Stokes' Theorem

There are three “big theorems” in vector calculus, namely the Divergence Theorem: \begin{equation} \int_R \grad\cdot\FF\,dV = \int_{\partial R} \FF\cdot d\AA \label{divthm} \end{equation} Stokes' Theorem: \begin{equation} \int_S \grad\times\FF\cdot d\AA = \int_{\partial S} \FF\cdot d\rr \label{curlthm} \end{equation} and \begin{equation} \int_A^B \grad f\cdot d\rr = f \bigg|_A^B \label{gradthm} \end{equation} where “$\partial$” stands for “the boundary of”. This latter theorem is sometimes called the Fundamental Theorem for Gradients, just as the first two can be thought of as fundamental theorems for the divergence and curl, respectively. All three of these theorems are based on the fundamental theorem of calculus, which says \begin{equation} \int_a^b \frac{df}{dx}\,dx = f(b) - f(a) \end{equation}

We rewrite each of these theorems in the language of differential forms. Beginning with $(\ref{gradthm})$, we have \begin{equation} \int_C df = \int_{\partial C} f \end{equation} where $C$ is any path from $A$ to $B$, and where we have used the integral notation introduced in the last section for function evaluation. Turning to ($\ref{curlthm}$), we have \begin{equation} \grad\times\FF\cdot d\AA = {*} \bigl({*}dF\bigr) = dF \end{equation} so Stokes' Theorem becomes \begin{equation} \int_S dF = \int_{\partial S} F \end{equation} Finally, turning to ($\ref{divthm}$), we have \begin{equation} \grad\cdot\FF\,dV = {*} \bigl({*}d{*}F\bigr) = d{*}F \end{equation} and the Divergence Theorem becomes \begin{equation} \int_R d{*}F = \int_{\partial R} {*}F \end{equation}

Comparing these three results, we can conjecture that for any differential form $\alpha$, in any dimension and signature we must have \begin{equation} \int_R d\alpha = \int_{\partial R} \alpha \end{equation} which turns out to be correct. The proof of this result, also called Stokes' Theorem, is beyond the scope of this book, although the only real subtlety is keeping track of the relative orientations of $R$ and its boundary $\partial R$.

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