Lagmental Vicfred

Stokes' Theorem Contains Several Integral Theorems at Once by Vicfred

Last updated: Sun 19 September 2021

tags: mathematics, smooth manifolds, tangent spaces, differential forms

The integral of an exterior derivative over a manifold equals the integral of the form over its oriented boundary. I will separate the object being defined from the consequence being claimed.

Notation

A smooth \(n\)-manifold is locally modeled on \(\mathbf R^n\) with smooth transition maps. A smooth map \(F:M\to N\) differentiates to linear maps between tangent spaces.

$$ M^n\ \text{oriented with boundary},\qquad\omega\in\Omega_c^{n-1}(M) $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \int_Md\omega=\int_{\partial M}\omega $$

Stress the formula

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ \underbrace{\int_\Omega(\nabla\times F)\cdot n\,dS}_{\int_\Omega d\omega}=\underbrace{\oint_{\partial\Omega}F\cdot dr}_{\int_{\partial\Omega}\omega} $$

Interpretation

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad M^n\ \text{oriented with boundary},\qquad\omega\in\Omega_c^{n-1}(M),\\[5pt] \mathsf{C}\;&:\quad \int_Md\omega=\int_{\partial M}\omega. \end{aligned} $$

Limit of the argument

Coordinates are computational tools, not intrinsic data. Tensorial formulas must transform correctly on chart overlaps.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \int_Md\omega=\int_{\partial M}\omega \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sat 20 March 2021. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.