Stokes and Divergence Theorem
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Stokes and Divergence Theorem | Proof
In this paper, I prove Stokes Theorem and the Divergence Theorem for functions of the form $\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$ which is abstractly written as…
\[\begin{equation} \mathbf{F} = \pmatrix{F_x \\ F_y \\ F_z} \end{equation}\]under the bases $\mathbb{e}_1, \mathbb{e}_2, \mathbb{e}_3$ abstractly in any orientation so long as all of them are unit and orthogonal to each other. By definition of differentials, we can identify the differential 1-form mapping as
\[\begin{equation} F_x\mathbb{e}_1 + F_y\mathbb{e}_2 + F_z\mathbb{e}_3 \to F_xdx + F_ydy + F_zdz \end{equation}\]for our reference, we always have…
\[\begin{equation} \omega_\mathbf{F} = F_x\mathbb{e}_1 + F_y\mathbb{e}_2 + F_z\mathbb{e}_3 \\ \text{d}\omega_{\mathbf{F}} = F_xdx + F_ydy + F_zdz \end{equation}\]So, $\omega_{\mathbf{F}}$ is always the 1-form and $\text{d}\omega_{\mathbf{F}}$ is always the differential 1-form. The derivative we have taken for the differential 1-form is the exterior derivative and uses the regular rectangular infinitesimal bases under no orientation. We note in Stokes formulation that $\omega_{\nabla \times \mathbf{F}}$ is the form for the circulation around any defined point in $\mathbf{F}$. If we take the Hodge star of this, we end up with $\star \omega_{\nabla \times \mathbf{F}}$ which is the resultant of the normal vectors to the circulation. This ends up measuring the differential 1-form again due to orthogonality, so we always have…
\[\begin{equation} \star\omega_{\nabla\times \mathbf{F}} = \text{d}\omega_{\mathbf{F}} \end{equation}\]Now, we can write the full derivation of Stokes Theorem…
\[\begin{equation} \oint_{\partial \Sigma} \mathbf{F} \cdot d\gamma = \oint_{\partial \Sigma} \omega_{\mathbf{F}} = \int_{\Sigma} \text{d}\omega_{\mathbf{F}} = \int_{\Sigma} \star\omega_{\nabla \times \mathbf{F}} = \oint_{\gamma} (\nabla \times \mathbf{F})\cdot d\gamma \\ = \iint_\Sigma (\nabla \times \mathbf{F}) \cdot d\Sigma \end{equation}\]which is Stokes Theorem. In the first two equalities, we used Greenʻs Theorem and from here, we used $(4)$ for the next one. The last two equalities come from understanding that we are looking at net circulation over the surface $\Sigma$. Now, using similar reasoning, we can derive the Divergence Theorem as well…
\[\begin{equation} \oint_{\gamma} (\nabla \cdot \mathbf{F}) \cdot d\gamma = \oint_{\partial \Sigma} \omega_{\mathbf{F}} = \int_{\Sigma} \text{d}\omega_{\mathbf{F}} = \int_{\Sigma} \omega_{\mathbf{F} \cdot \mathbb{n}} = \oint_{\gamma} (\mathbf{F}\cdot \mathbb{n})\cdot d\gamma \\ = \iiint_{V} (\mathbf{F}\cdot \mathbb{n}) \cdot dV \end{equation}\]In the first two equalities, we used Greenʻs Theorem and from here, we used the fact that $\text{d}\omega_{\mathbf{F}} = \omega_{\mathbf{F} \cdot \mathbb{n}}$ due to divergence being the same regardless of orientation so long as we always have a normal vector. The last two equalities come from understanding that we are looking at net divergence over the surface $\Sigma$. This derives both Stokes Theorem and Divergence Theorem, as desired. $\blacksquare$
These two proofs for the theorems are tributes for Madagascar for more ability to use Xenaʻs power!