17.9 SUMMARY

• Divergence of \({\bf F} = \langle F_1,F_2,F_3 \rangle\): \[ \hbox{div}({\bf F})= \nabla\cdot{\bf F} = \dfrac{\partial{F_1}}{\partial{x}}+\dfrac{\partial{F_2}}{\partial{y}}+ \dfrac{\partial{F_3}}{\partial{z}} \]
• The Divergence Theorem: If \(W\) is a region in \({\bf R}^3\) whose boundary \(\partial W\) is a surface, oriented by normal vectors pointing outside \(W\), then \[ \iint_{\partial W\,} {\bf F} \cdot d{\bf S} = \iiint_{W\,} \hbox{div}({\bf F})\, dV \]
• Corollary: If \(\hbox{div}({\bf F}) = 0\), then \({\bf F}\) has zero flux through the boundary \(\partial W\) of any \(W\) contained in the domain of \({\bf F}\).
• The divergence \(\hbox{div}({\bf F})\) is interpreted as “flux per unit volume,” which means that the flux through a small closed surface containing a point \(P\) is approximately equal to \(\hbox{div}({\bf F})(P)\) times the enclosed volume.
• Basic operations on functions and vector fields: \begin{array} \(f & \overset{\nabla}{\longrightarrow} & {\bf F} & \overset{\textrm{curl}}{\longrightarrow} & {\bf G} & \overset{\textrm{div}}{\longrightarrow} & g\\ \hbox{function}&&\hbox{vector field}&&\hbox{vector field}&&\hbox{function} \end{array}
• The result of two consecutive operations is zero: \[ {\bf curl}(\nabla(f)) = {\bf 0},\qquad\hbox{div}({\bf curl}({\bf F})) = 0 \]
• The inverse-square field \(\displaystyle{{\bf F}= {{\bf e}_r}/{r^2}}\), defined for \(r\ne 0\), satisfies \(\hbox{div}({\bf F})=0\). The flux of \({\bf F}\) through a closed surface \(S\) is \(4\pi\) if \(S\) contains the origin and is zero otherwise.