Divergence

From HvWiki

In rough terms, the divergence of a vector function describes the amount of flux created or destroyed at a point. It is often written as

\textrm{div} \textbf{A}\,

or

\nabla\cdot\textbf{A}.\,

In rectangular coordinates:

\nabla\cdot\textbf{A} = \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}.

In circular cylindrical coordinates:

\nabla\cdot\textbf{A} = \frac{1}{r}\frac{\partial}{\partial r} (r A_r) + \frac{1}{r}\frac{\partial A_\phi}{\partial \phi} + \frac{\partial A_z}{\partial z}.

In spherical coordinates:

\nabla\cdot\textbf{A} = \frac{1}{r^2}\frac{\partial}{\partial r}(r^2 A_r) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}(\sin\theta A_\theta) + \frac{1}{r\sin\theta}\frac{\partial A_\phi}{\partial \phi}.

In orthogonal curvilinear coordinates:

\nabla\cdot\textbf{A} = \frac{1}{h_1h_2h_3}\left [\frac{\partial}{\partial q_1}(h_2h_3A_1) + \frac{\partial}{\partial q_2}(h_1h_3A_2) + \frac{\partial}{\partial q_3}(h_1h_2A_3)\right ]

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