divergenceDivergence2006-08-28 23:46:392006-08-28 23:46:39Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands here\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 dot product and that of the \emph{del} operator
$$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $$
$$ \nabla =
\frac{\partial}{\partial x} {\bf \hat{i}} + \frac{\partial}{\partial y}{\bf \hat{j}} + \frac{\partial}{\partial z}{\bf \hat{z}}$$
carrying out the dot product with ${\bf V}$ then gives (1).
\subsection{Physical Meaning}
Building physical intuition about the divergence of a vector field can be gained by considering the flow of a fluid.
\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
$$\nabla \cdot {\bf V} = \frac{1}{r^2}\frac{\partial}{\partial r} (r^2 V_r) + \frac{1}{r sin \theta} \frac{\partial}{\partial \theta}(V_{\theta} sin \theta) + \frac{1}{r sin \theta}\frac{\partial V_{\phi}}{\partial \phi}$$