorphism/|section]] is a [[../Work/|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 [[../NeutrinoRestMass/|vector fields]] is a uniform [[../Velocity/|velocity]] [[../CosmologicalConstant/|field]] shown in below figure.

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

Mathematically, this field would be

$𝐕=5\widehat{𝐢}$

The divergence is then

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

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

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

Cylindrical Coordinates

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

Spherical Coordinates

$\mathrm{\nabla }\cdot 𝐕=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{V}_r)+\frac{1}{rsin\theta }\frac{\partial }{\partial \theta }(V_{\theta }sin\theta )+\frac{1}{rsin\theta }\frac{\partial V_{\varphi }}{\partial \varphi }$