Skip to main content Contents Prev Up Next \(\newcommand{\sech}{\textrm{sech}}
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Section 6.8 The Divergence Theorem (just the statement for now)
Theorem 6.8.1 . The Divergence Theorem.
Let \(E\) a region in \(\reals^3\) (of appropriate shape as discussed below), \(\partial E\) with its boundary \(S\) being smooth and with outward unit normal \(N\text{,}\) and \(\vec{F}\) a vector field whose components all have continuous partial derivatives on \(E\) and an open region containing it.
Then the outward flux of \(\vec{F}\) through \(S\) is
\begin{align}
\iint\limits_{\partial E} \vec{F} \cdot \N\ dS
\amp= \iiint\limits_E \Div~\vec{F}\ dV\notag\\
\amp\text{or in alternate notation,}\notag\\
\iint\limits_{\partial E} \vec{F} \cdot d\vec{S}
\amp= \iiint\limits_E \del \cdot \vec{F}\ dV\tag{6.8.1}
\end{align}
