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Section 20.3 Fundamental theorems in higher dimensions
Objectives
Connect Green’s theorem with its higher-dimensional analogs.
Interpret divergence and curl through Gauss’s and Stokes’ theorems.
Relate these theorems to the structure of Maxwell’s equations.
Analogs of Green’s theorems in higher dimensions
Theorem 13 . Gauss’s divergence theorem.
Let \(E\) be a simple solid region and let \(\bv{S}\) be the boundary surface of \(E\text{,}\) given with positive (outward) orientation. Let \(\bv F\) be a vector field whose component functions have continuous partial derivatives on an open region containing \(E\text{.}\) Then:
\begin{equation*}
\iint_{\bv{S}}\bv{F}\cdot\dd{\bv{S}}=\iiint_{E}\Div\bv{F}\dd{V}
\end{equation*}
Theorem 14 . Stokes’s curl theorem.
Let \(\bv{S}\) be an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve \(C\) with positive orientation. Let \(\bv F\) be a vector field whose component functions have continuous partial derivatives on an open region containing \(\bv{S}\text{.}\) Then:
\begin{equation*}
\oint_{C} \bv{F}\cdot\dd{\bv{r}}=\iint_{\bv{S}}(\Curl\bv{F})\cdot\dd{\bv{S}}
\end{equation*}
Relate to Maxwell’s equations in both integral and derivative form
